mathematical and statistical modelling of infectious ... · emma mcbryde mbbs (honours) university...

Mathematical and StatisticalModelling

of Infectious Diseases in Hospitals

Emma McBrydeMBBS (Honours)

University of QueenslandFRACP

A thesis submitted in partial fulfilment of the requirements for the degree ofDoctor of Philosophy

November 2006

Principal Supervisor: Professor Sean McElwainAssociate Supervisors: Professor Tony Pettitt and Dr Mike Whitby

Queensland University of TechnologySchool of Mathematical Sciences

Faculty of ScienceBrisbane, Queensland, 4001, AUSTRALIA

c© Copyright by Emma McBryde 2006All Rights Reserved

ii

I dedicate this work to Liam Eisen McBryde.

Acknowledgements

A number of people have assisted me in producing this work. My principal

supervisor, Professor Sean McElwain, has encouraged me from our first en-

counter, has been generous with his time and provided inspiration, technical

assistance and practical advice. Professor Tony Pettitt has been an exacting

and thorough supervisor imparting expertise in statistical methods.

A owe a number of people and groups thanks for assistance with data gath-

ering. The CHRISP group at the Princess Alexandra Hospital, for providing

data, my associate supervisor Dr Mike Whitby and Dr David Cook, Director

of Intensive Care at the Princess Alexandra Hospital. The Taiyuan Centre of

Disease prevention and control and coauthors Ms Zhang and Mr Zhao for the

Shanxi SARS dataset.

Thank you also to those who gave me practical assistance and support dur-

ing my time as a PhD candidate. In particular I would like to thank Jenny

Eisen my mother-in-law for devotedly looking after Liam and giving me an

opportunity to work.

Funding sources for this thesis include Australian Research Council Linkage

Scheme (LP347112), Australian Postgraduate Award and National Health

and Medical Research Council scholarship number 290541. Additionally the

Princess Alexandra Hospital provided me with a research scholarship for the

year 2003.

Thank you also to the Australian Mathematical Sciences Institute (AMSI)

for providing me with a room and equipment while I was collaborating

and working externally from Melbourne. The Royal Brisbane and Womens’

Hospital also assisted me by allowing dedicated research time within my

appointment as staff specialist from June 2004 to April 2005.

v

Abstract

Antibiotic resistant pathogens, such as methicillin-resistant Staphylococcus

aureus (MRSA), and vancomycin-resistant enterococci (VRE), are an increas-

ing burden on healthcare systems. Hospital acquired infections with these

organisms leads to higher morbidity and mortality compared with the sensi-

tive strains of the same species and both VRE and MRSA are on the rise world-

wide including in Australian hospitals. Emerging community infectious dis-

eases are also having an impact on hospitals. The Severe Acute Respiratory

Syndrome virus (SARS Co-V) was noted for its propensity to spread through-

out hospitals, and was contained largely through social distancing interven-

tions including hospital isolation. A detailed understanding of the transmis-

sion of these and other emerging pathogens is crucial for their containment.

The statistical inference and mathematical models used in this thesis aim

to improve understanding of pathogen transmission by estimating the

transmission rates of contagions and predicting the impact of interventions.

Datasets used for these studies come from the Princess Alexandra Hospital

in Brisbane, Australia and Shanxi province, mainland China.

Epidemiological data on infection outbreaks are challenging to analyse

due to the censored nature of infection transmission events. Most datasets

record the time on symptom onset, but the transmission time is not ob-

servable. There are many ways of managing censored data, in this study

we use Bayesian inference, with transmission times incorporated into the

augmented dataset as latent variables. Hospital infection surveillance data

is often much less detailed that data collected for epidemiological studies,

often consisting of serial incidence or prevalence of patient colonisation

with a resistant pathogen without individual patient event histories. Despite

the lack of detailed data, transmission characteristics can be inferred from

such a dataset using structured Hidden Markov Models (HMMs).

Each new transmission in an epidemic increases the infection pressure on

vii

those remaining susceptible, hence infection outbreak data are serially de-

pendent. Statistical methods that assume independence of infection events

are misleading and prone to over-estimating the impact of infection control

interventions. Structured mathematical models that include transmission

pressure are essential. Mathematical models can also give insights into

the potential impact of interventions. The complex interaction of different

infection control strategies, and their likely impact on transmission can be

predicted using mathematical models.

This dissertation uses modified or novel mathematical models that are

specific to the pathogen and dataset being analysed. The first study es-

timates MRSA transmission in an Intensive Care Unit, using a structured

four compartment model, Bayesian inference and a piecewise hazard meth-

ods. The model predicts the impact of interventions, such as changes to

staff/patient ratios, ward size and decolonisation. A comparison of results of

the stochastic and deterministic model is made and reason for differences

given. The second study constructs a Hidden Markov Model to describe

longitudinal data on weekly VRE prevalence. Transmission is assumed to be

either from patient to patient cross-transmission or sporadic (independent

of cross-transmission) and parameters for each mode of acquisition are

estimated from the data. The third study develops a new model with a

compartment representing an environmental reservoir. Parameters for

the model are gathered from literature sources and the implications of the

environmental reservoir are explored. The fourth study uses a modified

Susceptible-Exposed-Infectious-Removed (SEIR) model to analyse data from

a SARS outbreak in Shanxi province, China. Infectivity is determined before

and after interventions as well as separately for hospitalised and community

symptomatic SARS cases. Model diagnostics including sensitivity analysis,

model comparison and bootstrapping are implemented.

viii

Keywords

Bayesian inference

epidemic modelling

environmental reservoir

hidden Markov models

infectious diseases

mathematical modelling

methicillin resistant Staphylococcus aureus (MRSA)

severe acute respiratory syndrome (SARS)

statistical modelling

stochastic processes

vancomycin resistant enterococci (VRE)

epidemiology

public health

infectious disease

ix

List of Publications and Manuscripts

Presented in this Thesis

McBryde, E. S., Pettitt, A.N., McElwain, D. L.S., 2006c. A stochastic

mathematical model of methicillin resistant Staphylococcus aureus trans-

mission in an intensive care unit: Predicting the impact of interven-

tions. The Journal of Theoretical Biology, e-published November 2006

http://dx.doi.org/10.1016/j.jtbi.2006.11.008. (Chapter 3)

McBryde, E. S., Bradley, L. C., Whitby, M., McElwain, D. L.S., 2004. An inves-

tigation of contact transmission of methicillin-resistant Staphylococcus au-

reus. The Journal of Hospital Infection 58 (2), 104-8. (Chapter 4)

McBryde, E. S., Pettitt, A.N., Cooper, B.S., McElwain, D. L.S., 2006b. Char-

acterising outbreaks of vancomycin-resistant enterococci using a statistical

method. Submitted to Public Library of Science-Medicine, September 2006.

(Chapter 5)

McBryde, E. S., McElwain, D. L.S., 2006. A mathematical model investigat-

ing the impact of an environmental reservoir on the prevalence and control

of vancomycin-resistant enterococci. The Journal of Infectious Diseases 193

(10), 1473-4. (Chapter 6)

McBryde, E., Gibson, G., Pettitt, A.N., Zang, Y., Zhao, B., McElwain, D. L.S.,

2006a. Bayesian Modelling of an epidemic of Severe Acute Respiratory

Syndrome. The Bulletin of Mathematical Biology 68(4), 889-917. DOI:

10.1007/s11538-005-9005-4. (Chapter 7)

xi

Additional Publications & Manuscripts

involving Quantitative studies of

Infectious Diseases by the Candidate

during Ph.D. Candidature

McBryde, E.S., 2003. Severe Acute Respiratory Syndrome. Australian and

New Zealand Journal of Medicine 33, 10suppl(A70).

McBryde, E.S., 2004. Using mathematical models to investigate transmission

of methicillin-resistant Staphylococcus aureus (MRSA) in the healthcare set-

ting. Internal Medicine Journal 34, 9-10suppl (A68).

McBryde, E.S., 2004. An investigation of contact transmission of methicillin-

resistant Staphylococcus aureus (MRSA). Internal Medicine Journal 34,

9-10suppl (A71).

McBryde, E.S., 2004. Severe Acute Respiratory Syndrome: Predicting the epi-

demiology using mathematical modelling and data analysis techniques. In-

ternal Medicine Journal 34, 11suppl(A80).

McBryde, E.S., Tilse, M., McCormack, J., 2005. Comparison of contamination

rates of catheter-drawn and peripheral blood cultures. Journal of Hospital In-

fection 60, 118-121.

xiii

Contents

Acknowledgements v

Abstract vii

Keywords ix

Publications xi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Burden of antibiotic resistant bacteria . . . . . . . . . . 2

1.1.2 The role of mathematical and statistical modelling in

infection control research . . . . . . . . . . . . . . . . . 3

1.2 Overall objectives of the thesis . . . . . . . . . . . . . . . . . . . 5

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Literature review and outline of thesis 13

2.1 Review of pathogens discussed in this thesis . . . . . . . . . . . 13

2.1.1 Methicillin resistant Staphylococcus aureus . . . . . . . 13

2.1.2 Vancomycin-resistant enterococci . . . . . . . . . . . . 16

2.1.3 Environmental pathogens . . . . . . . . . . . . . . . . . 18

2.1.4 Severe Acute Respiratory Syndrome Coronavirus . . . . 19

2.2 Mathematical models of human infectious diseases . . . . . . 20

2.2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 The Susceptible-Infectious-Removed model . . . . . . 21

2.2.3 The Susceptible-Exposed-Infectious-Removed model . 23

2.2.4 The basic reproduction ratio, R0 . . . . . . . . . . . . . 24

2.2.5 Adaptation of the Ross-MacDonald model to the

healthcare setting . . . . . . . . . . . . . . . . . . . . . . 25

2.2.6 Single population models . . . . . . . . . . . . . . . . . 27

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2.2.7 Stochastic models . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Relationship of current literature to thesis . . . . . . . . . . . . 30

2.3.1 Studies based on the Ross-MacDonald model . . . . . . 30

2.3.2 Stochastic epidemic models based on the Susceptible-

Infectious model with migration . . . . . . . . . . . . . 35

2.3.3 Environmental models of transmission . . . . . . . . . 36

2.3.4 Epidemic models of Severe Acute Respiratory Syndrome 36

2.3.5 Other important models of transmission of nosocomial

pathogens . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Review of methodology used in stochastic epidemic modelling 42

2.4.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . 43

2.4.2 Methods used to manage serial dependence in infec-

tion data . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.3 Methods used to manage censored transmission data . 47

2.4.4 Hidden Markov models . . . . . . . . . . . . . . . . . . . 51

2.4.5 Assessing convergence of MCMC . . . . . . . . . . . . . 52

2.4.6 Model checking and improvement . . . . . . . . . . . . 53

2.4.7 Model selection and comparison . . . . . . . . . . . . . 54

2.5 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 Mathematical model of MRSA 73

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.1 Patients and Setting . . . . . . . . . . . . . . . . . . . . . 79

3.3.2 Surveillance of colonisation . . . . . . . . . . . . . . . . 79

3.3.3 Parameter estimates . . . . . . . . . . . . . . . . . . . . 79

3.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.1 Formula for daily hazard of MRSA cross-transmission . 80

3.4.2 Bayesian inference to estimate φ . . . . . . . . . . . . . 82

3.4.3 Estimates of the attack rate and the ward reproduction

ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4.4 Model for the impact of interventions . . . . . . . . . . 83

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5.1 Estimate ward transmission: attack rate and the ward

reproduction ratio . . . . . . . . . . . . . . . . . . . . . 86

3.5.2 Predicted impact of interventions . . . . . . . . . . . . . 86

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3.5.3 Model adequacy and sensitivity . . . . . . . . . . . . . . 87

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.A Bayesian estimation of the transmission parameter . . . . . . 94

3.A.1 Likelihood of the complete dataset . . . . . . . . . . . . 95

3.A.2 Gibbs update for the transmission parameter, φ . . . . 96

3.A.3 Latent variable imputation . . . . . . . . . . . . . . . . . 96

3.A.4 Incorporating uncertainty of model parameters . . . . 97

3.A.5 Markov chain Monte Carlo algorithm to estimate the

transmission parameter, φ . . . . . . . . . . . . . . . . . 98

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Contact transmission of MRSA 105

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.1 Hand sampling . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.2 Laboratory technique . . . . . . . . . . . . . . . . . . . . 108

4.2.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3.1 Detection of MRSA . . . . . . . . . . . . . . . . . . . . . 109

4.3.2 Pre-handwash sample . . . . . . . . . . . . . . . . . . . 110

4.3.3 Post-handwash sample . . . . . . . . . . . . . . . . . . . 110

4.3.4 Compliance with infection control procedures . . . . . 110

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Characterising outbreaks of VRE using statistical methods 117

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.1 Description of outbreak and infection control interven-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.2 Serial surveillance data used for statistical analysis . . . 121

5.2.3 Data used for cluster analysis . . . . . . . . . . . . . . . 121

5.2.4 Model of transmission . . . . . . . . . . . . . . . . . . . 122

5.2.5 Hidden Markov model . . . . . . . . . . . . . . . . . . . 124

5.2.6 Constructing a transition probability matrix . . . . . . . 125

5.2.7 Observation Model . . . . . . . . . . . . . . . . . . . . . 126

xvii

5.2.8 Bayesian framework . . . . . . . . . . . . . . . . . . . . 127

5.2.9 Comparison of cluster analysis results using genotyping

with statistical analysis . . . . . . . . . . . . . . . . . . . 128

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . 128

5.3.2 Comparison of statistical model and genotyping data . 129

5.3.3 Model selection and validation . . . . . . . . . . . . . . 129

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


5.A Likelihood computation . . . . . . . . . . . . . . . . . . . . . . 133

5.B Monte Carlo Markov chain algorithm . . . . . . . . . . . . . . . 134

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6 Environmental reservoir model for VRE 143

6.A Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.B Elaboration of Environment Model . . . . . . . . . . . . . . . . 146

6.B.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.B.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.B.3 Further discussion . . . . . . . . . . . . . . . . . . . . . 150

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Bayesian modelling of an epidemic of SARS 155

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model . . . . 159

7.3 SARS Data from Shanxi Province . . . . . . . . . . . . . . . . . 162

7.4 Challenges and specific aims of the study . . . . . . . . . . . . 166

7.5 Estimation of time to transmission and incubation period . . . 166

7.5.1 Bayesian approach to estimating incubation period . . 167

7.5.2 Results: Time to transmission and incubation period . 168

7.5.3 Discussion: Time to transmission and incubation period 170

7.6 Estimation of other transition periods . . . . . . . . . . . . . . 171

7.6.1 Results: Estimation of other transition periods . . . . . 172

7.7 Model for estimating coefficients of infectivity . . . . . . . . . 172

7.7.1 Bayesian approach to estimation of the transmission

coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.7.2 Prior specification . . . . . . . . . . . . . . . . . . . . . . 174

7.7.3 Likelihood estimation . . . . . . . . . . . . . . . . . . . 175

7.7.4 Results: Change point Estimation . . . . . . . . . . . . . 176

7.7.5 Results: Coefficients of Infectivity . . . . . . . . . . . . . 176

xviii

7.7.6 Results: Reproduction ratio . . . . . . . . . . . . . . . . 178

7.8 Individual Infectivity profiles . . . . . . . . . . . . . . . . . . . 179

7.9 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . 180


7.A Gantt chart of Shanxi epidemic . . . . . . . . . . . . . . . . . . 182

7.B Computations for time to transmission and incubation period 183

7.C Diagnostics: Convergence and Sensitivity analysis . . . . . . . 184

7.C.1 Sensitivity of estimate of incubation period to model

choice and hazard of transmission parameter . . . . . . 185

7.D Estimated values of shape and scale parameters for the

Gamma distributions . . . . . . . . . . . . . . . . . . . . . . . . 187

7.E Statistical inference used to estimate infectivity and change

points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.E.1 Augmented data . . . . . . . . . . . . . . . . . . . . . . . 187

7.E.2 Computations to determine posterior distributions of

the coefficients of infectivity and the change point . . . 188

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8 Conclusions and suggestions for future work 195

8.1 What has been achieved? . . . . . . . . . . . . . . . . . . . . . . 196

8.1.1 Estimation of basic reproduction ratio and cross-

transmission rates . . . . . . . . . . . . . . . . . . . . . . 196

8.1.2 Development of new models . . . . . . . . . . . . . . . 196

8.1.3 Using of models to inform health policy . . . . . . . . . 197

8.1.4 Methodological framework for future studies . . . . . . 197

8.1.5 Model comparison . . . . . . . . . . . . . . . . . . . . . 198

8.1.6 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . 198

8.2 Limitations and opportunities for extensions . . . . . . . . . . 198

8.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

xix

List of Figures

2.1 The application of the Ross-MacDonald model to the trans-

mission of nosocomial pathogens . . . . . . . . . . . . . . . . . 26

2.2 Single population nosocomial transmission model . . . . . . . 28

2.3 Predicted linear relationship between number of patients

colonised and number of healthcare workers colonised. . . . . 28

3.1 Four compartment model of nosocomial pathogen transmission 78

3.2 Data collected over period of study . . . . . . . . . . . . . . . . 85

3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.4 Effect of changing parameters on attack rate . . . . . . . . . . . 90

3.5 Effect of cohorting on attack rate . . . . . . . . . . . . . . . . . 90

3.6 Effect of ward size on attack rate . . . . . . . . . . . . . . . . . . 91

4.1 Flow diagram of study participants and results of MRSA testing. 109

4.2 Boxplot of time taken to wash hands, based on type of health-

care worker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Prevalence data for VRE over 157 weeks . . . . . . . . . . . . . 122

5.2 The transmission of bacterial pathogens in the hospital ward. 123

5.3 Hidden Markov model . . . . . . . . . . . . . . . . . . . . . . . 124

5.4 Posterior distribution of proportion of VRE acquisitions that

are due to ward transmission . . . . . . . . . . . . . . . . . . . 130

6.1 Comparison of environmental and standard model predictions. 146

6.2 Environmental model of VRE transmission in the hospital setting147

7.1 The schematic of the SEIR model. . . . . . . . . . . . . . . . . . 159

7.2 The schematic of the extended SEIHRD model . . . . . . . . . 160

7.3 Histogram of daily admissions to hospital. . . . . . . . . . . . . 162

7.4 Histogram of time from first exposure to another SARS case to

symptom onset. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.5 Recorded time interval from symptom onset to hospitalisation. 164

xxi

7.6 Recorded time interval from symptom onset to recovery. . . . 164

7.7 Recorded time interval from symptom onset to death. . . . . . 165

7.8 Posterior distribution for the hazard of transmission, λ. . . . . 169

7.9 Estimated distribution of the incubation period . . . . . . . . . 169

7.10 Estimated best fit Gamma distribution for time from symptom

onset to hospitalisation . . . . . . . . . . . . . . . . . . . . . . . 172


onset to recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 173


onset to death . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.13 Posterior distribution for change-point. . . . . . . . . . . . . . 176

7.14 Posterior distribution for the reproduction ratios prior to and

after March 29, 2003 . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.15 Infectivity profile versus time since symptom onset. . . . . . . 180

7.16 Gantt chart of epidemic . . . . . . . . . . . . . . . . . . . . . . . 183

7.17 Convergence of a series of Markov chains . . . . . . . . . . . . 185

7.18 Sensitivity of the incubation period estimate to model assump-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

xxii

List of Tables

2.1 Comparison of studies that applied the Ross-MacDonald

model to nosocomial transmission of hospital pathogens . . . 32

2.2 Comparison of studies that estimated infectivity of Severe

Acute Respiratory Syndrome . . . . . . . . . . . . . . . . . . . . 38

3.1 Parameters used in the model for MRSA transmission. . . . . 88

4.1 Compliance with glove use amongst different healthcare

worker groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.1 Parameters used in the model . . . . . . . . . . . . . . . . . . . 125

5.2 Comparison of different models using the Deviance Informa-

tion Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.1 Model assumptions and their justifications. . . . . . . . . . . . 151

6.2 Table of parameters, their symbols and default values . . . . . 152

7.1 The posterior mean and standard deviation (in days) of the

times to hospitalisation, hospital discharge and death. . . . . . 174

7.2 Table of transmission coefficients . . . . . . . . . . . . . . . . . 177

7.3 The estimated mean and standard deviation (in days) of the

incubation period comparing the estimates using the assump-

tion of constant hazard, used by the current study, and the as-

sumptions of uniform probability and immediate transmission. 186

7.4 Estimated values for the parameters of the Gamma distribu-

tions for sojourn times . . . . . . . . . . . . . . . . . . . . . . . 187

xxiii

Statement of Original Authorship

The work contained in this thesis has not been previously submitted

for a degree or diploma at any other higher education institution. To

the best of my knowledge and belief, the thesis contains no material

previously published or written by another person except where due

reference is made.

Signature:Emma McBryde

Date:

xxv

CHAPTER 1

Introduction

1.1 Motivation

Many infectious diseases owe their existence to the hospital environment.

Patients are vulnerable to infection for a number of reasons: intravenous

catheters and other devices breach fundamental barriers, illnesses and

medication can lead to impairment of both the innate and acquired immune

system and antibiotic exposure alters normal host flora. Healthcare workers

have been recognised as the agents of transmission of bacteria since the

treatise of Semmelweiss (1861).

Hospital acquired infections are serious, causing expense, morbidity

and mortality. The incidence of infections caused by antibiotic resistant

pathogens is increasing and often leads to greater morbidity and mortality

and hospital costs than their antibiotic-sensitive counterparts. Infection

control recommendations, aimed at reducing the burden of antibiotic resis-

tant pathogens, are rarely based on high level evidence such as randomised

controlled trials (Cooper et al., 2003).

In the current political environment, hospitals are strategic sites of contain-

ment of community epidemics. In planning policy for bed utilisation, iso-

lation facilities and quarantine, we can learn from past events including the

role of the hospital in the Severe Acute Respiratory Syndrome pandemic. This

viral agent (SARS Co-V) emerged in the community. Nevertheless, hospitals

became sites of intensified spreading in several countries and effective con-

tainment in others.

This thesis analyses healthcare-associated infections. Firstly, attention

is turned to antibiotic-resistant bacteria, arising from populations and

pressures that are peculiar to hospitals. The aim is to estimate the trans-

mission characteristics of methicillin-resistant Staphylococcus aureus and

2 Chapter 1. Introduction

vancomycin-resistant enterococci. Pathogen characteristics, available data

and questions being addressed determine the choice of model and method-

ology. Secondly, the thesis considers the role of the hospital in reflecting,

intensifying and potentially preventing community infection, using SARS as

an example.

1.1.1 Burden of antibiotic resistant bacteria

Staphylococcus aureus is a major cause of healthcare-associated bacteraemia

and post-operative wound infection (Mandell et al., 2005). This pathogen

has achieved resistance to successive antibiotics, such as aminoglycosides,

macrolides and tetracyclines following their introduction through the 1950s.

Methicillin-resistant S. aureus (MRSA) emerged in Europe in the early 1960s,

soon after methicillin was first used (Ericksen and Erichsen, 1963), has con-

tinued to increase in prevalence and is now found throughout the world. In-

fection with MRSA is associated with higher morbidity, mortality (Engemann

et al., 2003) and expense (Capitano et al., 2003) compared with methicillin-

sensitive S. aureus (MSSA). The rise of MRSA appears to be additive to the

underlying rate of MSSA (Cooper et al., 2004a).

Most hospital-acquired MRSA (HA-MRSA) isolates are also resistant to

other classes of antibiotics. During the 1990s, some MRSA isolates showed

reduced susceptibility to the “last line” antibiotics, linezolid (Tsiodras et al.,

2001; Meka et al., 2004) and vancomycin (Hiramatsu et al., 1997; MMWR,

2002; Chang et al., 2003). MRSA isolates with reduced susceptibility to

vancomycin have now been described in Australia (Gosbell et al., 2003).

This requires a rethink in the strategies for control of MRSA, with renewed

emphasis on prevention of cross-transmission of the pathogen whether

causing asymptomatic colonisation or infection.

Enterococci are part of the normal gastrointestinal flora; harmless colonis-

ers of healthy people. In contrast, hospitalised patients are at high risk of

invasive enterococcal infection, especially those with compromised immune

systems and breached integument. Enterococci are intrinsically resistant to a

number of antibiotics (Weinstein, 2005). Many strains are resistant to ampi-

cillin, leading to the reliance upon vancomycin as a commonly used effec-

tive bactericidal agent against enterococci for many decades. Vancomycin-

resistant enterococci (VRE) emerged in the late 1980s, long after the intro-

duction of vancomycin. Since its emergence, VRE has spread rapidly from

Europe to the United States. It was first identified in Australia in 1994 and

1.1 Motivation 3

its incidence is increasing (Nimmo et al., 2003). It is now regarded as one of

the most important nosocomial pathogens (Murray, 2005). Infection with en-

terococci carrying one of the vancomycin resistant genes is associated with

higher mortality (Lodise et al., 2002); in one study attributable mortality was

as high as 30% (Edmond et al., 1996). Strains of enterococci have been found

with resistance to all conventional antibiotics, including linezolid (Gonzales

et al., 2001). Of great concern is that the vanA resistance gene has been shown

to be able to cross genus into the methicillin-resistant S. aureus (Chang et al.,

2003; Tenover et al., 2004). In view of the failure of antibiotic development

to keep pace with VRE’s acquisition of resistance, VRE control strategies (as

with MRSA) must necessarily focus on containment and prevention.

Other antibiotic resistant pathogens are emerging and spreading within

and between hospitals. These include antibiotic resistant Gram-negative

pathogens such as Pseudomonas spp. and Acinetobacter spp. (Pimentel JD,

2005). These pathogens are not discussed further in this thesis; however

methods and models developed in this thesis could be applied to such

organisms.

1.1.2 The role of mathematical and statistical modelling in

infection control research

Hospital infection control is a relatively recent discipline. Recommended

infection control interventions include surveillance and patient isolation,

antibiotic restriction, ward cleaning, personal protective equipment for staff

and hand hygiene (CDC Guidelines, 1995). Infection control practices are

expensive and a burden for both patients and healthcare workers (Ridwan

et al., 2002). It is therefore essential that the efficacy of these be established.

Infection control strategies in place today frequently are not supported by

rigorous scientific evidence, with some exceptions (Cooper et al., 2004b).

A study by Pittet et al. (2000) showed improvements in hand hygiene was

accompanied by a reduction in MRSA “attack rate”1. Cepeda et al. (2005)

found that there was no reduction in MRSA transmission when patient

isolation was practised. It must be taken into account that in this study,

surveillance was conducted only weekly, a time interval larger than the

median length of stay in the unit and that laboratory turn around time was 4

days. Therefore, the proportion of time that each colonised patient spent in

1Attack rate was defined in this study as the number of transmissions of MRSA per 10 000uncolonised patient days.


isolation would be well under 50% which would dilute the effect of isolation

considerably. Nevertheless, the lag-time experienced in the study is likely

to be a realistic reflection of isolation practices outside of study conditions.

Grundmann et al. (2002) found that low staff/patient ratio was the only risk

factor for colonisation with MRSA, allowing for colonisation pressure in a

multivariate risk analysis. De la Cal et al. (2004) found that patients could

be decolonised through a combined approach of enteral vancomycin and

nasal mupiricin. This approach was criticised because of concerns about

increasing vancomycin resistance in enterococci and potentially MRSA as

well as methodological flaws (Daschner, 2005; Humphreys and Smyth, 2005).

The belief that antibiotic exposure predisposes patients to antibiotic-

resistant pathogen acquisition, while compelling, is not based on ran-

domised control trials. Antibiotic exposure has been found to be a risk

factor in a number of case control studies for VRE and MRSA (Rao, 1998;

Weinstein, 2005). Such studies are frequently subject to confounders and

rarely consider length of hospital stay, co-morbidities and colonisation

pressure. Antibiotics have been shown to increase the stool density and

environmental contamination of VRE (Donskey et al., 2000), suggesting that

antibiotic restriction aimed at colonised patients may be more effective than

those aimed at uncolonised patients. Prevalence of Healthcare Associated

(HA-) MRSA is lower in countries with strict antibiotic policies such as the

Netherlands and Scandinavian countries (Voss et al., 1994).

Many challenges face the infection control investigator. Logistical difficulties

and ethical considerations limit the application of randomised controlled

trials. The data available on hospital acquired infection or colonisation

are often time series data. Frequently this entails a retrospective, quasi-

experimental analysis of an outbreak, after multiple interventions have

taken place (Cooper et al., 2004b).

Interpretation of studies involving planned interrupted time series inter-

ventions requires caution. Data on any contagious disease by its nature is

not serially independent. Each infection or colonisation leads to greater

infection pressure on remaining susceptible individuals. This serial de-

pendence leads to both autocorrelation and overdispersion of colonisation

events. Analyses of serial colonisation data that use standard statistical

techniques, such as Cox proportional hazards survival analysis, applied

to serial infection data, may thus be misleading and are likely to result in

erroneous conclusions (Cooper et al., 2003).

1.2 Overall objectives of the thesis 5

Cluster-randomised control trials would be one way of achieving appropriate

comparison groups in the area of hospital infection control and some are cur-

rently underway or recently published (Cepeda et al., 2005). However, these

are expensive and lengthy and may not be sufficient to determine the opti-

mal infection control strategy. The number of events required to substantiate

an effect (such as a different isolation strategy) is larger in studies in which

the events are serially dependent and the outcomes are over-dispersed com-

pared with studies which can correctly assume serial independence. Such

studies also fail to capture dynamic interactions that contribute to the trans-

mission of bacteria in the healthcare setting (D’Agata et al., 2005), including

the likely impact of combined infection control strategies and the different

equilibria (steady state of proportion of patients colonised) that are possible

following infection control interventions, as found in a theoretical model by

Cooper et al. (2004a).

An additional challenge for the infection control investigator arises because

transmission events such as timing and chains of transmission, are not

wholly observable (Becker, 1989). Colonisation is asymptomatic and acts

as a carrier state for transmission antibiotic-resistant bacterial pathogens.

Serial infection data fail to capture the colonisation that underlies the small

number of patients who manifest disease. Statistical models ideally take

into account the interval-censored nature of hospital transmission data, the

potential for unobserved infectious cases.

Mathematical models can be useful in the area of hospital infection control

for two reasons. Firstly, they can be used to predict quantitatively the course

of an epidemic, predicting its total size, peak; and time to peak and the im-

pact of infection control interventions including nonlinear interactions that

occur when multiple interventions are undertaken. Secondly, they can in-

form the design of trials and structure statistical analyses to avoid assump-

tions of serial independence and difficulties with interval censoring and un-

known numbers of infectious cases.

1.2 Overall objectives of the thesis

The overall objectives of this thesis are to develop mathematical and statis-

tical models in order to improve the understanding of the transmission of

infectious agents in the hospital and to use these models to inform infection

control practitioners as to the likely impact of interventions.


The first part of the analyses in this thesis is the development of plausible

models based on biological and epidemiological knowledge of each or-

ganism with full recognition of model assumptions and limitations. The

second part is the analysis of datasets to determine parameters that govern

the underlying epidemic model. Statistical inference techniques used in

this thesis were mostly Bayesian, including the use of latent variables to

represent unobserved or missing data, with computation by Markov chain

Monte-Carlo techniques including the Metropolis-Hastings algorithm.

Expectation-Maximisation (EM) algorithms and piecewise constant hazard

models are also used throughout the thesis as are standard model selection

and model checking techniques. These are elaborated in Chapter 2. The

third part of the analysis is the prediction of the impact of infection control

procedures. By changing model parameters and assessing large scale behav-

iour of the model, we can make reasonable estimates of the likely outcome

of infection control interventions.

The strategy in each study is to a develop pathogen-specific mathematical

model and use this as the foundation of a statistical model to probe unpub-

lished datasets and quantify transmission. In two of the studies, described

in Chapters 5 and 7, infection control interventions occurred during the data

collection period, allowing estimates to be made of the impact of these in-

terventions on pathogen transmission. The methodology aims to account

for both serial dependence in infection data and the censored nature of the

transmission events. In Chapter 5, the thesis extends the notion of censored

data to develop a model and methodology that could make use of simple ser-

ial surveillance data in the absence of patient event data or perfect detection

of colonisation, using a hidden Markov model structure.

A major objective of this thesis, as in many studies in the area, is to estimate

the basic reproduction ratio (defined in Section 2.2.4). Estimates are not

confined to this measure however. In the study described in Chapter 3,

the transmission rate was low and the reproduction ratio was below unity.

Despite this, infection control interventions were shown to reduce trans-

mission, which could be quantified using the attack rate. Characterising

the source of infection (endemic or epidemic) using statistical methods was

an important outcome of the work described in Chapter 5. Estimation of

differences in infectivity following interventions (in Chapters 5 and 7) or

depending on the hospitalisation status of the individual (in Chapter 7) was

also undertaken.

1.2 Overall objectives of the thesis 7

One of the objectives of this thesis is to consider infection control interven-

tions that have not previously been considered by mathematical modellers.

Local policies and practices, for example local cohorting policy, HCW/patient

contact rates and ward size, were specifically considered in models devel-

oped. Recent research findings, for example the use of enteral vancomycin

to reduce MRSA colonisation as described by de la Cal et al. (2004), were in-

corporated into model predictions.

Both deterministic and stochastic models are considered in this thesis. The

results are compared and the reasons for any discrepancies are discussed.

Stochasticity is an important consideration given that the scale of the popu-

lations was small.

Where more than one model is plausible, several different models are com-

pared and the model with the best balance between parsimony and data fit,

as defined by model comparison criteria, is selected. Finally, this thesis aims

to validate, where possible, the models developed. This is achieved using ex-

ternal validation sources if available and internal validation to show that the

model techniques estimated the parameters without bias and with adequate

precision.

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Capitano, B., Leshem, O. A., Nightingale, C. H., Nicolau, D. P., 2003. Cost effect ofmanaging methicillin-resistant Staphylococcus aureus in a long-term care facility.J Am Geriatr Soc 51 (1), 10–16.

CDC Guidelines, 1995. Recommendations for preventing the spread of vancomycinresistance. Hospital Infection Control Practices Advisory Committee (HICPAC).Infect Control Hosp Epidemiol 16 (2), 105–13.

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Edmond, M. B., Ober, J. F., Dawson, J. D., Weinbaum, D. L., Wenzel, R. P., 1996.Vancomycin-resistant enterococcal bacteremia: natural history and attributablemortality. Clin Infect Dis 23 (6), 1234–9.

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Pittet, D., Hugonnet, S., Harbarth, S., Mourouga, P., Sauvan, V., Touveneau, S., Per-neger, T. V., 2000. Effectiveness of a hospital-wide programme to improve compli-ance with hand hygiene. Infection Control Programme. Lancet 356 (9238), 1307–12.

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Semmelweiss, I., 1861. Die tiologie, der Begriff und die Prophylaxis des Kindbet-tfiebers.

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Tsiodras, S., Gold, H., Sakoulas, G., Eliopoulos, G., Wennersten, C., Venkataraman,L., Moellering, R., Ferraro, M., July 2001. Linezolid resistance in a clinical isolate ofStaphylococcus aureus. Lancet 358 (9277), 207–208.

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CHAPTER 2

Literature review and outline of thesis

The critique of relevant literature begins with a review of the biology and epi-

demiology of the microorganisms discussed in this thesis. This is followed by

a general review of compartmental mathematical models applied to infec-

tious diseases transmission at a population level. Specific models relevant to

the studies in this thesis are then explored. Methods used in this thesis are

outlined with an emphasis on reasons for choice of methodology. This chap-

ter concludes with an outline of the original chapters of the thesis, including

their relationship with previous work, innovations and contributions to the

field of research.

2.1 Review of pathogens discussed in this thesis:biological and epidemiological features rele-vant to model development

2.1.1 Methicillin resistant Staphylococcus aureus (MRSA)

Summary

• MRSA is principally transmitted from patient to patient via the hands of

healthcare workers

• Healthcare worker carriage is usually transient and removed by hand

washing

• New MRSA colonisation is caused by cross-transmission of pre-existing

MRSA clones rather than spontaneous emergence of resistance during

exposure to antibiotics (Cooper et al., 2003)

• MRSA colonisation is asymptomatic and precedes infection

14 Chapter 2. Literature review and outline of thesis

• Colonisation of patients is long-term (weeks to months)

• While community MRSA is significant, it has different characteristics

and is readily distinguishable from healthcare-associated strains

S. aureus is a Gram-positive coccus, that is carried asymptomatically on the

skin or in the nares of approximately 30% of the population at a given time.

Carriage is more common on eczematous skin (Boyce, 2005), in people who

receive repeated injections such as insulin-dependent diabetics and infect-

ing drug users, and haemodialysis and peritoneal dialysis patients (Mandell

et al., 2005). The gastrointestinal tract is a recently discovered important site

of colonisation (Boyce et al., 2005).

S. aureus becomes pathogenic when it crosses the integumentary barrier and

can cause serious invasive infection including septicaemia, endocarditis,

pneumonia and severe soft tissue infections and bone and joint infections

(Mandell et al., 2005). Breaches in the skin (the use of central catheters,

surgical wounds, haemodialysis and peritoneal dialysis) increase this risk

of S. aureus infection. Other risk factors include diabetes, HIV and alcohol

abuse.

Beta-lactam antibiotics include the penicillins, cephalosporins and car-

bapenems and monobactams. Although the spectrum of activity of these

antibiotics differs, each kills bacteria by binding and inhibiting penicillin-

binding proteins (PBPs). PBPs are membrane bound transpeptidase and

transglycosidase enzymes essential in bacterial cell wall synthesis. When

penicillin was first introduced for therapeutic use in 1941, S. aureus was

highly sensitive to the antibiotic, but resistance emerged rapidly. By 1950,

isolates appeared that had acquired a plasmid encoded penicillinase en-

zyme, capable of hydrolysing the β−lactam ring of penicillin. By the mid

1950s 40% of isolates were penicillin resistant (Chambers, 2001).

After the emergence of penicillin-resistant Staphylococci, semisynthetic

penicillinase-resistant beta-lactams and cephalosporins were introduced

and these drugs remain the most effective therapy for sensitive staphylococci

(currently in Australia, flucloxacillin, dicloxacillin, cephalexin, cephalothin,

cephazolin are widely used). However, soon after the new class of antibi-

otics was introduced, methicillin-resistant Staphylococcus aureus appeared

(Ericksen and Erichsen, 1963). An absolute requirement for methicillin

resistance in S.aureus is the presence of the mecA gene. MecA is one part of

the staphylococcal chromosomal cassette (SCCmec), a large mobile chro-

mosomal element. Additional components include two regulatory genes

2.1 Review of pathogens discussed in this thesis 15

and five auxiliary genes that can control or modify gene expression (Lowrie,

2006). MecA encodes penicillin binding protein (PBP) 2a, a novel protein

that has reduced affinity to all β−lactam antibiotics, leading to resistance

to the entire class (Lowrie, 2006). DNA hybridisation studies suggest that

there are only five major clones of the mecA gene responsible for methicillin

resistance (Enright et al., 2002).

Like the methicillin susceptible strains, MRSA frequently causes only harm-

less colonisation (Boyce et al., 2005). The median MRSA patient carriage

has been estimated to be 8.5 months by Scanvic et al. (2001) and 40 months

by Sanford et al. (1994). It has been reported that in 30-60% of healthcare-

associated cases, MRSA colonisation proceeds to invasive disease (Boyce,

2005).

HA-MRSA are believed to spread via the transiently-contaminated hands

of healthcare workers (Boyce, 2001). Studies show that MRSA can exist on

healthcare workers hands for 3 hours, but that hand hygiene will almost

completely eradicate MRSA (Peacock et al., 1980; Thompson et al., 1982).

The environment has been found to be contaminated by MRSA (Boyce

et al., 1997), but the contribution of environmental sources of MRSA to

transmission is not well established.

MRSA has recently emerged in the community (CA-MRSA) but this is distin-

guishable from the hospital strains by antibiotic resistance patterns and the

presence of type IV SCCmec (HA-MRSA carry type I, II or III SCCmec) (Boyce,

2005). Molecular techniques also distinguished CA- from HA-MRSA isolates,

suggesting that community-associated disease is not typically due to spread

of nosocomial strains into the community (Naimi et al., 2003). Almost all CA-

MRSA isolates remained non-multiresistant, being sensitive to other classes

of antibiotics such as aminoglycosides (Nimmo et al., 2006).

The facts presented in this section underpin the assumptions made in

Chapter 3 in constructing a model for MRSA transmission. The frequency

of asymptomatic colonisation leads us to study both colonised and infected

patients. The highly complex mechanism of resistance described leads

to the assumption that no de novo resistance arises. The transient nature

of healthcare worker carriage compared with patient carriage leads to the

structure of the model used in Chapter 3. The straightforward differentiation

of community and hospital strains (by antibiogram) allows us to separate the

influence of CA-MRSA in the study on Intensive Care Unit patients.


2.1.2 Vancomycin-resistant enterococci (VRE)

Summary

• VRE colonisation of the gastrointestinal tract is often asymptomatic

• Apparent VRE acquisition may arise from the patient’s own gut flora

• Colonisation is a carrier state of VRE and precedes infection

• Colonisation is long term, lasting weeks to months

• Complex mobile genetic elements confer resistance and these can be

transferred to other enterococci, leading to resistance in unrelated en-

terococcal strains

• Environmental contamination may play a part in transmission in some

settings

Enterococci are Gram-positive, catalase negative cocci that are part of the

normal gut flora of humans and animals. E. fecalis and E. fecium are the two

species most likely to cause disease in humans. They are inherently resistant

to many antibiotics. Until the late 1980s, vancomycin was a reliable agent

for enterococcal strains with resistance to other agents. Vancomycin resis-

tance in enterococci was first identified in Europe, then spread to the USA

and reached Australia by 1994 (Bell et al., 1998). Its prevalence in Australian

both in the community and in hospitals remains low. The hospital prevalence

is rising with 0.3% of hospital strains found resistant in 1999 (Nimmo et al.,

2003) compared with none of the surveyed strains in 1995.

Enterococci are less virulent than S.aureus but can cause urinary tract in-

fection, bacteraemia and infective endocarditis. Invasive disease has a high

mortality rate, probably more so if the enterococcus is vancomycin resistant,

although many studies are confounded by patient risk factors (DiazGranados

et al., 2005). Most patients who develop VRE infection are debilitated and

their underlying disease contributes to their high mortality rate (Weinstein,

2005).

Unlike β−lactams, glycopeptides (vancomycin, teichoplanin) kill Gram-

positive bacteria by preventing cross-linking of the peptidoglycan compo-

nent in the cell wall by binding tightly to the terminal D-alanyl-D-alanine

residues of the pentapeptide stem. The genes encoding glycopeptide resis-

tance lead to replacement of the usual terminal D-alanyl-D-alanine residue

with a D-alanyl-D-lactate moiety. Vancomycin is unable to bind to this


peptide thus cross-linking of the peptidoglycan proceeds and resistance

results (Weinstein, 2005).

The vanA phenotype leads to resistance to both vancomycin and teicho-

planin, while the vanB phenotype leads to resistance to vancomycin only.

VanA resistance is conferred by a mobile genetic element, a transposon (Tn

1546) (Weinstein, 2005). This is a complex set of genes involving regulatory

genes, genes coding for the production of the new pentapeptide terminal

and some genes responsible for integration of the transposon into larger

genetic elements such as plasmids.

These mobile genetic elements can be transferred to other enterococcal

strains (horizontal transfer of resistance) and indeed interspecies transfer to

S.aureus has been described as discussed in Section 2.1.1. Epidemiological

studies that rely on strain typing to determine clonality of an outbreak of

resistance (such as Pulsed Field Gel Electrophoresis) could thus incorrectly

assess the degree of relatedness of the vancomycin resistance genes.

Colonised patients are asymptomatic, contribute to transmission (Bonten

et al., 1998) and may go on to develop invasive disease (Noskin et al., 1995a).

Colonisation of VRE is long term, potentially indefinite (Noskin et al., 1995a).

Exposure of patients to antibiotics may allow previously undetectable levels

of organism to multiply and become detectable (Donskey et al., 2002).

Prior hospitalisation is a risk factor for colonisation with antibiotic resistant

enterococci on admission to hospital (Weinstein et al., 1996).

There is an interesting dichotomy in the pattern of VRE in Europe and the

United States. In parts of Europe in the late 1990s, VRE colonised healthy

people who were not exposed to hospitalisation. Endtz et al. (1997), for ex-

ample, found that 2% of both hospitalised and community patients had VRE

colonisation in the Netherlands. In the United States, community VRE is rare

but VRE is much wider spread in hospitals. It is believed that successful an-

tibiotic stewardship in hospitals keeps hospital prevalence of VRE low, while

the use of avoparcin (an antibiotic of the same class as vancomycin) in animal

feed contributes to the spread of community VRE in the Netherlands (Ridwan

et al., 2002). Avoparcin was used in Australia until 2001, and the prevalence

of VRE in community volunteers was at a non-negligible level of 0.2% in 1997

(Padiglione et al., 2000).

From the evidence above, Australian patients on admission to hospital

may be colonised with VRE, either from exposure in the community or

prior hospitalisation. Colonisation is frequently asymptomatic and goes


undetected unless there is an active screening program in the healthcare

institution. Therefore, when colonisation or infection are detected, the time

of acquisition will not be clear. Additionally, swab sensitivity is considerably

less than 100% (D’Agata et al., 2002). Therefore, people who have tested

VRE-negative may be colonised with VRE, possibly with sub-detectable

levels. Exposure to antibiotics which suppress other gut flora may allow VRE

to reach detectable densities. Donskey et al. (2002) found that patients with

a history of VRE colonisation could become swab negative for VRE but on

exposure to antibiotics again become positive for VRE.

These facts and inferences underlie the structure of the model presented in

Chapter 5. The model is designed to distinguish between two types of VRE ac-

quisition, that which occurs due to cross-transmission, and that which arises

sporadically, such as might occur when a patient with VRE gut colonisation is

exposed to antibiotics. The model also assumes that VRE colonisation is long

term, with decolonisation rates negligible relative to the duration of hospi-

tal stay. It also considers the deficiencies of genotyping studies, given the

confusion that arises when horizontal transfer of genetic resistance elements

occurs.

2.1.3 Environmental pathogens

Summary

• A number of pathogens have been found in the environment

– VRE (Noskin et al., 1995b; Bonten et al., 1996)

– MRSA (Boyce et al., 1997)

– Clostridium difficile (Kim et al., 1981)

– Gram-negative bacilli, including coliforms and oxidase positive

(Dancer et al., 2006)

• the role of the environment in transmission has not been determined

VRE has been found frequently in patients’ environments and has been

shown to be viable on inanimate objects for some days (Noskin et al., 1995b).

It is not known to what degree environmental contamination contributes

to transmission, or whether it merely reflects stool density of the patient, as

found by Donskey et al. (2000). It is still believed that transmission occurs


principally via the hands of healthcare workers (Weinstein, 2005), although

the evidence for this is not as strong as is the case for MRSA.

These findings lead to the study in Chapter 6 which considers the potential

impact of an environmental reservoir on the transmission of VRE.

2.1.4 Severe Acute Respiratory Syndrome Coronavirus (SARS

Co-V)

Summary

• SARS Co-V has a distinct incubation period which is not directly observ-

able

• SARS Co-V is spread principally by direct person to person contact

• SARS Co-V is transmitted principally through contact and droplet

spread

• There is (at most) limited transmission of SARS Co-V from cases during

the incubation period

• Asymptomatic infection is rare

• Nosocomial transmission was a prominent feature of SARS during the

2002/3 pandemic epidemiology

Evidence suggests that infectivity does not precede symptom onset for SARS.

Early contact tracing studies found transmission occurred only to close con-

tacts of a symptomatic SARS case (Poutanen et al., 2003). This is consistent

with virological studies that found levels of viral shedding were low in the

early phase of illness (Cheng et al., 2004).

Sero-surveys of populations affected by SARS concluded that asymptomatic

infection is uncommon. Rainer et al. (2004) found that of the patients

presenting with mild respiratory symptoms during the SARS outbreak in

Hong Kong, only 0.7% had serological evidence of SARS. Leung et al. (2004a)

found a very low seropositivity (0.2%) of SARS antibodies in people who did

not have symptomatic SARS but who had close contact with SARS cases.

Estimation of the incubation period for SARS-CoV has proven to be a consid-

erable challenge. Numerous studies have attempted to make estimates (see

Donnelly et al., 2004, for review). It is currently believed to be around 5 days

with an upper limit of 10 days (Donnelly et al., 2004).


Early in the SARS pandemic, a majority of cases arose from hospital transmis-

sion in many places, including Toronto (Booth et al., 2003), Hong Kong (Ri-

ley et al., 2003; Wong et al., 2004) and Singapore (Gopalakrishna et al., 2004).

In Hong Kong almost half of cases were healthcare-associated (Leung et al.,

2004b). Later in the course of the epidemic, hospitals were effective sites of

containment of SARS (Gopalakrishna et al., 2004).

Clusters of cases from Hong Kong and Canada suggest that the SARS coron-

avirus (SARS-CoV) spreads directly from person-to-person. Acquisition from

face-to-face interaction rather than physical contact suggests droplet spread

(Donnelly et al., 2004; Hirsch, 2006). Other methods of spread including via

the faeces are also possible, as SARS-CoV is frequently recovered from the

feces (Cheng et al., 2004).

These features of SARS-CoV transmission are incorporated into the structure

of the model presented in Chapter 7.

2.2 Mathematical models of human infectious dis-

eases

2.2.1 History

The first attempts to apply mathematical methods to infectious disease data

involved descriptive statistics. Bernoulli (1760) and Farr (1840) evaluated

data on smallpox vaccination and deaths, respectively, using empirical

methods. Following the general acceptance of germ theory, Hamer (1906)

brought the principle of mass action, used in chemistry since Boyle (c. 1674),

into the transmission of infectious diseases. Hamer (1906) postulated that

the progress of an epidemic depends on the contact rate between suscep-

tible and infectious individuals, therefore being dependent on the density

of each in the population. At the time there was debate about the cause of

waning of epidemic waves, with many arguing that a decrease in virulence

of the contagion was responsible. Hamer argued that reduced density of

susceptibles was responsible for decline in epidemics. He also postulated

that births and loss of immunity would lead to increased susceptibility and

new epidemic waves.

Ross (1916) and Ross and Hudson (1916) developed the mass action con-

cept further, presenting continuous time epidemic equations. In 1927,

Kermack and McKendrick developed the equations of the compartmental

2.2 Mathematical models of human infectious diseases 21

Susceptible-Infectious-Removed (SIR) model, called the general epidemic

model, still widely used today.

2.2.2 The Susceptible-Infectious-Removed model

The general epidemic model assumes that people begin susceptible to an in-

fectious disease, may become infected by exposure to an infectious person,

becoming immediately infectious themselves and after a time period either

recover or die. Recovery constitutes immunity to further infection and they

are said to be removed.

The simplest version of this SIR model assumes homogenous mixing and a

fixed population size, N = S(t) + I(t) + R(t), where S(t), I(t), and R(t) are

the numbers in the population who are susceptible, infectious and removed

at time t. Each contact between a susceptible and an infectious patient has

a probability, p, of leading to transmission and contacts occur at a rate, c per

day. The classical system of ordinary differential equations is

dS

dt= −cpSI (2.1)

dI

dt= cpSI − γI

dR

dt= γI,

where c, and γ are positive constants and 0 < p ≤ 1. If one does not require

separate estimates of p and c, one can use β = cp. The behaviour of the sys-

tem is governed by the first two equations and the number of recovered, R,

can be determined as N = S(t) + I(t) + R(t).

The differential equation for the number of infectious, I(t), can be rewritten

as

dI

dt= βI(S − γ

β), (2.2)

which leads to a critical value of susceptibles. Following the introduction of

an infectious case, in order for an epidemic to proceed, the number of the

population that are susceptible must be greater than γ/β, and if the initial

value of βN is less than γ, an epidemic will not occur.

Linearising the system about a steady state (S, I) and putting S = S + s and


I = I + i leads to

(s

i

)=

(−βI −βS

βI βS − γ

)(s

i

). (2.3)

The dominant eigenvalue of the Jacobian at the steady state (S = N , I = 0)

gives the growth rate of the epidemic curve, namely βN − γ.

The classic epidemic SIR model has been extended in a number of ways, in-

cluding

• models that incorporate temporary immunity (SIRS) or no immunity

(SIS)

• models with vital dynamics (in which birth and death are included)

• SIR models with carriers (a carrier is one who spreads diseases but has

no symptoms)

• SIR models with vertical transmission

• SIR models with stratified populations

• models in which there is no recovery from infection (SI)

• host-vector-host models (contagion is passed alternately from one

species to another) also called the Ross-MacDonald model

• models in which an infected individual has a latent period before

becoming infectious; Susceptible-Exposed-Infectious-Removed (SEIR)

models

In the latter half of last century there was a greater emphasis on probabilis-

tic models (Bailey, 1975; Becker, 1989). Other extensions of classic SIR mod-

els of disease include the spatial spread of disease (Diekmann and Heester-

beek, 2000), understanding of recurrent epidemic waves, and heterogeneity

of spreading (Anderson and May, 1991) and the importance of households

and other social networks (Becker, 1989).

In this thesis, modified SIR models are developed according to the specifics of

the pathogen being considered. Chapter 3 develops a modified host-vector-

host model with migration (vital dynamics), appropriate to the transmission

characteristics of MRSA. Chapter 5 uses an SI model with migration, and


Chapter 7 develops an SEIR model, modified by relaxing the assumption of

negative exponential sojourn times within SEIR compartments.

2.2.3 The Susceptible-Exposed-Infectious-Removed model

Susceptible-Exposed-Infectious-Removed (SEIR) models include an addi-

tional class of latently-infected (exposed)1 individuals. These individuals

have acquired infection but are not yet infectious. A simple system of

equations can be used to describe this model

dS

dt= −βSI (2.4)

dE

dt= βSI − νE

dI

dt= νE − γI

dR

dt= γI,

where β, ν and γ are positive constants.

Linearising the system about the steady state (S, E, I) and putting S = S + s,

E = E + e and I = I + i leads to

s

e

i

=

−βI 0 −βS

βI −ν βS

0 ν −γ

s

e

i

. (2.5)

The dominant eigenvalue of the Jacobian at the steady state (S = N, E =

0, I = 0) gives the growth rate of the epidemic curve,

λ =−(γ + ν) +

√(ν − γ)2 + 4βNν

2. (2.6)

Note that the growth of the epidemic is dependent on the rate of transition

1The incubation period is the time from acquisition of a contagion until the person devel-ops symptoms. The latent period is the time from acquisition of a contagion until the personis infectious.


from the latent to the infectious period, ν. The rise in the early epidemic

curve, λ, is easily calculated during an outbreak. Inferences regarding β based

on the value of λ will be highly sensitive to ν which is often taken to be the

reciprocal of the mean duration of the incubation period (see, for example,

Chowell et al. (2003)).

Importance of compartmental sojourn times

The models using the system of ordinary differential equations (2.1) and

(2.4) implicitly assume that the exposed (latent) and the infectious periods

are negative exponentially distributed with parameters ν and γ, respectively.

Keeling and Grenfell (1997) showed that assumption of the exponential

distribution for the infectious period leads to underestimation of the critical

community size, the size necessary to sustain endemic transmission, for

measles. The over-prediction of fade-outs that occurred in standard (expo-

nential infectious periods) model was corrected by allowing the infectious

periods to be normally distributed in line with observed infectious period

distributions. Estimates of infectivity, particularly those based on the early

epidemic curve, are also highly sensitive to the shape of the survival curve

in the Exposed and Infectious compartments (Lloyd, 2001). Donnelly et al.

(2003) showed that the latent period for Severe Acute Respiratory Syndrome

is not exponential. Lloyd (2001) showed that non-exponential compart-

mental sojourn times lead to more realistic model predictions for the SIR

model.

2.2.4 The basic reproduction ratio, R0

The basic reproduction ratio (R0) is defined as “The average number of per-

sons directly infected by an infectious case during its entire infectious period,

after entering a totally susceptible population” (Giesecke, 1994).

The basic reproduction ratio is a function of daily infectivity and expected

duration of infectivity. The effective reproduction ratio is expected number

of persons directly infected by an infectious case without the assumption of a

fully susceptible population. In the SIR or SEIR model, with constant hazard

of transition between compartments and constant infectivity, the effective

reproduction ratio is Rt = βSt

γ. When the entire population is susceptible

(S(0) = N), this expression gives the basic reproduction ratio, R0 = βNγ

.


A more general expression is obtained when the infectivity does not remain

constant over the infectious period and the transit time over the infectious

period is not necessarily negative exponential.

Here

R0 =

∫ ∞

0

c(τ)p(τ)q(τ)dτ, (2.7)

where τ is the time since transmission occurred to an individual and q(τ) is

the probability of remaining infectious, c(τ) is the contact rate and p(τ) the

probability of transmission per contact, a time period τ from infection.

In Chapter 7, Bayesian inference is used to estimate the changes in infectivity

over the course of SARS infection. Such estimates are informative to infection

control practitioners and can be the basis of more realistic models.

2.2.5 Adaptation of the Ross-MacDonald model to the

healthcare setting

Originally used to describe the transmission of malaria, the host-vector-host

model was developed by Ronald Ross (1857-1932) and George MacDonald

(1903-1967).

The model is applicable to infections that are transiently carried by a vec-

tor2. In the original model, the vector population was a different species, the

Anopheles mosquito, and the infectious agent, malaria, was an obligate para-

site, requiring the vector to complete its lifecycle. A fundamental assumption

of the model is that all host-to-host transmission is indirect, via the vector.

The vector carries the pathogen transiently while the principal host is infec-

tious for longer.

The Ross-MacDonald model is readily modified to describe the transmission

of pathogens in the healthcare setting. Nosocomial pathogens are believed

to spread from patient to patient indirectly via the transiently-contaminated

hands of HCWs. In this sense the HCWs are analogous to the mosquito vec-

tors of the original model.

Figure 2.1 illustrates the model dynamics. Transmission occurs only during a

discordant contact: when a colonised HCW contacts an uncolonised patient

2Here vector is used in the medical sense: an organism that does not cause disease itselfbut which spreads infection by conveying pathogens from one host to another.


or when a colonised patient contacts an uncontaminated3 healthcare worker.

An important modification of the Ross-MacDonald model, is the incorpora-

tion of rapid migration of patients in and out of the ward. Uncolonised pa-

tients are discharged at a rate µx, while colonised patients are discharged at

a rate µy. A proportion, σ, of patients may be colonised on arrival. Transition

from contaminated back to susceptible is assumed to occur at a fixed rate for

healthcare workers, κ. The probability of transmission from HCW to patient

is denoted by php and patient to HCW pph. Contact rate per patient per HCW is

c. The number of admissions per day is given by Λ. The patients are labelled

p and the healthcare workers, h. Y indicates infectious and X susceptible.

hp p hcp X Y

y pYµ

hYκ

hY

pX

hX

pY

ph h pcp X Yx p

Xµ

(1 )σ− Λ σΛ

Figure 2.1: The application of the Ross-MacDonald model to the transmis-sion of nosocomial pathogens.

Each element, Gij , in the next generation matrix, G, is the expected number

of new infections transmitted to each population j by a single infective of

population i. The next generation matrix for the model shown in Figure 2.1 is

given by

G =

(0

cpphNh

µycphpNp

κ0

). (2.8)

The basic reproduction ratio is the square of the dominant eigenvalue of the

next generation matrix4.

3The term contaminated/uncontaminated will be used to describe the transient coloni-sation of healthcare workers, while the terms colonised/uncolonised will be used when re-ferring to patients.

4R0 is either the dominant eigenvalue or the square of the dominant eigenvalue depend-ing on definition of R0. The former definition, used by Diekmann and Heesterbeek (2000)


In this system, the basic reproduction ratio, R0, is given by

R0 =c2phppphNhNp

µyκ. (2.9)

This model was used in Chapter 3 of this thesis. The Ross-MacDonald model

is relatively complex with a number of parameters which useful for predicting

the impact of different infection control interventions. In the hospital there is

rapid migration, which must be incorporated into the model. Individual pa-

rameters (for example hand hygiene compliance or admission prevalence) or

combinations of parameters can be changed to model the effect of interven-

tions, as elaborated in Chapter 3.

2.2.6 Single population models

The large number of parameters and complexity of the Ross-MacDonald

model make it sensitive to uncertainties in parameter estimates and model

assumptions. A more parsimonious model is the two compartment model

in which the HCW compartments are replaced by a constant, using the

assumption Yh ≈ βYp, which leads to an SI model. The justification for

this assumption is given later in this section. The two compartment model

has fewer unknown values to estimate, which reduces collinearity between

parameters and improves precision of estimates. The assumption that all

acquisition of the infectious agent occurs via indirect transmission is relaxed.

The cross-transmission parameter, β, incorporates both direct and indirect

transmission.

The study presented in Chapter 5 is based on this model. The model includes

an additional parameter for VRE acquisition that was independent of cross-

transmission, ν, as shown in Figure 2.2. In this model, R0 is given by

R0 = βN/µc. (2.10)

is the expected number of the other population infected by a single infectious individualassuming a fully susceptible population. The latter definition, used in Chapter 3 of this the-sis, is the expected number of the same population infected by a single infectious individualassuming a fully susceptible population. The threshold value for R0 will be unity in eithercase.


CU

( )u

CU Uβ ν σµ+ +

(1 )c Cµ σ−

Figure 2.2: The single population model. Graphical representation of thetransmission of bacterial pathogens among patients in the hospital ward.Here C is the number of colonised patients, U is the number of uncolonisedpatients, N = U + C is the total number of ward patients (assumedfixed), µu is the discharge rate of uncolonised patients, µc is the dischargerate of colonised patients, ν is the colonisation rate independent of cross-transmission, σ is the admission prevalence and β is the cross transmissioncoefficient.

The ordinary differential equation governing this model is

dC

dt= βC(N − C) + (ν + µU(1− σ))(N − C)− µcC. (2.11)

Simplification of Ross-MacDonald compartment model

The reduction of the Ross-MacDonald model (used in Chapter 3) to a two

compartment, patient-only model (used in Chapter 5) can be justified on

the basis that in the four compartment model, the proportion of healthcare

workers colonised is directly proportional to the number of colonised pa-

tients. Figure 2.3 shows the relationship between the expected number of

colonised healthcare workers as the number of colonised patients changes

using realistic values (h = 0.59, Np = 15, pph = 0.13) derived from the study

presented in Chapter 3.

0 5 10 150

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Number of colonised patients

Figure 2.3: Predicted linear relationship between number of patientscolonised and number of healthcare workers colonised.


Therefore

Yh ≈ βYp, (2.12)

and Yh in Figure 2.1 can be replaced by βYp, leading to the much simpler two

compartment model.

2.2.7 Stochastic models

Deterministic models, which assume continuous changes in compartment

numbers and give results based only on initial conditions, model parameters

and structure, give a reasonable approximation for parameters when num-

bers in each model compartment are large, (Bailey, 1975). They fall down,

however, in small scale epidemics such as that encountered in the hospital

ward setting. They are also not reliable at the beginning and end of the epi-

demic when the number of infectives is small. Additionally, they are unable

to quantify uncertainty in the parameter estimates, nor the expected varia-

tion in the simulated epidemics based on those parameters. Here stochastic

models prove useful (Becker and Britton, 1999). Statistical models of infec-

tious disease transmission apply structured epidemic models, such as those

described in this section, to statistical analyses for parameter estimation, hy-

pothesis testing and study design (Becker and Britton, 1999).

Stochastic equations involve discrete changes in model variables. In contin-

uous time stochastic models (used in all studies of this thesis) compartment

counts increment or decrement by unit amounts. Over a brief time inter-

val, dt, unit changes in model compartments occur with a probability deter-

mined by the hazard of transmission (based on the current state of the model

compartments). For example, the stochastic version of the SEIR equations is


given by

PrS(t + dt) = i− 1, E(t + dt) = j + 1, I(t + dt) = k, R(t + dt) = l| (2.13)

S(t) = i, E(t) = j, I(t) = k,R(t) = l= βikdt + o(dt),

P rS(t + dt) = i, E(t + dt) = j − 1, I(t + dt) = k + 1, R(t + dt) = l| (2.14)

S(t) = i, E(t) = j, I(t) = k,R(t) = l= νjdt + o(dt),

P rS(t + dt) = i, E(t + dt) = j, I(t + dt) = k − 1, R(t + dt) = l + 1| (2.15)

S(t) = i, E(t) = j, I(t) = k,R(t) = l= γkdt + o(dt).

Similar stochastic equations can be derived for the other models.

2.3 Relationship of current literature to models

presented in this thesis

The preceding discussion was a general overview of the structure of mathe-

matical models that relate to work in this thesis and methodology used in this

thesis. The following discussion concentrates on specific recent published

research that share features with the models in this thesis.

2.3.1 Studies based on the Ross-MacDonald model applied to

nosocomial transmission of bacteria

A number of studies have applied an adapted Ross-MacDonald model to the

transmission of bacterial pathogens in the healthcare setting. Most made the

following assumptions: transmission occurs only via the hands of healthcare

workers; the populations are homogenous with regard to susceptibility and

infectiousness; mixing between HCWs and patients is homogenous; patient

colonisation is long term (greater than duration of hospital stay) while

healthcare worker colonisation is short-term. Table 2.1 gives an analysis of

studies which follow this model structure including the study presented in

Chapter 3 on this thesis.

The structure of the four compartment models allows simulations and

predictions of the likely efficacy of a number of different interventions

2.3 Relationship of current literature to thesis 31

including improved hand hygiene, removal/discharge of colonised patients

and changes in admission prevalence. Small changes in model structure

also allow models to predict the impact of patient cohorting and changes

in staff/patient ratio, patient isolation, antibiotic restriction and patient

decolonisation.

Sources of parameters for models generally fall into five categories:

1. expert opinion

2. literature review

3. observation on the ward as part of the study

4. epidemiological results from dataset

5. fitting remaining parameters to colonisation data.

All of the studies reviewed to date have used at least one parameter that was

not measured as part of the study, usually the transmission probability or

the duration of healthcare worker contamination. These were estimated or

derived from literature sources or expert opinion. This is a weakness in the

current body of literature on the subject as the model predictions are highly

sensitive to these parameters, although the values of R0, derived by Austin

et al. (1999), are not dependent on estimated model parameters.

32C

hap

ter2.L

iterature

reviewan

do

utlin

eo

fthesis

Study MRO Relaxation of Assumptions Interventions tested Main findings / predictions Source of parameter estimation Study design

Sebilleet al.(1997)

MRSA Patient to patient and HCWto HCW transmission con-sidered. Decolonisation ofpatients included.

Hand hygiene, antibioticpolicy, reduction in admis-sion prevalence

Hand hygiene had large impacton HCW contamination but notpatient colonisation. Antibi-otic policy had little impact onMRSA colonisation.

Ward data: mean length of stay of pa-tients (colonised and uncolonised treatedthe same). Expert opinion: all other para-meters.

Deterministic

Cooperet al.(1999)

non-specific

Hand hygiene, changes inlength of stay and transmis-sibility, improvements indetection of organism

Hand hygiene compliance andtransmission probability havethe greatest impact on trans-mission. Large variation in out-come can be expected due tostochastic effects.

Literature review and expert opinion esti-mates for all parameters.

Stochastic

Austinet al.(1999)

VRE Ward was not of fixedsize for simulations (butassumption of fixed sizeused to calculate the basicreproduction ratio).

Hand hygiene, patientcohorting and admissionprevalence

The effective reproduction ratiowas estimated to be 0.69. In theabsence of interventions thebasic reproduction ratio was es-timated to be 3.81.

Expert opinion: transmission probability,duration of HCW contamination. Database:length of stay, ward size, admission preva-lence.

Deterministic(for pa-rameterfitting) andstochastic forsimulations

Grundmannet al.(2002)

MRSA Ward was not of fixedsize for simulations (butassumption of fixed sizeused to calculate the basicreproduction ratio).

Hand hygiene, patient co-horting

Staff deficit was the only covari-ate associated with increasedMRSA transmission. Improve-ment in hand hygiene by 12%was predicted to make up forstaff deficit.

Expert opinion: transmission probability,duration of HCW contamination. Data-base: length of stay, ward size, admissionprevalence. Measured on the ward: con-tact rates, transmission probability, handhygiene compliance.

Cohort studywith covariateanalysis, Sto-chastic anddeterministic

D’Agataet al.(2005)

VRE Patient compartment nothomogenous. Divided intothose receiving and not re-ceiving antibiotics.

Hand hygiene, patientcohorting, staff/patientratio, length of stay,antibiotic policy, admissionprevalence

Reducing antibiotic exposure touncolonised patients is moreeffective than reducing antibi-otic exposure to colonised pa-tients. Increasing staff/patientratio led to reduction of coloni-sation.

Expert opinion: transmission probability.Database: length of stay, ward size, admis-sion prevalence, staff/patient ratio, antibi-otic treatment and cessation rates. Litera-ture sources: hand hygiene compliance.

Deterministic

Raboudet al.(2005)

MRSA Heterogeneity of colonisedpatients as isolated patientshave reduced transmissionof MRSA.

Hand hygiene, staff-patientratio, patient isolation, an-tibiotic policy, admissionprevalence

Hand hygiene and early detec-tion and isolation were pre-dicted to be the most effectiveinterventions.

Literature sources: Transmission rate, ef-ficacy of isolation, sensitivity of MRSAscreening. Database: admission preva-lence, length of stay, screening rate. Mea-sured on the ward: hand hygiene, contactrate.

Stochastic

Chapter 3 MRSA Ward size not fixed forestimate of transmission;homogenous mixing notassumed when cohortingconsidered

Hand hygiene, patient co-horting, changes in staff-patient ratio, changes inlength of stay, decolonisa-tion

Increase in staff levels couldincrease transmission if it ledto more contacts and if co-horting did not occur. Wardsize effects transmission rateindependently of other para-meters. Decolonisation usingenteral vancomycin is relativelyineffective at reducing trans-mission of MRSA.

Derived from database: ward size length ofstay admission prevalence. Measured onthe ward at the time of the study: hand hy-giene compliance, probability of transmis-sion patient to HCW. Final transmission pa-rameter estimated using data fitting.

Stochasticand Deter-ministic

Table 2.1: Comparison of studies that applied the Ross-MacDonald model to nosocomial transmission of hospital pathogens. MRO: multipleantibiotic-resistant organism.


Sebille et al. (1997) and Sebille and Valleron (1997) were early studies which

introduced simulation to the area of nosocomial infection control, using de-

terministic and stochastic models, respectively. Unlike models that followed,

these models included patient to patient and staff to staff transmission. The

parameter values in this study were either observed in the ward or estimated.

Some of the estimates differ markedly from those obtained from the current

state of knowledge, particularly the duration of colonisation of healthcare

workers, which was assumed to be 36 days in these studies, whereas it is

estimated to be 1 hour in most other studies. The main findings of these

investigators were that both hand hygiene compliance and antibiotic usage

policy had a limited effect on patient colonisation, but hand hygiene had

a marked effect on staff colonisation. The reason for the poor response

of patient colonisation to hand hygiene (this is the only study of this type

to make the conclusion) is that much of the transmission was patient to

patient and the parameter representing hand hygiene was not included in

the patient to patient transmission term. The model for antibiotic effect was

to assume that there were two antibiotics used to decolonise patients, the

first antibiotic, A1 was 80% effective for sensitive strains, the second was

30% effective for A1-resistant strains and otherwise ineffective. It appears

that all strains were resistant to at least one antibiotic. It is not surprising

then that the model predicted that the use of one antibiotic led to rapid

strain replacement and that no antibiotic usage policy made a substantial

difference to transmission.

Cooper et al. (1999) brought innovation to this research area by showing the

variability of outcomes that results from stochasticity. They predicted that

hand hygiene compliance and transmission probability have the greatest

impact on transmission. Another finding was that the apparent success

of a policy depends on the outcome being measured. For example, new

outbreaks could be more common in a ward that was experiencing good

control of MRSA compared with a ward that had poor infection control

and endemic colonisation. In this study, the transmission parameters were

not derived using a data source, coming from literature or expert opinion

sources.

Austin et al. (1999) developed a data-based model for VRE transmission

and derived a useful, non-dimensional relationship between admission

prevalence, ward prevalence and the reproductive ratio, R0. Estimates

of R0 were thus independent of some of the estimated parameters such

as transmission probability and duration of colonisation. To derive this


relationship, the investigators made the assumption that the reproduction

ratio of healthcare workers to patients (this equates to G2,1 in equation 2.8)

is negligible compared with the basic reproduction ratio. This assumption

would not be valid for wards in which hand hygiene compliance is low or

transmission rates are high. Austin et al. (1999) were the first to consider the

effects of patient cohorting, finding that both cohorting and hand hygiene

have a marked impact on R0 and on ward prevalence.

Grundmann et al. (2002) used the same model structure as Austin et al. (1999)

and incorporated it into an MRSA cohort study. The study included a num-

ber of covariates as putative risk factors for colonisation. The main findings

were that, of the risk factors studied, a staff deficit was the only one signif-

icantly associated with increased MRSA transmission. An improvement in

hand hygiene of 12% would be needed to compensate for the staff deficit, as

predicted by this model.

D’Agata et al. (2005), modelling VRE, introduced new patient compart-

ments, dividing the patient groups into those exposed and not exposed to

antibiotics. This model was not fitted to data. Compartments were treated

differently both with respect to contact rates and probability of colonisation

per contact, making it difficult to differentiate these effects. Within each

compartment, the assumptions of homogenous mixing held. The main

findings were that reducing antibiotic exposure to uncolonised patients was

more effective than reducing antibiotic exposure to colonised patients. This

follows from the author’s assumption that uncolonised patients not taking

antibiotics could never become colonised. Increasing antibiotic exposure to

colonised patients therefore had no impact on transmission from colonised

to uncolonised patients unless the uncolonised patients were antibiotic-

exposed (a minority). Increasing antibiotic exposure to uncolonised patients

led to a larger number of patients becoming susceptible. This model of

transmission is not likely to represent reality as antibiotic exposure is not a

pre-requisite for VRE colonisation. Further studies are needed to quantify

the relative vulnerability of antibiotic exposed and unexposed patients.

A new focus in the study by Raboud et al. (2005) was the role of isolation in

containing MRSA transmission. Citing a study by Jernigan et al. (1996), iso-

lation was assumed to reduce transmission by a factor of 16. The use of this

data was somewhat cavalier, as the study was based on only 16 MRSA trans-

missions, and was a neonatal intensive care unit unlikely to be extrapolative

to the adult ward setting. Additionally, determination of the source of MRSA


was subjectively based, performed by two observers on the ward. Not sur-

prisingly, assuming relative risk of isolation of 1/16, early detection and iso-

lation were found to be effective.

Grundmann and Hellriegel (2006) and Bonten et al. (2001) present reviews of

mathematical models of nosocomial infections. The scope of these reviews

differs from the scope of this thesis, with Bonten et al. including studies of the

dynamics of bacterial resistance explored by Lipsitch et al. (2000) and Grund-

mann and Hellriegel (2006) including game theory and economic models.

2.3.2 Stochastic epidemic models based on the Susceptible-

Infectious model with migration applied to transmis-

sion of nosocomial pathogens

This section reviews studies that fit models to data, aiming to differentiate

the sources of acquisition of antibiotic-resistant pathogens in the healthcare

setting. Three previous studies have been published that use models simi-

lar the that described Section 2.2.6, the single population model. Unlike the

Ross-MacDonald based models, these studies did not assume that all coloni-

sation arises through cross-transmission on the ward. Each of the studies

used the data to estimate parameters associated with the different sources of

colonisation, rather than making assumptions or estimates based on expert

opinion or literature sources.

Pelupessy et al. (2002) used a longitudinal time series dataset. They differ-

entiated “spontaneous” colonisation from cross-transmission in their study.

The study assumed that the time of all events was known and the exact num-

ber of colonised patients was known throughout the study. This allowed the

direct application of a Markov chain algorithm to determine the likelihood.

The study used genotype data to validate the method.

Cooper and Lipsitch (2004) examined time series data on monthly inci-

dence of infection. Unstructured and structured hidden Markov models

(HMMs) were compared with models that assumed colonisation was serially

independent. The structured HMM assumed that there were two sources

of patient colonisation, similar to the model shown in Figure 2.2. Cooper

and Lipsitch (2004) omitted spontaneous colonisation, but included a term

for colonisation on admission, σµu. The relationship between the hidden

states (the number of colonised patients) and the observations (the monthly

incidence of infection) was assumed to be Poisson.


Pelupessy et al. (2002) and Cooper and Lipsitch (2004) both had some identi-

fiability problems in their analyses due to collinearity of the parameters rep-

resenting different sources of transmission. This is not surprising as trans-

mission is a monotonically increasing function of both parameters.

Forrester and Pettitt (2005) used time series data on MRSA acquisition to esti-

mate “background” sources of colonisation and cross-transmission sources,

finding that background sources were much higher than cross-transmission.

The authors extended the model further by dividing the colonised group into

those who were isolated and not. The fitted value for the cross transmission

for isolated patients was lower than for non-isolated colonised patients; how-

ever this was not statistically significant (p=0.1). The methodology in this

study took into account interval censoring of data but not incomplete de-

tection (compliance with swabbing and swab sensitivity was assumed to be

100%).

2.3.3 Environmental models of transmission

To date, no study has modelled the impact of an environmental reservoir on

the transmission of nosocomial pathogens. The model developed in Chapter

6 includes a new compartment representing the environmental reservoir.

This model is analogous to the models for schistosomiasis in which there

were reservoir compartments representing alternative mammalian hosts

(Williams et al., 2002). In the schistosomiasis models only the definitive

host infects the reservoir compartment, whereas in the model of Chapter

6, both populations (patients and healthcare workers) contaminate the

environment.

2.3.4 Epidemic models of Severe Acute Respiratory Syn-

drome

Many mathematical models have been published of the SARS epidemics of

2003. The aims are various: to estimate the infectivity of SARS, to describe

the sojourn times through the incubation and other disease stages, to predict

the effect of interventions and to estimate the case fatality rate.


Studies that estimate sojourn times

Estimates of the full probability density of the length of the incubation pe-

riod and other transitional periods were critical information both for infec-

tion control practitioners (for quarantine and contact tracing strategies) and

to develop realistic mathematical models. A challenge to accurate estima-

tion was the left-censored nature of the data. Exposure times to index SARS

cases were frequently available but the time of transmission can never be ob-

served. A common strategy used by SARS investigators to deal with the unob-

served transmission times was to assume a uniform probability of transmis-

sion across the exposure period and fit incubation period models to the data.

Donnelly et al. (2003) used a parametric approach, fitting a Gamma distrib-

ution to the incubation period; Farewell et al. (2005) examined a number of

parametric distributions, finding the log-gamma the best fit, while Meltzer

(2004) used a nonparametric approach. An alternative assumption, imme-

diate transmission upon exposure to a known symptomatic SARS case, was

employed by Lee et al. (2003).

Models that estimate transmission

Table 2.2 summarises some of the major SARS modelling studies published

to date. Not all of these appear in Chapter 7 which was submitted for publi-

cation in December 2004.

38C

hap

ter2.L

iterature

reviewan

do

utlin

eo

fthesis

Study Data source Innovations/ Methodology Outcomes Limitations

Chowell et al.(2003)

Canada, HongKong andSingapore

SEIRHD model which also divides susceptiblesinto two groups.

Basic reproduction ratio=1.1-1.2. Impact ofisolation

Homogenous mixing assumed.

Lipsitch et al.(2003)

Singapore andHong Kong

SEIRHD model. Weibull distributions fitted tosojourn times.

Basic reproduction ratio ≈ 3 for Singa-pore. Probability of epidemic depends onthe reproduction ratio, the heterogeneity ofspread and the number of index SARS cases.Quarantine has a dramatic effect on trans-mission.

Basic reproduction ratio estimated from theexponential growth curve therefore sensi-tive to incubation period. Unreported casesnot considered.

Lloyd-Smithet al. (2003)

Loosely based onHong Kong andSingapore

Stochastic SEIR model using Erlang sojourntimes for latent and symptomatic periods. Arange of interventions was modelled, includingisolation, contact tracing and quarantine.

Healthcare workers are critical targets ininterventions. Hospital wide precautionshave the largest impact on reducing the ba-sic reproduction ratio, followed by targetedprecautions eg isolation of SARS patients.

Assumptions of homogenous mixing(within compartment groups) andhomogeneity with respect to spreading.

Riley et al.(2003)

Hong Kong Gamma distributions fitted to sojourn times.Super spreading events incorporated determin-istically into model. Spatially stratified, using amixing matrix for 18 districts.

Basic reproduction ratio = 2.7 for HongKong, excluding super-spreading episodes.

Uncertainty regarding the number of casessecondary to super-spreading event leadsto wide margins of estimates for reproduc-tion ratio. Unreported cases not consid-ered.

Wallinga andTeunis (2004)

Hong Kong, VietNam, Canada,Singapore

Develops a likelihood-based computation forthe basic reproduction ratio based on inferenceof infection networks.

Estimated reproductive ratios were similaracross different countries, between 2 and 3prior to interventions and around 0.7 afterinterventions.

Assumes no unreported cases and nochange in the generation interval over time.Also poor estimates for small epidemics insimulations.

Hsieh et al.(2004)

Taiwan SIR model with a “suspected SARS” group in-cluded.

Effective reproduction ratio 4.23 Limited data prevented analysis of time de-pendence in infectivity parameters. Latentperiod not included in the model.

Wang andRuan (2004)

Beijing, China SEIRHD model with a “suspected SARS” groupincluded.

Reproduction ratio range 1.1-3.3 Owing to limited data, parameter valueswere fitted only to a (simplified) single com-partment model.

Lloyd-Smithet al. (2005)

Singapore forSARS and severalother virusesincluded

Branching process models Data suggests the distribution of the num-ber of secondary cases of SARS is highlyskewed. Incorporation of this heterogene-ity of spreading leads to very different esti-mates of SARS outbreaks with stochastic ex-tinction more likely and outbreaks less fre-quent.

Small datasets from published literaturetherefore subject to publication bias.

Glass et al.(2006)

Hong Kong,Singapore,Taiwan, Canada

Bayesian Markov chain Monte Carlo approachis used to fit a model of infectious disease trans-mission that takes account of undiagnosedcases

Basic reproduction ratio from differentcountries fell into same range 1.5-3 priorto intervention and 0.36-0.6 after interven-tions

Data broken into discrete generations, ne-glecting the likelihood of overlap. Deple-tion of susceptibles was not factored intothe model, which is reasonable on the scaleof the epidemics modelled.

Table 2.2: Comparison of studies that estimated infectivity of Severe Acute Respiratory Syndrome.


Two studies were reported in the same edition of Science, Lipsitch et al.

(2003) and Riley et al. (2003). Each extended the Susceptible-Exposed-

Infectious-Removed (SEIR) model to allow for isolation of patients as

they reached hospital leading to the Susceptible-Exposed-Infectious-

Hospitalised-Recovered/Death (SEIRHD) model. Both models allowed for

non-exponential sojourn times; Riley et al. (2003), following Donnelly et al.

(2003), used the Gamma distribution, while Lipsitch et al. (2003) used the

Weibull distribution. Each study took a different approach to determining

the basic reproduction ratio.

Riley et al. (2003), working on data from Hong Kong, were faced with two

super-spreading events that occurred at the Prince of Wales hospital and

in the Amoy Gardens complex. This was incorporated into the model in a

deterministic way. Because the incidence of SARS varied markedly among

districts, this group used a stochastic meta-population approach, allowing

different contact rates between people of the same district, adjacent districts

and more remote districts. Rather than incorporating heterogeneity of

spreading into the model in a stochastic way, the authors seeded the model

with infectious patients (to account for the Prince of Wales outbreak) and

restricted the model to one further super-spreading event (to account for

the Amoy Gardens outbreak). The conclusion of the study was that the

basic reproduction ratio was 2.7 excluding super-spreading events and that

transmission was highly regional. They also found that transmission per

symptomatic case per day was four times higher in the community than

in hospital and that the reproduction ratio dropped over the second and

third months of the study. Riley et al. (2003) used estimated parameters to

simulate the epidemic and effect of interventions such as reduced time to

hospital admission, reduced population movement and reduced contact

rate.

Lipsitch et al. (2003) used the early exponential curves of infectious cases in

a number of settings and the generation time to estimate the basic reproduc-

tion ratio. These authors investigated the outcome of heterogeneity number

of secondary cases per primary case. By increasing the variance to mean ratio

using different negative binomial distributions to model the number of sec-

ondary cases, they showed that the higher the variance to mean ratio (het-

erogeneity) of secondary cases, the less likely it would be that an epidemic

occur. Greater basic reproduction ratios and larger numbers of seeding cases

had the opposite effect. The authors went on to predict the effects of quaran-

tining contacts and reducing the effective infectious period through isolation


on the reproduction ratio. An paradoxical finding was that as quarantine be-

came more pervasive, less total case-time was spent in quarantine, reflecting

reduction in the size of the outbreak.

While these two studies were comprehensive accounts of early epidemics

of SARS, further advances have been made. Chowell et al. (2003) incorpo-

rated heterogeneity of susceptibles, which could account for the observed

difference in infection rates in children. Lloyd-Smith et al. (2003) simulated

a number of interventions specifically relevant to the hospital environment.

Hsieh et al. (2004) and Wang and Ruan (2004) incorporated a “suspected

SARS” group into their studies. Those who meet some but not all criteria for

SARS are an important group for logistic and economic models of SARS in

the hospital setting as they consume resources and bed capacity, which are

critical in preparing for pandemics.

Lloyd-Smith et al. (2005) extended the work of Lipsitch et al. (2003) to exam-

ine the effect of heterogeneity of secondary cases of a number of different

contagions including SARS, measles and influenza. The findings are that as

the heterogeneity of spreading increases, the likelihood of an outbreak de-

creases, the likelihood of stochastic fade-out increases and the expected im-

pact of control efforts increases. If an epidemic does occur, however, it is

more likely to progress rapidly.

Wallinga and Teunis (2004) used a different approach to estimate the basic

reproduction ratio. They calculate the likelihood of “who infected whom”

based on the time difference in presenting symptoms, and the distribution

of the serial interval. The likelihood methodology applied in that study did

not require an underlying structural model such as the SEIR or SEIRHD and

therefore does not take on these models’ assumptions and weaknesses. Glass

et al. (2006) considered hidden infectious patients in their study. Using a

Bayesian approach and modelling generations of SARS cases, these authors

allowed for a proportion of undiagnosed cases.

2.3.5 Other important models of transmission of nosocomial

pathogens

Cooper et al. (2004) extended the nosocomial transmission model by exam-

ining the effects of the inclusion of compartments representing a population

outside the healthcare setting. In this model, patients could be colonised or

uncolonised, isolated or in hospital without isolation and in the community.


Healthcare worker compartments were not included in this model. Patients

were followed after discharge and could be re-admitted prior to decoloni-

sation. The study showed that, while the basic reproduction ratio of a single

admission might be less than unity, the overall reproduction ratio, taking into

account readmission, could be greater than unity. The effect of that phenom-

enon was that as the level of MRSA increased in the community, isolation

facilities reached capacity and control of transmission was lost. A notable

finding of this study was that two steady states for the proportion of patients

colonised were possible under conditions of limited isolation capacity; one

in which isolation capacity was exceeded and patients were forced into reg-

ular ward beds and one in which isolation continued to contain colonised

patients and reduce transmission.

Smith et al. (2005) extend this model by performing a utility analysis of in-

fection control strategies of single hospitals within a group of hospitals shar-

ing the same population. The authors assumed that patients who acquired

MRSA at one hospital, and were colonised on discharge would be readmitted

randomly to any of the other hospitals. Not surprisingly, results of this study

predict that, economically, the outcome for a particular hospital is best if it

is surrounded by other hospitals investing in infection control and that the

contribution of a hospital to its own colonisation prevalence diminishes as

the number of hospitals in a region increases. The study also predicts that as

transmissibility of an agent increases, the value of infection control reduces.

Perencevich et al. (2004) used a single population model to investigate the

value of active surveillance of VRE. Patients were divided into several groups

depending on their isolation and colonisation status. The study found that

active surveillance was predicted to reduce the transmission of VRE, while

passive surveillance had minimal impact. A limitation in this study was that

it assumed that patients could have their colonisation status determined

within one day, a result rarely achieved in clinical practice. In reality patient

colonisation status is often determined after the patient has left the ward.

Bootsma et al. (2006) performed a simulation study of MRSA eradication us-

ing isolation and decolonisation. A traditional microbiological culture de-

tection approach was compared with a rapid detection approach. The study

showed that the use of rapid screening could reduce prevalence even if the

screening was less sensitive and specific compared with culture. It was pre-

dicted to take years to reach a new equilibrium prevalence after the strategy

was implemented. This study assumed that all patients were screened on ad-

mission, a protocol that has variable adherence in Australian hospitals. It also


assumed that isolation was 100% effective, which has not been proven, and

unfortunately, evidence to the contrary is beginning to emerge (Cepeda et al.,

2005; Forrester and Pettitt, 2005). Like the studies by Cooper et al. (2004) and

Smith et al. (2005), this study examined the effects of groups of hospitals who

share patients, and the impact on the behaviour of one on the outcomes of

the others.

There are many other relevant examples of mathematical models of hospital

acquired infectious diseases. Compartmental models have been developed

to predict the ecological consequences of antibiotic exposure on the compe-

tition between susceptible and resistant bacteria (see, for example, Lipsitch

et al. (2000)). Huovinen (2005) and Magee (2005) have developed models to

predict the impact of antibiotic cycling.

This thesis has limited its scope to models that predict the transmission of

pathogens from host to host in the healthcare setting. Other areas being ex-

plored by mathematical models include community-based strategies from

infectious diseases such as H5N1 influenza, HIV, tuberculosis and schistoso-

miasis to name a few. Intra-host dynamics of chlamydia (Wilson et al., 2004),

HIV (Davenport et al., 2006) and antibiotic resistant pathogens (Austin and

Anderson, 1999) is another area of active research.

2.4 Review of methodology used in stochastic epi-

demic modelling

The section reviews statistical methods applicable to stochastic epidemic

modelling, highlighting the needs for such methodology and the reasons

for the choice of methods made in this thesis. Some of valid methods not

utilised in this thesis are also reviewed.

There are several challenges and solutions in stochastic epidemic modelling.

1. The data are serially dependent, owing to the change of transmission

pressure with each successive infection event. Methodology used to in-

corporate serial dependence is reviewed in Section 2.4.2. This method

is utilised in Chapters 3 and 7.

2. Events are censored; data being either missing or unobservable (in the

case of transmission data). Data augmentation is used in Chapters 3

and 7 to “stand in” for missing data. A review of methods for manag-

ing censored data is given in Section 2.4.3. The hidden Markov model


structure can be applied to very sparse datasets. The likelihood can be

calculated in the absence of individual event data (exposure times on-

set of symptoms). Chapter 5 applies a hidden Markov model to serial

prevalence data. The applications of HMMs is reviewed in Section 2.4.4.

3. Bayesian inference was used in Chapters 3, 5 and 7. Following Bayesian

methodology has the advantages of providing a full posterior probabil-

ity distribution of model parameters, and complex functions of model

parameters. It also provides a framework for data augmentation and

Monte-Carlo Markov chain (MCMC) integration, allowing otherwise in-

tractable integrals to be numerically evaluated. Bayesian inference is

introduced in Section 2.4.1 and further discussed in Section 2.4.3

4. When MCMC integration is used, one needs to ensure that the parame-

ter space has been fully explored. This can be tested by assessing for

convergence, as described in Section 2.4.5 and applied to Chapters 3

and 5.

5. Model adequacy (ability to fit the data) can be assessed through a num-

ber of methods, reviewed in Section 2.4.6.

6. Model selection and comparison helps improve statistical inference.

The study in Chapter 5 used model selection to infer the source of

VRE acquisition and whether transmission rates changed over time.

Chapter 7 used model selection to infer the shape of the individual

infectivity profile for SARS.

2.4.1 Bayesian inference

There is a large philosophical difference between the frequentist view of

parameter estimation and the Bayesian view. The Bayesian framework treats

model parameters as random whereas frequentists regard parameters as

fixed and express uncertainty in terms of data replicates.

Prior information on the parameters, p(θ) is required to calculate the

Bayesian posterior distribution

p(θ|y) =p(y)p(y|θ)

p(y)(2.16)

∝ p(y)p(y|θ). (2.17)


The denominator in Expression (2.16) is not a function of the parameters.

When making inference about the parameters, it is therefore sufficient to use

the proportion (2.17).

One of the advantages of Bayesian inference is that it derives the full proba-

bility distribution of the parameters, rather than just the standard errors, as

in frequentist approaches. Additionally, Bayesian inference can be used to

determine the posterior distribution of functions of parameters.

E[f(θ)|y] =

∫f(θ)p(θ)p(y|θ) dθ∫

p(θ)p(y|θ) dθ. (2.18)

The function may be a simple point summary such as the mean or median, or

may be more complex, such as the proportion used in Chapter 5 Expression

(5.7). Because the integrals in Expression (2.18) are rarely able to be evaluated

analytically, we may evaluate E[f(θ)|y] by Monte-Carlo integration, drawing

samples θk, k = 1, ..., m from p(θ|y) and approximating

E[f(θ)|Y ] ≈ 1

m

m∑

k=1

f(θk), (2.19)

(Gilks et al. (1996, Chapter 1)).

Draws from the posterior probability distribution can be realised with

the Gibbs sampler, used in Chapters 3, 5 and 7; and the Metropolis and

Metropolis-Hastings algorithms, used in Chapters 5 and 7. The Metropolis-

Hastings algorithm entails initialising θ, proposing successive new values of

the model parameters, θ′, determining the likelihood, p(y|θ′), and accepting

the new value θk+1 = θ′ according to the acceptance probability

pacc = min

(1,

p(θ′)p(y|θ′)q(θ′ → θk)

p(θk)p(y|θk)q(θk → θ′)

), (2.20)

where q(.) is the proposal probability. If the proposed value is rejected, θk+1 =

θk otherwise θk+1 = θ′. Successive values of θk+1 form a Markov chain with

a transition kernel p(θk+1|θk) that guaranties that the target density, p(θ|y), is

the stationary distribution of the Markov chain, see Gilks et al. (1996, Chapter

1) for the proof.

When p(θ)p(y|θ) is easily determined, we can propose new θ′ with a proposal

distribution

q(θk → θ′) ∝ p(θ)p(y|θ) (2.21)


which leads to pacc = 1 so that the proposal is always accepted. This is the

Gibbs step. We can also make the proposal distribution symmetrical (q(θk →θ′) = q(θ′ → θk)) leading to the Metropolis algorithm.

Updating all of the model parameters at the same time can lead to very low

acceptance probabilities and is unnecessary. Rather, each component of θ

can be updated individually, using Metropolis Hastings or Gibbs steps based

on the full conditional distribution for each component.

Some potential problems which may be encountered using Bayesian infer-

ence are inefficient mixing of the Markov chains of the model parameters and

sensitivity of model outcomes to choice of prior probability distributions.

2.4.2 Methods used to manage serial dependence in infec-

tion data

The piecewise constant hazard model incorporates the phenomenon of

serial dependence in time series data based on the changing number of

infectious and susceptible individuals as each transmission event occurs.

The hazard of transmission for each individual is updated as each newly

infectious case arises. The method is flexible and has weak parametric

assumptions, (Lindsey and Ryan, 1998).

Taking the approach of Aslanidou et al. (1998), the time interval over which

a person is at risk of acquiring an infectious disease can be broken into sub-

intervals ij = (τj−1, τj] for j = 1, ..., J , assuming the event occurs at the end

of the sub-interval and there is, therefore, a constant hazard within each sub-

interval, λ(t) = λj for t ∈ ij. Hence the survival function from the first in-

terval to the beginning of interval J is given by

S(t) = e−PJ−1

j=1 λj(τj−τj−1) (2.22)

and the likelihood contribution of a “failure” occurring in interval J is given

by

q(t) = λJe−PJ

j=1 λj(τj−τj−1). (2.23)

This structure has the advantage of being able to incorporate covariates;

λ(t|z) = λjeβZ , where β is the vector of coefficients and Z is the vector of

covariates for each individual.


Greater than one “failure” event can easily be accommodated. Let λ1J de-

note event one and λ2J denote event 2. The survival function from the first

interval to the beginning of interval J is now given by

S(t) = e−PJ−1

j=1 (λ1J+λ2J )(τj−τj−1) (2.24)

The likelihood contribution of a “failure” occurring due to event one in inter-

val J is given by

q(t) = λ1Je−PJ

j=1(λ1J+λ2J )(τj−τj−1), (2.25)

while the likelihood contribution of a “failure” occurring due to event two in

interval J is given by

q(t) = λ2Je−PJ

j=1(λ1J+λ2J )(τj−τj−1). (2.26)

The structure is readily applied to the stochastic epidemic models described

in this thesis, as long as the number of infectious and susceptibles are per-

fectly observed. In mathematical models we assume the hazard of acquisi-

tion of a contagion is a function of the number of susceptibles, Sj , and the

number of infectives, Ij , at time j, that is

λ1j = f(SjIj). (2.27)

A simple algorithm applicable to SEIR, SIR, SIS and SI models with migra-

tion is that a transmission event occurs with a hazard λ1j = βSjIj , while a

removal event occurs with a hazard λ2j = γIj , where γ is the rate of removal

of infectives. If each removal and infection transmission event is observed,

then likelihood calculation is relatively straightforward,

L(β, γ) =n∏

i=1

(γI(ti))(Zrem(ti))(βS(ti)I(ti))

(Ztrans(ti))e−(γ+βS(ti))I(ti)(ti−ti−1), (2.28)

where Zrem(ti) = 1, Ztrans(ti) = 0 indicates that a removal event occurred

at time, ti, and Zrem(ti) = 0, Ztrans(ti) = 1 indicates that an infection event


occurred at time t. The maximum likelihood estimate for β is given by

β =

∑ni=1 Ztrans(ti)∑n

i=1 S(ti)I(ti)(ti − ti−1). (2.29)

This solution holds for SIR, SEIR and SI models and is applicable to models

that allow for migration. Expression (2.28) implies that during time inter-

vals in which there are no transmission events, each individual is subject to a

transmission hazard which is constant and conditionally independent.

The relationship between λ, Sj and Ij may be more complex, incorporating

personal risk factors, environmental contamination or alternative models

such as the Greenwood and Reed-Frost assumptions (Becker, 1989).

2.4.3 Methods used to manage censored transmission data

If transmission data are unknown but onset and resolution of symptoms is

perfectly observed, one approach to determining the likelihood of the data is

to assume a constant latent period, µ (Becker, 1989). Each transmission time,

tx, is assumed to occur at time tx = ts−µ, where ts is the time of symptom on-

set. The likelihood function can be calculated by determining the number of

susceptibles at all transmission times tx (all patients who have not displayed

symptoms prior to time tx + µ). The likelihood given in Expression 2.28 can

then be applied.

A drawback of this method is that it requires fully observed symptom onset

times. These may either be missing or yet to occur as in the case of infectious

diseases with long latent periods. Additionally, some model predictions are

highly sensitive to assumptions made regarding latent periods (Lloyd, 2001)

and latent periods of fixed duration are not realistic.

In the absence of exactly observed transmission data, Martingale methods

can be used (Becker and Britton, 1999). These methods allow the derivation

of the mean and standard error of β without knowledge of the epidemic

curve. There exists a Martingale-derived closed-form solution for the mean

and standard error of β when only the initial conditions and final state of the

population (that is S(0), R(0), S(∞) and R(∞)) are known, given by Becker

and Britton.

The Martingale method under incomplete observation, as described by

Becker and Britton, is applicable to single epidemic waves, but is not ap-

plicable in populations with endemic infection due to migration of both


infectious and susceptible individuals, as is the case in populations in this

thesis. Additionally, the datasets in this thesis, while incomplete, have more

data that just the initial and final states of the population, so methods

employed make use of this information.

Chain-binomial methods, also described by Becker and Britton, can be used

to describe incompletely observed epidemics. This model uses generations

of infectious cases, defined by the number of predecessors in the chain trac-

ing back to the index case. In order to infer the infectivity using such models,

it is necessary to be able to separate the generations of the epidemic, which

proves difficult unless the incubation period is long compared with the in-

fectious period or unless contacts between susceptibles and infectives are

easily defined, such as with sexually transmitted diseases. Additionally, esti-

mating time dependencies in parameters, such as infectivity, is not possible

if there is overlap in time between generations. The chain-binomial method

also breaks down when unobserved new introductions of infectious cases oc-

curs as occurred for example in the study described in Chapter 5 of this thesis.

Latent Variables

This section summarises work by Gilks et al. (1996, Chapter 15.2 and 15.3),

Gelman et al. (2004, Chapter 12) and Ridall (2005, Chapter 2).

Missing data in stochastic epidemic models can be imputed using latent vari-

ables. A set of latent variables, z, and a set of observations, y, form an aug-

mented dataset. Latent variables could be either truly missing data, an unob-

servable process or an auxiliary variable introduced into the model for con-

venience.

The probability of observations, given the augmented data and the model pa-

rameters, p(y|z,θ), is called the conditional probability of the observations.

The joint probability of the unobserved data and the observations, the com-

plete likelihood, is given by

p(y,z|θ) = p(y|z,θ)p(z|θ). (2.30)

The marginal distribution of y is given by

p(y|θ) =

∫p(y,z|θ)dz. (2.31)


The value of introducing the latent variable z into the model is clear when

the complete likelihood, p(y,z|θ), has a much simpler form than the mar-

ginal likelihood, p(y|θ), as is the case when there are missing data and one

wishes to apply the piecewise constant hazard to determine the likelihood of

a dataset.

Latent variables can be used to extend the range of distributions that can

be modelled (Damien et al., 1999) and simplify model computations. They

have also been shown to enhance convergence (Besag and Green, 1993).

This thesis uses latent variables to represent unobserved MRSA acquisition

times (Chapter 3), unknown number of VRE colonised patients (Chapter 5)

and missing hospitalisation data and unobserved SARS transmission times

(Chapter 7).

The integral required to evaluate the marginal distribution is often difficult or

intractable. The following discussion reviews methods used to tackle latent

variable problems.

The Expectation-Maximisation algorithm

The EM algorithm aims to maximise p(y|θ) in the presence of latent variables,

z. It involves iterating through successive expectation and maximisation

steps until convergence is apparent. The expectation step evaluates the

expected value of the log likelihood of the complete dataset, and the current

value of the model parameters, θm, which is denoted by

Q(θ|θm) = E|θm,y[log p(y,z|θ)|y]. (2.32)

Here the expectation is with respect to p(z|θm,y). In other words

Q(θ|θm) =

∫p(z|θm,y) log p(y,z|θ)dz, (2.33)

in the continuous case or

Q(θ|θm) =∑

z

p(z|θm,y) log p(y,z|θ), (2.34)

in the discrete case. The Maximisation step involves maximising Q(θ|θm) to

find θm+1.


One shortcoming of the EM algorithm is that it iterates to monotonically in-

crease the likelihood function and hence may converge to a local maximum.

Additionally, the E-step and the M-step do not in general have analytical solu-

tions. The integral in Expression (2.35) may be intractable particularly when

the number of dimensions of z or θ is large.

Stochastic EM (Gilks et al., 1996, Chapter 15.3)

The stochastic EM algorithm was introduced by Celeux and Diebolt (1985)

to manage the intractable nature of the EM algorithm in some settings. The

E-step is replaced by a “Simulation Step” which involves imputing missing

variables using plausible values given the observations and current model

parameters (Gilks et al., 1996, Chapter 15.3). Instead of finding the expecta-

tion of the complete log-likelihood with respect to p(z|θm,y), a single draw

from p(z|θm,y) is made. This “pseudo-complete” sample of z can then be

used in the M-step, which is maximised for fixed z, based on the random

draw, hence avoiding integral (2.35). Each successive value of θ becomes a

step in a Markov chain which converges to an approximately stationary dis-

tribution.

Stochastic EM has the advantage of leading to rapid convergence and

tractable computations on occasions when Equation(2.35) is intractable.

The stationary distribution can generate a plausible range for θ, giving a

measure of uncertainty of the estimate. In addition the imputed values of z

give estimates of missing data or auxiliary variables. Neal and Hinton (1999)

demonstrated that simply finding a new estimate of the model parameters,

θm+1, that gives some increase in the value of Q(θ|θm) over its current value

(rather than maximizing Q(θ|θm)) will also ensure successful convergence.

The Bayesian approach to EM

In the fully Bayesian context, the aim is to find the posterior probability dis-

tribution of the parameters p(θ|y), rather than the marginal likelihood p(y|θ).

The posterior probability distribution can be determined by integrating over

the latent variables

p(θ|y) =

∫p(zθ|y)dz. (2.35)


In the Bayesian framework, latent variables are treated in the same way as

the model parameters, each initialised and updated in turn using detailed

balance acceptance equations (Gelman et al., 2004).

This results in the full posterior probability distributions of latent variables,

model parameters and functions of each being available. This was the ap-

proach used in Chapters 3, 5 and 7 in this thesis. The full details of the ap-

proach are outlined in each chapter.

2.4.4 Hidden Markov models

Hidden Markov models (HMMs) are useful in dealing with data in which an

underlying process cannot directly be observed, but there is a clear, defin-

able relationship between the underlying state and the observations. HMMs

consist of hidden states, X, observations, Y , a model describing the transi-

tion between the hidden states, Pr(Xi = j|Xi−1 = i), a model describing the

relationship between the observations and the hidden states, Pr(Yi|Xi), and

the parameters that make up the models. The underlying process must be

expressed as a sequence of states which evolve in a Markov manner, that is

Pr(Xi|X1, X2, ..., Xi−1) = Pr(Xi|Xi−1). Observations must be dependent on

the underlying hidden states only.

HMMs were developed by Baum (1966). One of the first applications of

HMMs was speech and character recognition (Baker, 1975). More recently,

HMMs have been used in medical fields including bioinformatics (Boys

et al., 2000), spatial disease mapping (Green and Richardson, 2002), and

neurophysiology (de Gunst et al., 2001).

A number of algorithms have been developed to explore the properties of

HMMs. The Viterbi algorithm is ideal for exploring hidden states as is re-

quired for speech recognition (Viterbi, 1967). In other applications, the hid-

den states are of less interest and the aim is estimation of model parameters.

The Welsh-Baum algorithm is useful for determining the marginal likelihood

of the data when hidden states are not required (Baum et al., 1970). Details

of the Welsh-Baum algorithm and its application to the model developed in

Chapter 5 are given in the methods section of that chapter. Chapter 5 also

made use of work by Scott (2002), who adapted the forward backward re-

cursion to the Bayesian setting, implementing Gibbs updates of the hidden

states, so that hidden states and parameters could be updated sequentially.


2.4.5 Assessing convergence of Markov Chain Monte Carlo

algorithms, adapted from Gelman et al. (2004, Chapter

11.6)

Using MCMC integration to estimate the posterior distributions of para-

meters or functions of parameters has potential difficulties. If the iterative

process has not been sufficiently long, the parameter space may not have

been adequately explored and the simulations drawn from the MCMC may

not represent the true posterior distribution. Assessment of convergence

of the Markov chains can be performed either visually (taking different

starting points throughout the parameter space and observing the mixing

of the chains) or quantitatively. The latter method of assessing convergence

compares the intra-chain variance with the inter-chain variance. When

these two measured are approximately equal, it is assumed that the target

distribution has been reached.

To evaluate convergence, one approach is to simulate a number of Markov

chains, say m chains of length n (n being the length of the chain after the

burn-in period is excluded). Each simulation drawn from the Markov chains

is labelled φij (i = 1, ..., n; j = 1, ..., m), where φ is the estimand of interest. Let1nφ.j =

∑ni=1 φij and 1

nmφ.. =

∑mj=1

∑ni=1 φij . The convergence of each scalar

estimand of the model can be measured by

R =

√n− 1

n+

A

B, (2.36)

where

A =1

m− 1

m∑j=1

(φ.j − φ..)2 (2.37)

is the variance of the chain means and

B =1

m

m∑j=1

1

n− 1

n∑i=1

(φij − φ.j)2, (2.38)

is the mean of the chain variances. As n → ∞, we expect R → 1; however,

under usual conditions, values of below 1.1 suggest reasonable convergence

(Gelman et al., 2004, Chapter 11.6).

This thesis employs the visual method of assessing convergence in Chapter 7

and the quantitative method in Chapters 3 and 5.


2.4.6 Model checking and improvement

A number of methods for determining model adequacy have been described.

Each works on the basis that if a model fits the data well, replicated data gen-

erated under the model should look similar to observed data. Below is a brief

review of methods applicable to Bayesian model testing.

Cross-validation

In this method, subsamples of the data are used to estimate the model pa-

rameters, the remainder of the data is used for model validation. Different

approaches include; randomly omitting data, and using that data for train-

ing, dividing the data into equal subsamples, each of which are used in turn

as the validation data, and leaving out a single observation at a time and us-

ing the remainder of the data as training data.

Posterior predictive assessment

This method involves simulating data using model parameters and compar-

ing observed and simulated data sets (Gelman et al., 2000). Test quantities

can be either based on data alone, T (y) or on data and model parameters,

T (y,θ). The discrepancy between the actual data and the simulated datasets

can be measured in a number of ways such as a scatterplot of T (yobs, θ) ver-

sus T (yrep,θ) or a histogram of the differences. If a scatterplot is used, one

would expect an equal number of values to fall above and below the line of

unit slope, if a model is adequate. The proportion of values that lie above the

slope is the Bayesian p-value. The choice of test quantity should reflect the

scientific purpose of the model inference (Gelman et al., 2004). A number

of discrepancy measures have been used including standardised residuals,

Pearson chi-squared discrepancy (Gelman et al., 2000) and deviance. Poste-

rior predictive assessment can be used to extend the model inference, for ex-

ample, by correlation of model residuals with putative explanatory variables.

Posteriors from the simulation

This method, described by Dey and Vlachos (1995), simulates data using pa-

rameter estimates and determines the posterior probability distribution for

the parameters based on the simulated dataset. The posterior distributions

as estimated by the actual and simulated datasets can then be compared.

This method was used to test model adequacy in Chapters 3 and 5.


External validation

Ideally, models are also validated using independently collected information.

In Chapters 5 and 7, model results were compared with biological data.

2.4.7 Model selection and comparison

The objective of model selection is to optimise the quality of model inference.

The principle of parsimony means that the simplest possible model should

be chosen. Excessive numbers of parameters in a model, leads to poor preci-

sion, obscuring true effects or identifying effects that are spurious while too

few variables lead to model bias (Burnham and Anderson, 2004).

The model deviance is a measure of model fit, given by

D(y, θ) = −2 log(p(y|θ)). (2.39)

Using the deviance as the sole model selection criterion will favour the se-

lection of the highest dimension model when models are nested. Therefore

model comparison methods must include a penalty term for the number of

model parameters. All methods of model comparison discussed in this sec-

tion aim to derive the deviance of a model as well as a measure of model

complexity.

Different approaches to model selection and their relative merits is an area

of intense research. The following is a very brief discussion on three model

selection criteria, the Akaike Information Criterion (AIC) the Bayesian Infor-

mation Criterion (BIC), and the Deviance Information Criterion (DIC) and

their uses. Both DIC and AIC were used in this thesis in Chapters 5 and 7

respectively.

The Akaike Information Criterion and the Bayesian Information Criterion

The Akaike Information Criterion (Akaike, 1974) is used as a measure of the

predictive power of a model. It is given by

AIC = D(y, θ) + 2p, (2.40)

where p is the number of parameters in the model. It is used by frequentists

and gives similar results to the DIC when prior information is negligible. It


is not useful when the number of model parameters is not clearly defined

or when prior distributions, collinearity or complex hierarchical models re-

duced the number of effective model parameters (Spiegelhalter et al., 2002).

The AIC was used in Chapter 7 of this thesis to select among different mod-

els for the individual infectivity profiles. The AIC was an appropriate model

comparison tool in this context because there was little information on the

prior distributions of the model parameters and the models were not com-

plex, therefore the number of parameters was easily determined.

Supposing there are m models to choose from; in the Bayesian context, model

Mi has a posterior probability given by

p(Mi|y) =p(y|Mi)p(Mi)∑m

j=1 p(y|Mj)p(Mj), (2.41)

Bayes factor is the ratio of the marginal likelihoods of two models given by

Bij =p(y|Mi)

p(y|Mj). (2.42)

This model comparison is used with the aim of finding which of two models

is most probable; that is, it assumes that there exists one “true” model. Scharz

(1978) showed that asymptotically (for large number of observations, N),

−2log(Bij) = −2 logp(y|θi, i)

p(y|θj, j)− (ni − nj) log N, (2.43)

where ni and nj are the number of parameters in models Mi and Mj respec-

tively.

An approximation to Bayes Factor, the BIC, is defined as

BIC = D(y, θ) + p log N, (2.44)

where p is the number of model parameters and N is the number of observa-

tions in the model.

Deviance Information Criteria

The advantages offered by the DIC are that it allows for the reduction in the

complexity of a model when variables are correlated (and therefore the ef-

fective number of parameters is reduced). It also allows for the calculation


of the effective number of model parameters (Spiegelhalter et al., 2002). For

this reason it is often the best method of comparison of models in which the

number of parameters is not readily identified.

The posterior mean of the deviance is obtained by averaging the deviance

over the posterior distribution of the model parameters and is given by

D(θ) =1

L

L∑

l=1

D(y, θl), (2.45)

where θ1, ..., θL are the components of the Markov chain in stationary distri-

bution representing the posterior distribution of θ.

The deviance at a point estimate of θ is given by

D(θ) = D(y, θ). (2.46)

The value used for the point estimate is often the mean, but the mode or me-

dian could also be used. The effective number of parameters is given by

pD = D(θ)−D(θ). (2.47)

The DIC is defined as

DIC = D(θ) + pD. (2.48)

The increase in likelihood, or reduction in deviance, that occurs with in-

creased model parameters is thus compensated for by the term, pD, which

is the effective number of parameters (Gelman et al., 2004, Chapter 5.7). An

advantage of using the DIC is that calculation is trivial once an appropriate

Markov chain Monte Carlo algorithm is in place (Spiegelhalter et al., 2002).

The DIC was chosen as the method of model comparison for Chapter 5

of the thesis because the effective number of parameters was not readily

calculated.

Other Model Selection techniques

The AIC, BIC and DIC all require setting up and running a finite number

of models and comparing the deviance and number of parameters of each.

2.5 Outline of thesis 57

Some model comparison methods circumvent this by allowing the Markov

chain itself to traverse model space. Such techniques include the metropo-

lised Carlin and Chibb algorithm (Carlin and Chib, 1995; Dellaportas et al.,

1998), the reversible jump MCMC (Green, 1995) and birth death MCMC

(Stephens, 2000).

2.5 Outline of Thesis: Account of research progress

linking the papers

Contribution of Chapter 3 to area of research

The data in Chapter 3 comes from the Princess Alexandra Hospital, an 800

bed tertiary referral public Australian teaching hospital. The dataset was

derived from the APACHE IIITM database (admission and discharge dates),

AUSLABTMdatabase (microbiology results), eICATTMdatabase (record of new

or old MRSA colonisation) and, where necessary, patient notes. Consecutive

patient admissions (from 8th August 2001 to 3rd March 2004) to the 16 bed

Intensive Care Unit. Other inclusion criteria were inclusion in the APACHE

IIITM data base.

The study modified the Ross-MacDonald model to predict the impact of

interventions that are relevant to current local practice, proposed practice

or potential future practice and not yet considered by other studies. These

include the predicted impact of decolonisation using enteral vancomycin,

the effect of ward size and the impact of HCW-patient ratios assuming fixed

numbers of contacts per HCW. The study in Chapter 3 was the first of the

Ross-MacDonald models to include model parameters estimated solely

from ward observations or the time series data itself. No expert opinion or

literature estimates were used in the study.

An important difference between the structure of the model presented

in Chapter 3 and the model that forms the basis of the studies of Austin

et al. (1999), Grundmann et al. (2002) and Raboud et al. (2005) is that the

decontamination of HCWs in the aforementioned studies was assumed to

occur at an arbitrary, estimated rate, with a mean of one hour. In these

studies, decontamination was assumed to be independent of hand hygiene

compliance. This assumption leads to an underestimates of the impact of

hand hygiene on transmission. In the study in Chapter 3, decontamination

occurs at a rate dictated by hand hygiene (as does the model presented by


Cooper et al. (1999)). This assumption has the additional advantage that

it avoids unnecessary guesswork in parameter estimation as hand hygiene

compliance can be measured, as it was during the study in Chapter 3.

Two of the parameters used in this study derived from observational stud-

ies performed on the ward during the study period. One study estimated the

hand hygiene compliance, via a series of covert observation periods (Whitby

and McLaws, 2004), the other estimated the probability of MRSA transmis-

sion per discordant contact (McBryde et al., 2004). Chapter 4 gives an ac-

count of the latter study.

The data used in the study in Chapter 3 were interval-censored, and serially-

dependent. Transmission, an unobserved processes, was incorporated into

the model as a latent variable in a Bayesian context making use of Markov

chain Monte Carlo (MCMC) integration. The transmission parameter was es-

timated within the structure of the model, using a piecewise constant hazard

formula, as described in detail in Chapter 3.

Estimates were made of the basic reproduction ratio and attack rate. Outputs

of the model include predictions of the effect of hand hygiene, HCW/patient

ratios, decolonisation and ward size on MRSA transmission.

An important advance in the model presented in Chapter 3 was the way in

which HCW/patient ratio is examined. The study by D’Agata et al. (2005)

predicted the impact of HCW/patient ratio assuming each patient received

a fixed number of contacts. With patient cohorting and fixed contact rates

per patient, increase in staff was predicted to reduce transmission. Chapter

3 investigated an alternative scenario, that the staff make a fixed number of

contacts per day. This assumption led to completely different predictions as

discussed in Chapter 3. We used the latter assumption because staff in the

hospital under study describe a saturated work environment in which there

is always more work to do, that is, more contacts to make. We therefore as-

sume that more staff would lead to more contacts.

The study described in Chapter 3 did not select among alternative models

and assumed that all MRSA acquisition took place as a result of indirect ward

transmission. These deficiencies were addressed in the Chapter 5.



Chapter 4 is an original study that took part on the hospital ward. The aim

of the study was to estimate an important parameter used in the transmis-

sion model, the probability of transmission of MRSA during a single patient.

The study observed routine contacts between HCWs and patients known to

be MRSA colonised. Using standard “glove juice” methods, the study deter-

mined whether contamination of healthcare workers gloves occurred. It also

measures the compliance of different groups of healthcare workers to infec-

tion control protocol (glove use).

The measurement of this parameter was important firstly because it had

not specifically been measured previously and was in itself of interest for

infection control. Secondly, it enabled more accurate modelling of the MRSA

transmission data as outlined in Chapter 3, including the development of a

model in which all parameters were estimated from the ward or fitted to the

data and no parameters were derived through expert opinion or guesswork.


The dataset in Chapter 5 was serial VRE colonisation prevalence data, col-

lected as part of a period of intensive surveillance across three wards in the

Princess Alexandra Hospital. The aim of the study was to determine the pro-

portion of VRE acquisitions that was due to cross-transmission.

Chapter 5 applied the two compartment, single population model de-

scribed in Section 2.2.6 and allowed for two sources of VRE acquisition,

cross-transmission and a sporadic source, independent of ward cross-

transmission. The added complexity was necessary because VRE colonisa-

tions are known to arise from the patients endogenous flora or from sources

outside the ward such as other wards within the hospital, other healthcare

institutions and the community. This model would also be highly applicable

to organisms that readily acquire de novo resistance such as Pseudomonas

aeruginosa (for quinolones).

Because the exact number of patients who were colonised was not directly

observed in the dataset available, a hidden Markov model structure was used

to estimate transmission characteristics. The data are weekly prevalence

data, reflecting the practice of weekly swabs (rather than continuous preva-

lence data as in the study by Pelupessy et al. (2002)). The model developed

in the study allows that many events (colonisation, discharge, readmission)


may have taken place between observations (weekly prevalence checks). It

also takes into account that prevalence checks do not detect all colonised

patients, accounting for imperfect swab sensitivity and incomplete testing

of patients. Unlike the HMM presented by Cooper and Lipsitch (2004), the

relationship between the hidden state (true number of colonised patients)

and the observations (observed prevalence of colonised patients) has a clear

interpretation. Each colonised patient has a probability of detection leading

to a binomial relationship between number colonised (the hidden state) and

number detected (the observation).

Unlike the studies by Pelupessy et al. (2002) and Cooper and Lipsitch (2004),

the study in Chapter 5 did not encounter difficulties with collinearity of the

parameters. This is likely to be because of the larger dataset used in this study

and because the data consisted of a long period of little colonisation, followed

by a large outbreak of colonisation. Most of the information for estimating

the “sporadic colonisation” parameter would have been acquired during the

former period, while the information used to estimate the cross-transmission

parameter would mostly have been acquired from the latter data.

Chapter 5 used a model comparison technique, the Deviance Information

Criterion (DIC), as described in Section 2.4.7 to select among a number of dif-

ferent putative models. Models considered included one in which the cross-

transmission term was omitted, a model in which the sporadic colonisation

term was omitted and models in which the cross-transmission term was time

dependent. The study in Chapter 5 used external validation through compar-

ison with genotyping data, internal validation using simulation and model

selection in order to optimise and validate the model.

The advantage of this model is that it can be applied to imperfect datasets

of the type often collected for infection control surveillance. Vast amounts

of data are now being collected on nosocomial pathogens such as MRSA and

VRE, but little of it is complete or produced for statistical analysis. The model

described in Chapter 5 did not take into account the possibility of an envi-

ronmental reservoir. This was addressed in Chapter 6.


A new model is proposed in Chapter 6 which includes an environmental

reservoir compartment and explores the impact of this reservoir on predic-

tions regarding infection control interventions. This model used parameters


derived from a study of VRE transmission (D’Agata et al., 2005), but is

potentially applicable to other pathogens with a substantial environmental

reservoir.

The study predicts that the presence of an environmental reservoir would re-

duce the effect of a number of infection control measures and under some

conditions will lead to endemic ward colonisation of nosocomial pathogens

despite control measures which would otherwise be predicted to eliminate

transmission. The first part of Chapter 6 presents the paper verbatim as it

appeared in the Journal of Infectious Diseases; the second part of Chapter 6

gives the full details of the model.


Data collected by the Taiyuan Centre for Disease Prevention and Control in

Shanxi province, China was made available for the work described in Chapter

7. This is a unique database that has not been published elsewhere.

To estimate the incubation period, a parametric approach was used, fitting a

Gamma distribution. The study departs from the approach by Donnelly et al.

(2003) and Meltzer (2004) by including a model for time to transmission dur-

ing exposure of an uninfected person to a known SARS case. The assumption

of a constant hazard of transmission during a contact with a SARS case has a

biological basis (as compared with a uniform probability of transmission that

was implicitly assumed in the other studies).

To estimate infectivity, the SEIR model was adopted with some modifica-

tions:

• the proportion of susceptibles is assumed to remain at unity

• the survival times in the compartments is not negative exponential, in-

stead fitted to data using Gamma distributions

• the population is stratified into 2 subpopulations, those in and those

out of hospital

• the epidemic is divided into two time periods, corresponding to the

waxing and the waning of the epidemic.

The latter two modifications were necessary to answer crucial questions.

How did hospitalisation and interventions impact on infectiousness?


Three different models of infectivity profiles over the course of SARS-CoV in-

fection were considered in this study. The model considering a Gamma shape

for infectivity appeared statistically slightly superior to the model assuming

uniform infectivity, using the Akaike Information Criterion (AIC) for model

comparison (see Section 2.4.7). Of interest is that the estimated peak infec-

tivity occurs on the ninth day following symptom onset. This is consistent

with virological results of Peiris et al. (2003) and Cheng et al. (2004) .

The Gamma distribution was used for sojourn times in the SEIHRD model of

Chapter 7. Other authors use alternative distributions such as Weibull (Lip-

sitch et al., 2003), and lognormal (Farewell et al., 2005). The Gamma distri-

bution was chosen because it is relatively parsimonious, flexible and readily

adapted simulations (by using a series of α compartments, each with expo-

nential sojourn times with mean length 1/beta to represent the Gamma(α, β)

distribution). Hence the model can be used to predict the impact of infection

control interventions.

Chapter 8 describes what has been achieved by this thesis, limitations in the

studies presented and directions for future work.

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CHAPTER 3

A mathematical model of methicillin

resistant Staphylococcus aureus

transmission in an Intensive Care Unit:

Predicting the impact of interventions

Statement of joint authorshipEmma McBryde wrote the manuscript, constructed the dataset, developed

the mathematical model, analysed the data, wrote code for model extensions

including interventions and acted as corresponding author.

Tony Pettitt assisted with the analysis of data and the application of Bayesian

inference and the piecewise hazard model, proof read and critically reviewed

the manuscript.

Sean McElwain initiated the concept for the manuscript, assisted with

the development of the mathematical model and proof read and critically

reviewed the manuscript.

74 Chapter 3. Mathematical model of MRSA

AbstractObjectives: To estimate the transmission rate of MRSA in an intensive care

unit in an 800 bed Australian teaching hospital and predict the impact of in-

fection control interventions.

Methods: A mathematical model was developed which consisted of four

compartments: colonised and uncolonised patients and contaminated and

uncontaminated healthcare workers (HCWs). Patient movements, MRSA

acquisition and daily prevalence data were collected from an Intensive Care

Unit (ICU) over 939 days. Hand hygiene compliance and the probability of

MRSA transmission from patient to HCW per discordant contact were mea-

sured during the study. Attack rate and reproduction ratio were estimated

using Bayesian methods. The impact of a number of interventions on attack

rate was estimated using both stochastic and deterministic versions of the

model.

Results: The mean number of secondary cases arising from the ICU admis-

sion of colonised patients, the ward reproduction ratio, Rw, was estimated to

be 0.50 (95% CI 0.39-0.62 ). The attack rate was one MRSA transmission per

160 (95% CI 130-210) uncolonised-patient days.

Hand hygiene was predicted to be the most effective intervention. Decoloni-

sation was predicted to be relatively ineffective. Increasing HCW numbers

was predicted to increase MRSA transmission, in the absence of patient co-

horting. The predictions of the stochastic model differed from those of the

deterministic model, with lower levels of colonisation predicted by the sto-

chastic model.

Conclusions: The number of secondary cases of MRSA colonisation within

the ICU in this study was below unity. Transmission of MRSA was sustained

through admission of colonised patients. Stochastic model simulations

give more realistic predictions in hospital ward settings than deterministic

models. Increasing staff does not necessarily lead to reduced transmission

of nosocomial pathogens.

3.1 Introduction

Infections caused by antibiotic-resistant bacterial pathogens in the health-

care setting are detrimental to patients and place a large burden on health-

care institutions. Staphylococcus aureus is a common cause of hospital ac-

quired blood stream infection and wound infection. Methicillin-resistant S.

3.1 Introduction 75

aureus (MRSA) leads to a higher mortality, morbidity (Engemann et al., 2003)

and cost (Capitano et al., 2003) compared with methicillin-sensitive S. aureus

(MSSA).

The proportion of isolates of S. aureus that are methicillin-resistant is

increasing in many countries including Australia (Nimmo et al., 2003). It is

likely that the increase in MRSA does not represent replacement of MSSA,

but is an additional burden (Cooper and Lipsitch, 2004).

Methicillin resistance developed in S. aureus soon after this class of antibi-

otics was introduced (Ericksen and Erichsen, 1963). Most strains of Health-

care Associated (HA) MRSA are also resistant to other classes of antibiotics

including aminoglycosides and macrolides. Of even more concern is the re-

cent observation that some MRSA isolates have been found to be resistant to

glycopeptides (Bartley, 2002) and oxalidinones (Meka et al., 2004), the major

alternative therapies for MRSA infection.

Antibiotic-resistant bacteria are believed to spread from patient to patient,

principally via the hands of healthcare workers. Colonisation with MRSA fre-

quently precedes infection. This transmissible, asymptomatic state will not

be detected unless an active surveillance program is in place. Thus, halting

the institutional spread of MRSA requires measures that affect colonised pa-

tients as well as those with overt infection.

Recommendations for the control of MRSA transmission include isolation

(Garner, 1996) active surveillance cultures (Muto et al., 2003) and hand hy-

giene. While these guidelines are based on the best available evidence, few of

the studies of hospital acquired infectious diseases use sound methodology

(Cooper et al., 2003). The increase in the proportion of S.aureus isolates

that are methicillin resistant in the face of infection control measures led

to pessimism about their efficacy (Teare and Barrett, 1997). A recent study

found that moving patients into single rooms or cohorted bays did not

reduce MRSA acquisition (Cepeda et al., 2005), however this study screened

for MRSA only weekly which may have led to long delays before colonised

patients were removed from the general ward, diluting any benefit of

isolation.

Mathematical models provide a means of predicting the likely impact of an

intervention or the interaction of multiple interventions, capturing nonlin-

ear transmission dynamics. Stochastic models have the additional advan-

tage of predicting the expected variation in outcomes, which may be marked

in small populations such as hospital wards. Statistical methods based on


structured models provide a means of estimating transmission parameters

from data.

In modelling community epidemics and emerging infectious diseases, the

emphasis of model-informed infection control measures has been to achieve

an effective reproduction ratio (the number of cases that occur due to the in-

troduction of one infectious case, assuming a fully susceptible population)

below unity. In the case of hospital associated pathogens such as MRSA, the

mean number of secondary cases that arise within a ward during a single hos-

pital admission (which we call the ward reproduction ratio, Rw) may be below

unity, but colonised patients may go on to transmit MRSA in other wards and

during subsequent hospital admissions leading to an overall reproduction ra-

tio above unity (see Cooper et al. 2004 for full explanation).

In this study, we find a low ward reproduction ratio, Rw = 0.50. Frequent re-

introductions of MRSA maintain the endemic prevalence. We therefore use

attack rate, defined as the number of MRSA transmissions per uncolonised

patient day, as our outcome measure when predicting the impact of inter-

ventions.

This study differentiates imported cases of MRSA from those that occur dur-

ing ward stay. All new cases are assumed to arise from other colonised pa-

tients via the hands of healthcare workers (cross-transmission). We utilised

a mathematical model to quantify MRSA cross-transmission in an Australian

Intensive Care Unit. We collected data on admission, discharge and colonisa-

tion events as well as other critical model parameters, hand hygiene compli-

ance and transmission per contact, to estimate the MRSA attack rate and the

ward reproduction ratio. We overcame the challenge of unobserved events

by using a Bayesian framework and considering the MRSA acquisition date

as a latent variable. Stochastic and deterministic realisations of the model

gave predictions of the likely impact of interventions including changes in

health-care worker/patient ratio, patient cohorting, hand hygiene, length of

stay, admission prevalence, decolonisation and ward size on the attack rate.

This study extends previous models because all parameters used to estimate

transmission were derived through ward observation directly or fitted to

acquisition data. Ward observations running in parallel to the data collection

gave us realistic values for hand hygiene compliance and probability of

MRSA transmission from a colonised patient to healthcare worker. For the

simulation component of the study, we incorporate ward size as a parameter,

not previously considered, and predict the impact of increases in staff levels

3.2 Model 77

if this leads to increased contact rates. The study later considers the effect

of decolonisation based on parameters derived from an experimental study

(de la Cal et al., 2004).

3.2 Model

Our ward transmission model was a modification of the Susceptible-

Infectious (SI) model with migration, described by Bailey (1975). Versions of

this model have been used previously to analyse nosocomial transmission

data (Sebille and Valleron, 1997; Sebille et al., 1997; Cooper et al., 1999;

Austin et al., 1999; Grundmann et al., 2002; Raboud et al., 2005).

Model description and assumptions

Figure 3.1 illustrates the model for transmission of MRSA in an intensive

care unit. It was assumed that transmission will occur with a probability,

php when an MRSA contaminated health-care worker (HCW) contacts an

uncolonised patient and a probability, pph, when an MRSA colonised patient

was contacted by an uncontaminated HCW. Given that patients carry MRSA

for a long duration (the median MRSA patient carriage has been estimated

to be 8.5 months (Scanvic et al., 2001) or 40 months (Sanford et al., 1994))

compared with their length of ICU stay (4 days observed in the current

study) we made the simplifying assumption that the decolonisation rate, γ,

is zero in the absence of interventions. In contrast, HCWs were assumed to

be contaminated only until their next hand hygiene activity (which occurs

at a rate, κ). Patients arrive at the ward at a rate, Ω, and a proportion, σ are

colonised on arrival. Uncolonised and colonised patients are discharged at

rates µX

and µY

, respectively. The contact rate c is the number of contacts

per patient per HCW.

The assumption of transient contamination of HCWs is justified by the es-

tablished efficacy of hand hygiene activities for removing carriage (McBryde

et al., 2004) and the fact that health-care worker carriage of MRSA is usually

short term (Cookson et al., 1989). In this model we assumed that there was no

direct patient to patient or HCW to HCW transmission. It was also assumed

that there was no environmental reservoir contributing to transmission and

that all patients who were colonised on admission were detected. While en-

vironmental sites have been shown to become contaminated by MRSA, it is

uncertain whether this represents a significant source of MRSA transmission


(Boyce et al., 1997). The HCW/patient ratio was assumed to be unity through-

out the study and that the number of HCW did not vary over each 24 hour

period. This was in keeping with ward policy of providing at least one clini-

cal nurse per patient. We also assumed homogenous mixing of patients and

HCWs and time invariance of model parameters.

Figure 3.1 illustrates the mathematical model of MRSA transmission. The pa-

rameters of the model are given in Table 3.1.

Patients

HCWs

Uncolonised Colonised

Xp Yp

Xh Yh

hp p hcp X Y

ph h pcp X Y

hY

(1 )

Y pYX pXpY

Figure 3.1: Four compartment model of nosocomial pathogen transmission.Here Xp is the number of uncolonised patients, Yp the number of colonisedpatients, Xh the number of uncontaminated healthcare workers, Yh the num-ber of contaminated healthcare workers. The parameters and their symbolsare given in Table 3.1.

3.3 Data 79

MRSA cases can arise from ward transmission (at a rate cphpXpYh) or from

admission of newly colonised patients (at a rate σΩ). Healthcare workers ac-

quire MRSA via an interaction with a colonised patient at a rate proportional

to the number of contacts with patients cpphXhYp; they are decontaminated

at a rate dictated by hand hygiene, κYh.

3.3 Data

3.3.1 Patients and Setting

This study included all patients admitted to the Intensive Care Unit (ICU) of a

800 bed tertiary referral teaching hospital (Princess Alexandra Hospital, Bris-

bane, Australia) from 8th August 2001 to 3rd March 2004 (939 days inclusive).

The ICU bed capacity varied during the study from 16 to 22.

3.3.2 Surveillance of colonisation

During the investigation period, all patient admissions were recorded in the

Apache IIIT M database. The mean number of inpatients for the study each

day was 15 (median 16). Ward policy was to swab all patients on admission,

on discharge from the unit and twice weekly for MRSA surveillance. Newly

colonised patients were defined as those negative on admission who had

a positive swab attributed to ICU stay (more than 48 hours following ICU

admission and less than 48 hours following ICU discharge). An MRSA

colonisation database was collected using pathology reports and record of

prior colonisation on admission. For each of the 939 days of the study, the

number of uncolonised patients, colonised patients and new colonisations

were recorded. Following discharge, each patient was categorised as not

known to be colonised, known to be colonised prior to admission or newly

colonised.

3.3.3 Parameter estimates

The admission prevalence of known MRSA colonised patients, σ was 3% in

this study. At the time of data collection, we estimated model parameters

through ward observation. The hand hygiene compliance, h, was estimated

to be 59% (395 hand hygiene episodes out of 668 hand hygiene opportunities


observed in the study population (Whitby and McLaws, 2004)). This has a re-

lationship with the hand hygiene rate as described in Section 3.4.1. The prob-

ability of MRSA transmission during a single contact between a colonised pa-

tient and an uncolonised healthcare worker was estimated to be 13% during

the study period (17 positive hand cultures out of 129 patient visits found by

McBryde et al. (2004)). We used the data on all patient contacts (anyone who

enters a patient bay) rather than strictly clinical contacts, which has a trans-

mission probability of 17%, as measured by McBryde et al. (2004).

3.4 Methods

To quantify cross-transmission of MRSA in our study population, we esti-

mated the attack rate (number of transmissions per uncolonised patient

day) and the ward reproduction ratio, Rw.

We have no direct estimate of contact rate, c, or probability of transmission

from healthcare worker to patient, php. These two parameters are inseparable

in the model, so we estimate the value of their product, the transmission pa-

rameter, φ = cphp. The admission and discharge dates of patients are directly

observed in this study and are thus incorporated deterministically. Colonisa-

tion status on admission is known (assumed to be perfectly observed).

In Section 3.4.1 we derive a form of the model equations that leaves only the

transmission parameter, φ, to be estimated. Section 3.4.2 explains how φ was

inferred from the data. Section 3.4.3 describes how φ can be used to estimate

Rw and attack rate. Section 3.4.4 describes how the model structure can be

used to predict the impact of interventions.

3.4.1 Formula for daily hazard of MRSA cross-transmission

The daily hazard of MRSA cross-transmission, λ, is given by

λ = φXpYh. (3.1)

We do not have direct observations of Yh, however we derived a formula for

Yh based on observable model parameters. Firstly, we assumed dYh

dt= 0 . We

base the assumption on the fact that decontamination of healthcare workers

3.4 Methods 81

is known to be rapid (minutes to hours) compared with discharge or spon-

taneous decolonisation of patients (days to years, Boyce, 2005). The quasi-

equilibrium value for the number of contaminated HCW, Yh, which we de-

note by Y h is given by

Y h = NhcpphYp

κ + cpphYp

, (3.2)

where Nh = Xh + Yh, is the number of HCWs.

During the study, we measured the pre-contact hand hygiene compliance,

h. This was the proportion of patient contacts that were preceded by either

hand washing or the use of a disinfectant hand spray or gel. A relationship

between hand hygiene compliance and hand hygiene rate, κ, was derived by

Cooper et al. (1999), namely,

h =κ

κ + cNp

, (3.3)

where Np is the total number of patients.

Solving Equation (3.3) for κ and substituting this into Equation (3.2) gives

Y h = NhpphYp

hNp

1−h+ pphYp

. (3.4)

Noting that in this study Nh = Np = Xp + Yp, we have a revised expression for

the rate of MRSA transmission to uncolonised patients, λ, given by

λ =φpphXpYp(Xp + Yp)

h(Xp+Yp)

1−h+ pphYp

. (3.5)

The hand hygiene compliance, h, and the probability of MRSA transmission

from patient to HCW, pph, were measured on the ward at the time of the study,

leaving only one unknown value, the transmission parameter φ, which was

fitted to the data.


3.4.2 Bayesian inference to estimate φ

Estimates of MRSA cross-transmission were complicated by interval censor-

ing of colonisation times. Colonisation events are asymptomatic so obser-

vations of MRSA acquisition consisted of the time of first detection, via rou-

tine swabs or clinical isolates. Assuming 100% swab sensitivity, transmission

could have occurred at any point between the last negative swab or ICU ad-

mission (whichever was later) and the first positive swab or discharge from

the ICU (whichever was sooner). We used a Bayesian framework to estimate

the posterior probability density of the transmission parameter, φ, given in

the Appendix.

3.4.3 Estimates of the attack rate and the ward reproduction

ratio.

In this context, the ward reproduction ratio, Rw, is the expected number of

MRSA cross-transmissions resulting from a single colonised patient, assum-

ing all other patients on the ward are susceptible. The model used in this

study was a two population model in which there was no direct transmission

between people of the same population type. The ward reproduction ratio is

therefore the product of the expected number of transmissions from a single

colonised patient to healthcare workers (HCWs), Rph, and the expected

number of transmissions from a single contaminated HCW to patients, Rhp.

Each component of Rw can be calculated by multiplying the daily transmis-

sion probability by the expected duration of colonisation/contamination.

Therefore

Rw =c2phppph(Np − 1)Nh

µYκ

. (3.6)

By solving Equation (3.3) for the hand hygiene rate, κ, and substituting it into

equation 3.6 and using φ = cphp we get

Rw =φpph(1− h)(Np − 1)

µYh

, (3.7)

where Nh and Np(= Nh) are the number of healthcare workers and patients in

the ward, respectively. Therefore, the ward reproduction ratio will vary from

day to day as the number of patients and healthcare workers changes, under

3.4 Methods 83

the principle of pseudo-mass action. The estimated ward reproduction ratio

was taken as the mean over the study period.

The hazard (rate) of transmission in the ward on day t is φYh(t)Xp(t). There-

fore the attack rate (the rate of transmission per uncolonised patient day)

over the study period is given by

AR = φ

n∑t=1

Y (t). (3.8)

3.4.4 Model for the impact of interventions

We used attack rate as the outcome measure to model the effect of a num-

ber of interventions: improving hand hygiene compliance, decolonisation,

HCW/patient ratios with and without patient cohorting, ward size and pa-

tient discharge rate on the attack rate. We examined both deterministic and

stochastic model predictions.

Estimated means of the parameters derived from the data were used as the

default parameters. The ward size in the study was not fixed, however the

ward ran at near maximum capacity much of the time, therefore new admis-

sions were often limited by the rate of patient discharge. This justified the use

of a simplifying assumption of fixed ward size to estimate the impact of inter-

ventions. We used the mean occupancy derived from the data to determine

the number of patients in the ward, np = 15 (here we used a fixed value of

occupancy as a parameter, np, rather than the variable, Np). We also assumed

that Nh = ρnp, where ρ is the health-care/patient ratio. This simplifies the

mathematical equations to

dYp

dt= cphp(np − Yp)Yh − (γ + µ

Y(1− σ))Yp + µ

Xσ(np − Yp),

dYh

dt= cpph(ρnp − Yh)Yp − κYh. (3.9)

Note that we have now allowed decolonisation of patients, γ, to be non-zero.

The equilibrium attack rate is given by

AR = cphpY he , (3.10)

where Y he is the equilibrium value for Yh, obtained when dYp

dt= dYh

dt= 0.


In the stochastic version of the model, the probability during a small time in-

terval, δ, of transiting from one state to another is described by the equations

Pr(Yp(t + δ) = i + 1|Yp(t) = i) = cphp(np − i)Y hδ + µXσ(np − i)δ + o(δ)

Pr(Yp(t + δ) = i− 1|Yp(t) = i) = (γ + µY(1− σ))iδ + o(δ)

Pr(Yp(t + δ) = i|Yp(t) = i) = 1− cphp(np − i)Y hδ − µXσ(np − i)δ

− (γ + µY(1− σ))iδ + o(δ),

(3.11)

where o(δ) is the Landau symbol, denoting lower order terms of δ. It was as-

sumed that dYh

dt= 0. All other probabilities are o(δ).

The default value for the HCW/patient ratio, ρ, was unity. The default value

for the decolonisation rate, γ, was zero. Other default values were admis-

sion prevalence, σ = 0.03, discharge rate of colonised patients, µY

= 1/10.6,

corresponding to a length of stay of 10.6 days, discharge rate of uncolonised

patients, µX

= 1/4, corresponding to a length of stay of 4 days, probabil-

ity of transmission from colonised patient to healthcare worker per contact,

pph = 0.13, hand hygiene compliance, h = 0.59. In the simulations, for each

set of parameters, the ward was assumed to start with no colonised patients,

the burn-in period was 1000 days and the predicted attack rate was derived

from the next 939 simulated days. Stochastic results were based on 1000 sim-

ulations for each set of parameters, and the 2.5-97.5 percentile ranges were

determined.

By leaving all other parameters at their default values and modifying h, µY

,

µX

and σ, we simulated the effects of changes in hand hygiene compliance,

discharge rate of colonised and uncolonised patients and admission preva-

lence respectively. By changing γ from zero to 0.05, we simulated the effect of

decolonisation. The latter decolonisation rate was chosen based on a study

by de la Cal et al. (2004) in which patients were given enteral vancomycin in

an attempt to eradicate MRSA.

Cohorting was simulated by reducing the number of “effective contacts”. We

assumed that cohorting was non-selective. That is that HCWs cared for a co-

hort of patients who could be a mix of colonised and uncolonised patients.

The smaller the group in the cohort, the more likely that a given contact is

a return contact and thus not an “effective contact”. When maximum co-

horting is taking place, we assume that a proportion of contacts equal to the

3.5 Results 85

HCW/patient ratio, ρ, pose no risk (when ρ ≥ 1 all cohorted contacts pose no

risk).

Our model defined c as the number of contacts per patient per HCW. By ex-

amining the effect of increasing staff patient ratio, ρ, we assume that each

HCW has a fixed number of contacts and increasing staff increases contacts.

To extend this simulation to allow for changes in patient numbers but con-

tinuing to assume a fixed number of contacts per HCW, one could modify

the contact rate, c∗ = c np

Np, where np is the default number of patients and Np

is the actual number of patients. We could alternatively simulate a situation

where patients have a fixed number of contacts and increasing staff does not

increase contacts. Such a simulation would require modifying the contact

rate to c∗, where c∗ = c/ρ.

3.5 Results

The study included 3329 patients. Of these, 100 patients were known to be

colonised on admission and 77 met the criteria for new colonisation. Figure

3.2 summarises the data.

0 100 200 300 400 500 600 700 800 9000

5

10

15

20

25

Day of study

Nu

mb

er o

f p

atie

nts

UncolonisedColonisedNew cases

Figure 3.2: Data collected over period of study. The grey bar plot indicates thenumber of uncolonised patients on each day, the black line plot indicates thenumber of colonised patients and the white bar plot the new acquisitions.


3.5.1 Estimate ward transmission: attack rate and the ward

reproduction ratio

The posterior probability distribution of the ward reproduction ratio, Rw, is

shown in Figure 3.3(a). The estimated mean value of the reproduction ratio

was 0.50 (95% CI 0.39-0.62 ). The posterior probability distribution of the at-

tack rate is shown in Figure 3.3(b). The estimated mean was 0.0062 transmis-

sions per uncolonised patient day (95% CI 0.0048-0.0076), or approximately

one new acquisition per 160 uncolonised patient days.

3.5.2 Predicted impact of interventions

Figure 3.4 shows the predicted impact of ward interventions. The model pre-

dicts that the attack rate would increase dramatically should the hand hy-

giene compliance fall below 40%. A hand hygiene compliance of 48% would

increase the ward reproduction ratio to unity.

Figure 3.4(b) shows the effect of changing the discharge rate of colonised pa-

tients, µY

, leading to a reciprocal change in expected duration of stay. The

response curve was sigmoidal in shape. Increasing the mean time on ward

following colonisation to 21 days would lead to the ward reproduction ratio

exceeding unity. Increasing length of stay of all patients, Figure 3.4(c), also

increases attack rate but less dramatically.

The response of attack rate to doubling the admission prevalence from the

current 3% to 6% is a predicted increase in attack rate from one transmission

per 160 uncolonised patient days to one per 105 uncolonised patient days

(Figure 3.4(d)).

We compared no decolonisation with decolonisation at a rate of 0.05 per day,

using the results of de la Cal et al. (2004). The reduction in attack rate was

modest, from 0.0061 to 0.0034, with overlapping 95% ranges for the stochas-

tic simulations.

We investigated the predicted impact of changing the HCW/patient ratio. In

the upper curve of Figure 3.5, there is no patient cohorting and increasing

HCW numbers increases cross-transmission. In the other curves, we as-

sumed that HCWs can be assigned to a fixed group of patients. Successively

lower curves in Figure 3.5 represent greater proportion of HCWs involved in

cohorting. The lower curve in Figure 3.5 gives the predicted change in attack

3.5 Results 87

rate as HCW/patient ratios change when 100% of HCWs practise cohorting.

Once the HCW/patient ratio reached 1:1 there was no MRSA transmission.

Figure 3.6 shows the effect of reducing ward size on attack rate. The de-

terministic curve is compared with the interquartile range in the boxplots

of 1000 stochastic simulation results. The attack rate in the deterministic

model, unsurprisingly, does not change with ward size. The stochastic model

shows reduced median attack rate, particularly when the ward size reduces

below 10 patients. This reflects an increased proportion of time spent in

stochastic fade-out in small wards.

In several plots, the attack rate predicted by the deterministic model was

higher than that predicted by the stochastic model. Often, the determin-

istic predictions were outside the stochastic 95% variability range of the

corresponding stochastic model.

3.5.3 Model adequacy and sensitivity

A parametric bootstrap analysis was used to determine model adequacy.

This process involves simulating data from the model, using ward observa-

tions (number of uncolonised patients and admission of known colonised

patients) and and the estimated transmission parameter, φ. The method-

ology described in this paper was then applied to the simulated data to

estimate the mean of the marginal posterior distribution of the transmission

parameter. The study found that this gave an unbiased estimate of the

transmission parameter.


Parameter Symbol Unitscontact rate c contacts pt−1HCW−1 day−1

decolonisation rate γ patient−1 day−1

admission prevalence σ -admission rate (pt per day) Ω pt day−1

discharge rate of colonised pt µY

day−1

discharge rate of uncolonised pt µX

day−1

transmission pt→HCW per contact pph colonisation contact−1

transmission HCW→ pt per contact php colonisation HCW−1

hand hygiene rate per HCW κ HCW−1day−1

Table 3.1: Parameters used in the model for MRSA transmission. Key: pt pa-tient, HCW healthcare worker.

3.5 Results 89

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

Ward reproduction ratio(a) Posterior probability density of ward reproduction ratio

0 0.002 0.004 0.006 0.008 0.01 0.0120

50

100

150

200

250

300

350

Attack rate(b) Posterior probability density for the attack rate per uncolonised patient day.

Figure 3.3:


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

Hand hygiene compliance

Att

ack

rate

(a) Effect of hand hygiene complianceon attack rate

0 10 20 30 400

0.005

0.01

0.015

0.02

0.025

0.03

Mean time on ward following colonisation (days)

Att

ack

rate

(b) Effect of length of stay of colonisedpatients on attack rate

0 5 10 15 200

1

2

3

4

5

6

7

8x 10

−3

Length of stay all patients (days)

Att

ack

rate

(c) Effect of length of stay of all pa-tients on attack rate

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Admission prevalence, σ

Att

ack

rate

(d) Effect of admission prevalence onattack rate

Figure 3.4: Effect of changing parameters on attack rate. The bold line repre-sents the prediction of the deterministic model, the feint line represents themean of the stochastic model predictions with error bars giving the 2.5%-97.5% interval.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7x 10

−3

HCW/patient ratio

Atta

ck r

ate

proportion of HCW withan assigned cohort

0.2

0.4

0.6

0.8

1

0

Figure 3.5: Effect of cohorting on attack rate. The impact of HCW patient ratiovaries depending on the proportion of contacts that are able to be cohorted.The deterministic results only are given here.

3.5 Results 91

2 4 8 12 14

0

1

2

3

4

5

x 10−4

Atta

ck r

ate

Ward size

Figure 3.6: Effect of ward size on attack rate. The deterministic value (hor-izontal line) is compared with the median (broken line) and interquartilerange (boxplots) of 1000 stochastic simulation results.


3.6 Discussion

We used a Bayesian framework to quantify MRSA transmission and estimate

the ward reproduction ratio of MRSA in an Intensive Care Unit in a large

teaching hospital. The Bayesian methodology allowed us to incorporate

unseen events, namely, the time of MRSA transmission.

This study used a four compartment modified Susceptible-Infectious (SI)

model with migration. Ward observations of hand hygiene compliance and

transmission probability per contact gave us estimates of all but one model

parameter, which was readily fitted to the data.

We found that, in the Intensive Care Unit under investigation, the ward re-

production ratio was below unity (0.50, 95% CI 0.39-0.62). This compares

with the finding by Grundmann et al. (2002), also studying MRSA in an ICU,

of a ward reproduction ratio of 1.52, when interventions were included. The

hand hygiene compliance in the study by Grundmann et al. was similar to

the current study, however the length of stay of colonised patients was con-

siderably longer, possibly accounting for some of the difference. A study by

Austin et al. (1999) on vancomycin-resistant enterococci transmission found

a ward reproduction ratio of 0.7 when infection control interventions were

in place. The study by Austin et al. found a hand hygiene compliance of

50% and length of stay of colonised patients of around 15 days, both of which

would be expected to lead to higher reproduction ratios than that found in

the current study.

This study found that the predicted transmission rate did not dramatically

change as the ward reproduction ratio went above unity in simulations

involving changing the hand hygiene compliance and duration of stay.

This finding differs from studies of community epidemics in which the

basic reproduction ratio represents a threshold value, below which only

very limited transmission occurs. When there is continued migration of

colonised patients, as occurs in most hospitals with MRSA, the reproduction

ratio does not discriminate between high levels and low levels of transmis-

sion; nor does it quantify risk of colonisation to individual patients. We

therefore recommend that the attack rate be used as a measure of efficacy of

interventions in this setting.

Our model predicted that improving hand hygiene compliance would be the

most effective method of preventing MRSA transmission. Small increments

in compliance resulted in large nonlinear reductions in attack rate. If the

compliance were to fall below 40%, there would be a dramatic rise in attack

3.6 Discussion 93

rate, as predicted by our model. Such a rate is commonly encountered in

hand hygiene studies, prior to interventions (Johnson et al., 2005; Amazian

et al., 2005; Wong and Tam, 2005).

The model predicted that patient decolonisation would not be as effective as

hand hygiene. This is because the time frame to decolonisation (mean of 20

days) was long relative to the mean length of stay. The time to decolonisation

was estimated from a paper by de la Cal et al. (2004) with a decolonisation

rate of 0.05 per day. Our model did not account for the possibility that the

transmissibility may be reduced following a decolonisation intervention even

in patients who remain colonised. If this were to occur, the impact on trans-

mission of decolonisation would be greater than predicted by this model.

A finding in this study which differed from previous studies (for example

D’Agata et al. (2005)) was that increasing health-care worker levels may lead

to an increase in attack rate. An analysis of intensive care workload studies

found that, in the presence of a staff deficit, some studies report that the

productivity of staff reaches a limit leading to inability to complete tasks

involving patient care (Carayon and Gurses, 2005). In this circumstance, the

number of contacts per day is determined by the number of available staff

rather than the number of patients, and the number of contacts will increase

as staff level increases. We aimed to capture this circumstance in our

model. When the contacts were not cohorted, increased staff/patient ratios

resulted in a dramatic rise in attack rate, as predicted by the model. When

cohorting was introduced, the model predicted an initial rise in attack rate as

health-care worker numbers increase, followed by a decline as the increased

health-care worker ratio permitted greater cohorting. Other factors such as

improved compliance with hand hygiene could mitigate against increase

in attack rate and could explain the increased transmission associated with

staff deficit in the study by Grundmann et al. (2002).

When considering patient cohorting, one needs to keep in mind that cohort-

ing measures are usually only able to be carried out by nurses. We found in

our hospital that doctors, who are not involved in cohorting, have a lower

than average hand hygiene compliance rate (McBryde et al., 2004). The im-

pact of this on transmission could be predicted by relaxing the assumption of

uniformity of behaviour within the healthcare worker group.

The stochastic version of our model gave different results from the determin-

istic version. This is accounted for by frequent “fade outs” in MRSA coloni-

sation leading to episodes in which transmission cannot take place in the


stochastic model. These findings are not unexpected in models of small pop-

ulations but reinforce the need to incorporate stochasticity in simulations of

interventions.

We found that the ward reproduction ratio was below unity in the Intensive

Care Unit in our study. Therefore in order for MRSA to persist it needs to

be imported. This leads us to question why MRSA continues to be problem-

atic in the healthcare facility in the study. The answer probably lies in the

fact that patients continue to transmit MRSA after leaving ICU. Colonised pa-

tients may transmit MRSA in other hospital wards, in nursing homes, in the

community and on readmission to hospital (Cooper et al., 2004). Although

the ward reproduction ratio within the ICU was less than unity, the overall

reproduction ratio of MRSA colonisation could be greater than unity.

In future studies, other possible modes of transmission need to be consid-

ered, including an environmental reservoir, transmission from healthcare

workers with chronic MRSA carriage, or unobserved colonisation events,

including patients harboring MRSA on admission without being detected.

Economic modelling of the cost and utility of different interventions would

be a useful adjunct to future studies in this area.

Appendix

3.A Bayesian estimation of the transmission para-

meter

Transmission events were treated as latent variables in the model. The full

conditional probability of the transmission parameter, φ, given a augmented

dataset, D, including daily numbers of uncolonised, colonised patients and

transmission events (latent variables) is given by

p(φ |D) ∝ π(φ)L(D|φ), (3.12)

where π(φ) is the prior probability of φ and L(D|φ) is the likelihood.

The marginal posterior probability density of the transmission parameter, φ,

can be obtained by summation over all possible values of the latent variables

3.A Bayesian estimation of the transmission parameter 95

p(φ |O) ∝ π(φ)∑

A

L(D|φ), (3.13)

where O is the observed data and A is the vector of latent variables (D =

O,A). Here, we take A to be the exact day of MRSA acquisition and the

resulting numbers of colonised and uncolonised patients on each day.

A Markov chain Monte-Carlo (MCMC) approach was used to estimate the

posterior probability density of the transmission parameter, φ. A prior prob-

ability, π(φ), was assigned to φ, the likelihood of the data, L(D|φ), was calcu-

lated. Latent variables and the transmission parameter were updated using a

Gibbs steps and the process was iterated.

Each of the component of the MCMC is explained in turn.

3.A.1 Likelihood of the complete dataset

We used a piecewise constant hazard assumption (Aslanidou et al., 1998) to

calculate the likelihood of the complete dataset. The complete dataset con-

sisted of the daily number of MRSA colonised patients, uncolonised patients

and the number of MRSA cross-transmissions. We assumed that events on

the same day were conditionally independent (given the known number of

colonised patients on the ward at the end of the previous day). We assumed

that a newly colonised patient did not become colonised until the end of the

time interval (one day), and therefore could not cause transmissions until the

following day.

The complete dataset D consists of three vectors, [Xp, Yp,Z], where Xp is the

vector of the number of uncolonised patients on each day of the study, Yp is

the vector of the number of colonised patients on each day of the study, and

Z is the vector of the number of new acquisitions on each day of the study.

New acquisitions of MRSA were assumed to follow a Poisson process with a

rate that was constant over each time increment of one day. This rate was the

daily hazard of transmission, λ(t), calculated using Equation (3.5).

Let

a(t) =pphXp(t)Yp(t)(Xp(t) + Yp(t))

h(Xp(t)+Yp(t))

1−h+ pphYp(t)

, (3.14)


so that λ(t) = φa(t). The likelihood of the complete data over the duration of

the study (n = 939 days) is determined using

L(D |φ) ∝ φPn

t=1 Z(t) e−φPn

t=1 a(t). (3.15)

Here, multiple events were allowed to occur during a given time increment

and the likelihood was calculated at integer times (days) making no specific

allowance for this being an approximation for continuous time. Becker (1989,

Chapter 6.3) suggests that this approximation is sufficiently accurate for ap-

plications where the value of the rate parameter is relatively small.

3.A.2 Gibbs update for the transmission parameter, φ

The Gamma prior distribution is a conjugate prior to the likelihood calcu-

lation given in Equation 3.15. The posterior probability of the transmission

parameter, φ, given the complete dataset, and assuming a Gamma(α, β) prior

for φ, is given by

φ|D ∼ Gamma(α + z, β +n∑

t=1

a(t)), (3.16)

where z is the total number of cross-transmissions over the duration of the

study.

3.A.3 Latent variable imputation

The vector of latent variables, A, consists of the MRSA acquisitions times for

the 77 newly colonised patients, as well as the number of colonised and un-

colonised patients each day that are dependent on those acquisition times.

For each iteration of the Markov chain, the vector was updated by drawing

new values from the full conditional distribution.

For each newly colonised patient, the date of admission to the Intensive Care

Unit or last negative swab (whichever was later) was taken to be the earliest

possible day on which MRSA acquisitions could have occurred (tmin) and the

discharge date or date of first positive swab (whichever was sooner) was taken

to be the latest possible day on which MRSA acquisitions could have occurred


(tmax). The likelihood of acquisitions occurring on each of these days was cal-

culated. An inferred day of acquisition was drawn from the weighted likeli-

hoods.

Let Ti be the day on which patient i acquires MRSA. Then Ti can take the val-

ues (tmin, ..., tmax). Transmission can take place only once and, using discrete

time intervals of one day, Ti has tmax − tmin + 1 possible values. The full con-

ditional posterior distribution for Tik is given by

p(Ti = k |Xs,k, Ys,k, Zs,k, φ) ∝n∏

t=1

[λs,k(t)]Zs,k(t)e−λs,k(t), (3.17)

where Xs,k(t), Ys,k(t), Zs,k(t) and λs,k(t) are the numbers of uncolonised,

colonised, acquisitions and daily hazard function respectively on day t, given

that acquisition for patient i occurs on day k and given the current state, s, of

values of Tj, j 6= i . Only part of the complete likelihood involves Ti, therefore

the likelihood that patient i acquired MRSA on day k, Lik, is given by

Lik ∝tmax∏

t=tmin

λs,k(t)Zs,k(t)e−λs,k(t). (3.18)

The sampling distribution for Ti, p(Ti = k), in the MCMC update for Ti, is

proportional to the likelihood given by the right hand side of Equation(3.18).

3.A.4 Incorporating uncertainty of model parameters

Because two of the three parameters in the study (hand hygiene compliance,

h, and the transmission from colonised patient to healthcare worker, pph)

were estimated by direct observation on the ward, there is uncertainty in

these estimates. The transmission parameter, φ, was estimated from the

data. We need to incorporate the uncertainty of the measured parameters

into the estimate for the transmission parameter, φ.

The posterior probability density for hand hygiene compliance, h, was

derived using a Beta(1, 1) conjugate prior probability density (Gelman et al.,

2004) and the data available from ward observations (the sufficient statistics

were the total number of hand hygiene opportunities observed, m and the

number in which hand hygiene compliance occurred, l). We assumed that

each hand hygiene opportunity was an independent Bernoulli trial. The

posterior probability density for the hand hygiene compliance is given by


h|l,m ∼ Beta(1 + l, 1 + m− l). (3.19)

The posterior probability density for the probability of transmission from

a colonised patient to a healthcare worker was derived using the same

methodology. Whitby and McLaws (2004) observed hand washing compli-

ance in 395 out of a total of 668 opportunities during the period of the current

study. We therefore drew the value for hand washing compliance from the

Beta (396, 274) distribution. McBryde et al. (2004) found transmission of

MRSA to the hands of healthcare workers in 17 out of 129 observed patient

care episodes. We therefore drew the probability of transmission from the

Beta (18, 113) distribution.

3.A.5 Markov chain Monte Carlo algorithm to estimate the

transmission parameter, φ

In order to determine the posterior probability density for the transmission

parameter, φ, we developed an MCMC algorithm, to explore the joint poste-

rior distribution of the augmented data and φ. The process consisted of the

following steps:

1. Determine the prior probability π(φ); an vague Gamma(0.001, 0.001)

distribution was chosen as little was known about transmission of

MRSA from HCW to patient.

2. Draw values of h and pph from their respective beta distributions

3. Update the vector of latent variables. Use a Gibbs steps to update Xp, Yp

and Z by sampling new values of Ti from the distribution given by the

RHS of Expression (3.18). With each iteration, all 77 cross-transmission

events were updated.

4. Update φ using a Gibbs step, sampling from the Gamma distribution

given in Expression (3.16).

5. Perform 10 000 iterations of steps 2-4, using a “burn in” period of 5 000

iterations, collecting the final 5 000 values of the Markov chain for φ and

Rw to contribute to the posterior probability density.

6. Repeat steps 2-5 to construct 10 Markov chains. Intra and inter chain

variance tests showed very good convergence. R = 1.0001 for both Rw


and φ (see Gelman et al. (2004, Chapter 11.6) for discussion on conver-

gence and R values).

Acknowledgements

This work was partially supported by a grant under the Australian Research

Council Linkage Scheme (LP0347112) and NHMRC scholarship number

290541. The authors would like to thank Dr M. Whitby for providing advice

and data. The authors would like to acknowledge the helpful comments of

the anonymous referees.

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CHAPTER 4

An investigation of contact transmission of

methicillin-resistant Staphylococcus

aureus

Statement of joint authorshipEmma McBryde initiated the concept of the study, wrote the manuscript,

collected the data, analysed the data and acted as corresponding author.

Lisa Bradley assisted in data collected, proofread the manuscript.

Mike Whitby critically reviewed the manuscript.

Sean McElwain assisted in data analysis and critically reviewed the manu-

script.

106 Chapter 4. Contact transmission of MRSA

Summary

Hand hygiene is critical in the healthcare setting and it is believed that

MRSA, for example, is transmitted from patient to patient largely via the

hands of health professionals. A study has been carried out at a large busy

teaching hospital to estimate how often the gloves of a healthcare worker

are contaminated with MRSA after contact with a colonised patient. The

effectiveness of handwashing procedures to decontaminate the health

professionals’ hands was also investigated together with how well different

health care professional groups complied with hand washing procedures.

The study show that about 17%(9%-25%) of contacts between a health care

worker and a MRSA-colonised patient leads to transmission of MRSA from

a patient to the gloves of a healthcare worker. Different health professional

groups have quite different rates of compliance with infection control pro-

cedures. Non-contact staff (cleaners, food-services) had the shortest hand

washing times. In this study, glove use compliance rates were above 75%

in all healthcare worker groups except doctors whose compliance was only

27%.

4.1 Introduction

Methicillin-resistant Staphylococcus aureus (MRSA) was first identified in

the 1960s by Ericksen and Erichsen (1963) in Europe and subsequently

spread throughout the world. It is believed that the primary mechanism of

MRSA spread throughout hospitals is via direct contact between patients

and healthcare workers (Pittet et al., 2000) however few studies have been

performed to quantify this transmission.

The transmission rate per contact is an important epidemiologic determi-

nant of MRSA. We have defined contact transmission rate in this study as the

probability that a healthcare worker who is uncolonised prior to contacting a

patient acquires MRSA on their hands or gloves during the contact. An accu-

rate estimate of the transmission rate of MRSA is essential for development

of realistic mathematical models of transmission which can be used to pre-

dict outcomes of interventions and their cost-effectiveness (see, for example

Cooper et al. (1999) and Austin et al. (1999) and references therein).

To date, as far as the authors are aware, there have been few attempts to

measure contact transmission rates for resistant bacteria. Most studies

on bacterial transmission have relied on self-inoculation (Foster, 1960)

4.2 Methods 107

artificial contamination (Rotter, 2001) and random sampling of staff on

wards (Casewell and Phillips, 1977) to determine transmission of organisms

to healthcare workers’ hands. One investigation examined nurses’ hands

after a clinical care episode (Trick et al., 2003) although this study did not

look specifically at MRSA-positive patients.

Other important epidemiological determinants of MRSA transmission are

the effects of different infection control interventions on the contact trans-

mission rate. Again, few studies have addressed this specifically. While it

has been shown that the use of a combination of infection control measures

can reduce the incidence of MRSA (Srinivasan et al., 2002) there have been

no reports to date that have demonstrated the efficacy of glove use alone in

reducing MRSA transmission.

4.2 Methods

4.2.1 Hand sampling

All healthcare workers on duty at Princess Alexandra Hospital (Brisbane, Aus-

tralia) between July 2003 and December 2003 entering the room of a patient

with MRSA were eligible for the study. Healthcare workers were intercepted

following a patient care episode and asked to participate in the study. Once

consent was obtained, a modified glove juice hand culture was performed on

the healthcare worker in accordance with published methodologies (Larson

et al., 2002). The hands of any healthcare worker were sampled no more

than twice throughout the study. A pre-wash sample was obtained from

one hand and a post-wash sample from the other hand. The choice of hand

was determined by a coin toss. Participants inserted one hand into a sterile

polyethylene bag with 50ml of sampling solution, containing 0.3% lecithin,

0.1% polysorbate 80 and 0.1% sodium thiosulphate. A timed one minute

hand massage was performed through the wall of the bag. The HCW was

then asked to remove both gloves and wash and dry hands as normal. A

second identically-conducted one minute hand massage was performed

on the other hand. The liquid from the hand massages was transferred to

separate sterile jars before transport to the microbiology laboratory.

Data were also collected regarding glove use, wearing of rings, type of pa-

tient contact (direct contact with patient skin, body fluids, patient transfer,

intravenous line care, tracheostomy care, wound care) and role of healthcare

worker. Hand washing agent was recorded (Microshield TM and Microshield


2TM were available) and hand wash was timed. Information was recorded on

the patient, including most recent MRSA positive culture, presence of infec-

tion or colonisation and site of colonisation or infection.

Additionally, 100 gloves taken from cubicles around patients with MRSA were

tested for the presence of MRSA, using the same glove juice formula as was

used for the post-contact samples.

4.2.2 Laboratory technique

Dilutions : 500µL of the original sample was used to prepare three 10-fold di-

lutions, in glove juice sampling medium, described elsewhere.10 A sample of

100µL from each dilution (including the original sample) was plated to man-

nitol salt agar with oxacillin and incubated in air at 350C for 48hrs. Colony

counts were performed, selecting plates with 10-100 colonies (except undi-

luted sample where plates with 1-100 colonies were counted). Colony counts

were expressed in colony forming units per mL of glove juice.

MRSA Identification : the specimens were plated onto oxacillin-impregnated

agar at four different dilutions. Colony counts were performed for all growth

and a coagulase test was used to establish the presence of S. aureus. Sus-

pected S. aureus colonies were sub-cultured onto colistin-naladixic acid agar

and incubated in air at 350C for 24hr. Catalase and slide coagulase positive

colonies were tested using the Bio Merieux VitekTM gps-431 gram-positive

susceptibility card, to confirm methicillin resistance. An antibiogram was

performed on each of the MRSA isolates. Non-multiresistant MRSA was de-

fined as a S. aureus isolate which was resistant to penicillin, methicillin, (and

therefore cephalothin) and was resistant to no more than two of the remain-

ing antibiotics tested, namely to gentamicin, tetracycline, erythromycin,

ciprofloxacin, fusidic acid, rifampicin, and clindamycin.

4.2.3 Data analysis

The Pearson correlation coefficient was used to compare hand washing time

with post- and pre-wash counts. A Mann-Whitney test was used to compare

discrete variables such as glove use, presence of rings and type of hand wash-

ing liquid, against continuous data such as colony counts. For all analyses of

discrete variables and discrete outcomes, such as compliance with glove use

versus MRSA acquisition, χ2 tests were used. Analysis of hand washing time

4.3 Results 109

versus type of healthcare worker colony counts utilised analysis of variance

(ANOVA) techniques.

4.3 Results

4.3.1 Detection of MRSA

Samples were taken from 129 healthcare workers. Of these 36 (28%) were not

wearing gloves. Of the 93 staff members who put gloves on prior to the con-

tact, 12 (13%) tested positive for MRSA in the post-contact pre-wash sample.

When healthcare workers who had no contact with the patient, bed or pa-

tient or bed clothes were excluded from the analysis 12 out of 70 tested posi-

tive leading to an estimate of a transmission rate of 17% (CI 9-25%). Of those

who had positive pre-wash samples, 11 of 12 were negative for MRSA in the

post-wash sample. One staff member who initially tested negative for MRSA

in the pre-wash sample, tested positive following removal of gloves and hand

washing. Of the 36 HCW who did not wear gloves, 5 (14%) tested positive in

the pre-wash for MRSA and two of these also had positive post-wash results.

The relative risk of MRSA following hand washing if gloves were not worn

compared with those who wore gloves in this study was 5.2, however this was

not statistically significant (p=0.3).

Sampling Episode

(n=129)

No gloves worn

(n=36)

Gloves worn

(n=93)

MRSA+

Pre-wash

(n=5)

MRSA-

Pre-wash

(n=31)

MRSA-

Pre-wash

(n=81)

MRSA+

Pre-wash

(n=12)

MRSA+

Post-wash

(n=2)

MRSA-

Post-wash

(n=3)

MRSA+

Pre-wash

(n=0)

MRSA-

Pre-wash

(n=31)

MRSA+

Pre-wash

(n=1)

MRSA-

Pre-wash

(n=11)

MRSA+

Pre-wash

(n=1)

MRSA-

Pre-wash

(n=80)

Figure 4.1: Flow diagram of study participants and results of MRSA testing.


4.3.2 Pre-handwash sample: oxacillin-resistant colony

counts

A number of factors were investigated, looking for an association with

pre-wash colony count. The only factor that had a significant association

with colony count was glove use. The group that used gloves (N=93) had a

median colony count of 30 colonies/ml whereas those without gloves (N=36)

had a median colony count of 930 colonies/ml. The difference between the

two groups was highly significant (p=0.0005). All other factors investigated

including; healthcare worker, presence of rings on the pre-wash hand and

type of contact had no statistically significant association with prewash

colony count.

4.3.3 Post-handwash sample: oxacillin-resistant colony

counts

There was a trend towards lower post wash colony counts when Microshield

2TM was used compared with Microshield TM with mean post wash counts

of 5030 colonies/mL and 12010 colonies/mL respectively (p=0.063). Hand

washing time, glove use, type of healthcare worker and presence of rings

on the post-wash hand did not have significant effects on post-wash colony

counts.

4.3.4 Compliance with infection control procedures

The median hand washing time was 26s, but there was considerable variation

as shown in Figure 4.2. Nurses and ward assistants had the highest median

hand washing time, with food assistants and cleaners having the lowest.

Table 4.1 shows the compliance rate for glove use in each of the healthcare

worker groups. Compliance with glove use was 75% or more in all healthcare

worker groups except doctors in whom it was 27% (p=0.003).

The difference in hand washing time amongst different healthcare groups is

statistically significant (p=0.019 using one way ANOVA).

4.4 Discussion 111

Figure 4.2: Boxplot of time taken to wash hands, based on type of healthcareworker.

Type of Healthcare Worker Compliancewith glove use(%)

Nurse 76Doctor 27Physiotherapist 83Ward Assistant 91Food Services 75Cleaner 75

Table 4.1: Compliance with glove use amongst different healthcare workergroups.

4.4 Discussion

This is the first study, as far as the authors are aware, that has attempted to

measure the rate of transmission of MRSA from patient to healthcare worker

during a single routine contact between a healthcare worker and an MRSA

positive patient prior to infection control procedures.

Trick et al. (2003) performed cultures of samples of nurses hands after a

patient contact, aiming to determine the prevalence of contamination of

healthcare workers’ hands with a variety of organisms. The investigation did

not aim to determine the specific transmission probability following a single

contact as gloves were removed prior to pre-wash testing. In that study,

gloves were only worn in 55 of the 282 health workers sampled, making it

difficult to attribute the presence of organisms to the particular contact.

The current study attempts to measure the transmission of MRSA during a


single contact. To achieve this, healthcare workers were intercepted after

a patient contact and cultures were taken from their gloved hands, before

hand washing had occurred. A potential source of contamination in this

study were the gloves worn by healthcare workers. To assess the likelihood

of glove contamination, 100 gloves were taken from the cubicles of MRSA

colonised patients and cultured for MRSA. None of gloves tested positive.

The rate of contamination of healthcare workers gloved hands with MRSA

following a contact with patient, patient’s clothes or patient’s bed was esti-

mated to be 17%. Interestingly, the type of healthcare worker, type of contact

and use of gloves did not have an impact on transmission rate. There was

a trend towards persistence of MRSA after hand washing if gloves were not

worn, but this did not reach statistical significance. A future study with larger

numbers is needed to confirm the association.

The only factor that significantly correlated with number of oxacillin

colonies on pre-wash specimens was glove use. Those wearing gloves had

lower colony counts than those not wearing gloves. Following glove removal

and hand washing, there was no difference between the two groups.

There was a trend toward fewer colonies in the post-wash sample when

Microshield 2TM was used rather than Microshield ∗TM . All other factors

tested were not associated with differences in post-wash colony counts.

Trick et al. (2003) found an association between ring-wearing and presence

of organisms on nurses hands for all organisms tested except methicillin-

resistant coagulase-negative staphylococci. The results of the current study

were consistent with these findings by Trick et al. (2003) as in the current

study only oxacillin resistant flora were measured.

In summary, the results of this study show that about 17%(9%-25%) of

contacts between a healthcare worker and a MRSA-colonised patient leads

to transmission of MRSA from a patient the gloves of a healthcare worker.

In addition, different healthcare workers exhibited different behaviour

with adherence to infection control measures. Non-contact staff (cleaners,

food-services) had the shortest hand washing times. In this study, glove use

compliance rates were above 75% in all healthcare worker groups except

doctors whose compliance was only 27 %.

4.5 Acknowledgement 113

4.5 Acknowledgement


Council Linkage Scheme (LP347112).

Bibliography


Casewell, M. W., Phillips, I., 1977. Hands as a route of transmission for klebsiellaspecies. Brit Med J 2, 1315–1317.



Foster, W., 1960. Experimental staphylococcal infections in man. Lancet ii, 1373–1376.

Larson, E., Gomez-Duarte, C., Lee, L., Della-Latta, P., Kain, D. J., Keswick, B. H., 2002.Microbial flora on the hands of homemakers. Am J Infect Control 31, 72–79.

Pittet, D., Hugonnet, S., Harbarth, S., Mourouga, P., Sauvan, V., Touveneau, S., Per-neger, T. V., 2000. Effectiveness of a hospital-wide programme to improve compli-ance with hand hygiene. Infection Control Programme. Lancet 356 (9238), 1307–12.

Rotter, M., 2001. Arguments for alcoholic hand disinfection. J Hosp Infect 48 (supplA), s4–8.

Srinivasan, A., Song, X., Ross, T., Merz, W., Brower, R., Perl, T. M., 2002. A prospectivestudy to determine whether cover gowns in addition to gloves decrease nosoco-mial transmission of vancomycin-resistant enterococci in an intensive care unit.Infect Control Hosp Epidemiol 23 (8), 424–8.

Trick, W., Vernon, M., Hayes, R., Nathan, C., Rice, T., Peterson, B. J., Segreti, J., Welbel,S., Solomon, S., Weinstein, R. A., 2003. Impact of ring wearing on hand contami-nation and comparison of hygiene agents in a hospital. Clin Infect Dis 36 (1 June),1383–1390.

CHAPTER 5

Characterising outbreaks of

vancomycin-resistant enterococci using

statistical methods

Statement of joint authorshipEmma McBryde wrote the manuscript, developed the mathematical model,

designed the analysis and acted as corresponding author.

Tony Pettitt assisted with the analysis of data, proof read and critically re-

viewed the manuscript.

Sean McElwain assisted with the development of the mathematical model

and proof read and critically reviewed the manuscript.

Ben Cooper proof read and critically reviewed the manuscript. Assisted with

design of the analysis.

118 Chapter 5. Characterising outbreaks of VRE using statisticalmethods

AbstractBackground Antibiotic-resistant nosocomial pathogens, such as van-

comycin resistant enterococci (VRE), can arise in epidemic clusters or

sporadically. Genotyping, to determine the number of colonising strains,

is commonly used to distinguish epidemic from sporadic VRE. We aim to

develop a statistical method to determine the transmission characteristics of

VRE.

Methods and Findings A structured continuous-time hidden Markov model

(HMM) was developed. The hidden states were the number of VRE-colonised

patients (both detected and undetected) at a series of time points. The in-

put for this study was weekly prevalence data; 157 weeks of VRE prevalence

observations from an Australian teaching hospital. We estimated 2 parame-

ters; one to quantify the cross-transmission of VRE (epidemic component)

and one to quantify the level of VRE colonisation from sources other than

cross-transmission (sporadic component). We compared the results to those

obtained by concomitant genotyping and phenotyping.

We estimated that 89% of transmissions were due to ward cross-transmission

while 11% were sporadic. This concordes with the findings that 90% were

identical with respect to glycopeptide resistance genotype and 84% were

identical or nearly identical on Pulsed-Field Gel Electrophoresis (PFGE). We

were also able to estimate the underlying colonisation prevalence (including

those not detected). There was some evidence, based on model selection

criteria, that the cross-transmission parameter changed throughout the

study period. The model that allowed for a change in transmission just

prior to the outbreak and again at the peak of the outbreak was superior

to other models. This model estimated that cross-transmission increased

at week 120 and declined after week 135, coinciding with environmental

decontamination.

Significance We found that HMMs can be applied to serial prevalence data

to estimate the characteristics of acquisition of nosocomial pathogens and

distinguish between epidemic and sporadic acquisition. Our methodology

required only serial prevalence and length of hospital stay data. This model

was able to estimate transmission parameters despite imperfect detection of

the organism. The results of this model were validated against PFGE and gly-

copeptide resistance genotype data and produced very similar results. Ad-

ditionally, HMMs can provide information about unobserved events such as

undetected colonisation.

5.1 Introduction 119

5.1 Introduction

There has been an alarming world-wide increase in the rate of infection

from vancomycin-resistant enterococci (VRE) in the last 15 years (Murray,

2006). Enterococci are part of the normal gastrointestinal flora and VRE

colonisation often is asymptomatic and undetected. However, in patients

with compromised immune systems and breached integument, enterococci

can become pathogenic, causing, for example, urinary tract infection, bac-

teraemia, and endocarditis. Large teaching hospitals and intensive care units

have the highest rate of infection with VRE (Weinstein, 2005). Infection with

enterococci harbouring a vancomycin resistance gene is associated with

higher mortality (Lodise et al., 2002) and many strains of VRE are resistant to

all known antibiotics.

Acquisitions of VRE colonisation can be broadly grouped into those that

come from cross-transmission within the ward, which we call transmitted,

and VRE that comes from other sources, which we call sporadic. Ward

transmission of multi-resistant organisms (MROs) is believed to be pre-

dominantly from patient to patient via the transiently contaminated hands

of health care workers (Boyce, 2001). The sources of sporadic VRE include

patients gastrointestinal tract, prior colonisation with VRE and transmission

from outside the ward. The presence of VRE on admission is often initially

not detected owing to infrequent swabbing, poor sensitivity of swabs or un-

detectable quantities of organism. VRE may exist in sub-detectable numbers

in human gut so that exposure of patients to antibiotics which facilitate VRE

growth (Donskey et al., 2002) may lead to an apparently new case of VRE.

VRE is also known to spread from other hospital wards via patient and staff

movements (Trick et al., 1999).

To select the most appropriate infection control interventions, one needs

to be able to estimate how much of the new acquisition is transmitted and

how much is sporadic. Restricting antibiotic exposure is thought to control

sporadic VRE, by reducing selection pressure in patients endogenous flora,

while hand hygiene, cohorting, patient isolation and limiting admission of

colonised patients are thought to impact on transmitted VRE.

Outbreak investigation often involves time intensive methods to characterise

the mode of VRE acquisition. Genotyping techniques such as pulsed-field gel

electrophoresis (PFGE), distinguish clonal outbreaks, which are presumed to

be due to transmitted VRE, from multiple new strain introductions, which


are presumed to be due to sporadic VRE. There are occasions when this tech-

nique breaks down, when horizontal transfer of the resistance gene, vanA or

vanB, can lead to several different genotypes being detected when in fact a

single transposon is being transmitted (Suppola et al., 1999; Weinstein, 2005;

Bradley, 2002).

Attempts have been made to distinguish between the two processes of

colonisation based on statistical analysis of surveillance data. Pelupessy

et al. (2002) used a Markov model, without hidden states, to estimate

transmission parameters, finding estimates were similar to those using

full event data and genotyping (PFGE). Cooper and Lipsitch (2004) used

structured and unstructured hidden Markov models (HMMs) to describe

infection incidence time series data, and to estimate transmission para-

meters. Collinearity between parameter estimates, failure of convergence

and computational difficulties were identified as potential problems using

HMMs for sparse data such as is typically found in time series infection

control data. Forrester and Pettitt (2005) compared background rates to

cross-transmission rates of methicillin-resistance Staphylococcus aureus,

finding background rates were larger than cross-transmission rates. Esti-

mating transmission coefficients using hospital infection control data has

a number of challenges. There are unobserved processes occurring; the

time of new acquisition of colonisation is not observed. Additionally, when

relying on routine swabs to determine the number of colonised patients, the

sensitivity of swabs is less than 100%.

This study uses an epidemic model structure to characterise transmission

of vancomycin-resistant enterococci during an outbreak at an 800 bed Aus-

tralian teaching hospital. The current paper extends the work by Pelupessy

et al. (2002) by estimating epidemiological parameters in the presence of

suboptimal swab sensitivities. It also allows delays in detection of VRE. We

use a hidden Markov model structure to estimate transmission in the face of

incomplete datasets and unobserved events. This framework distinguishes

between rates of transmitted and sporadic VRE acquisitions. This study also

considers that the transmission rates may change over time. Section 5.2.1

describes the data used to estimate VRE epidemic determinants. Section

5.2.4 describes the model of VRE transmission, while Section 5.2.5 describes

the HMM and the methodology behind it. Section 5.3 gives the results of the

parameter estimates, comparison of model estimates and genotyping data

and model selection.

5.2 Methods 121

5.2 Methods

5.2.1 Description of outbreak and infection control inter-

ventions

VRE was first isolated at the Princess Alexandra Hospital in October 1996 and

a VRE screening programme commenced in January 1997, the beginning

of the data collection period for this study. Data used in this study are VRE

colonisation data from the Intensive Care Unit (ICU), Renal and Infectious

Diseases Units. VRE colonised patients were identified by clinical isolates,

weekly routine screening and contact tracing swabs. Infection control

interventions introduced from the start of the study period were restriction

of vancomycin and third-generation cephalosporin use and isolation of

colonised patients. From week 125 of this study, infection control teams

were aware of an increased prevalence of VRE and further measures were

taken. Dedicated equipment was used in patient rooms and patients were

cohorted. VRE patients requiring haemodialysis used a dialysis facility

within the infection control unit. Medical and nursing staff wore disposable

aprons and latex gloves for patient contacts. An environmental audit was

performed in August 1999, approximately week 135 of the study period, and

an aggressive cleaning programme was instituted (Bartley et al., 2001).

5.2.2 Serial surveillance data used for statistical analysis

Input data for the statistical model in this study were

• weekly prevalence data for VRE colonisation

• mean length of stay of colonised patients; 15 days.

• the total number of beds in the wards; N = 68.

The data were collected from 1st January, 1997 until 31st December, 1999. The

weekly prevalence data are shown in Figure 5.1.

5.2.3 Data used for cluster analysis

Microbiological and clinical data were collected, including admission dates

and discharge dates of VRE colonised patients, as well as date of first positive

isolate. Additionally, we had information on the colonisation status on ad-

mission of three of the patients transferred from other hospitals. Genotype


0 50 100 1500

2

4

6

8

10

12

14

16

18

Week of Study

Week 120 Week 135

Figure 5.1: Prevalence data for VRE over 157 weeks. Arrows show times inwhich changes in transmission rates may have taken place.

data, both Pulsed Field Gel Electrophoresis (PFGE) and glycopeptide resis-

tance genotyping, were compared with the results of the statistical analysis

as part of the study validation. Presumptive VRE colonies were identified us-

ing standard techniques. Speciation (distinguishing E.fecium and E.fecalis)

was initially achieved by carbohydrate fermentation reactions of arabinose,

mannose and raffinose then confirmed by a multiplex PCR assay based on

specific detection of genes encoding D-alanine: D-alanine ligases (Bartley

et al., 2001). VRE phenotype was identified based on vancomycin and teicho-

planin MICs (mean inhibitory concentrations) using the E-test method. This

presumptively distinguishes vanA VRE, resistant to both vancomycin and te-

ichoplanin, from vanB VRE, resistant to vancomycin but sensitive to teicho-

planin. This presumptive phenotype result was confirmed by glycopeptide

resistance genotyping, achieved through a modified multiplex PCR assay, de-

scribed in detail in Bartley et al. (2001).

In the study on this outbreak by Bartley et al. (2001), isolates were also charac-

terised using PFGE. Electrophoretic band patterns were analysed according

to the criteria established by Tenover et al. (1995). Computer comparison us-

ing Gel Compar version 4.1 (Applied Maths Kortrijk, Belgium) was based on

the algorithm of the unweighted pair group method for arithmetic averages

and using the Dice coefficient with 1.5% band tolerance (Bartley et al., 2001).

This information was used to estimate the proportion of isolates that were

from the same strain.

5.2.4 Model of transmission

We base our ward transmission model on the Susceptible-Infected (SI) model

with migration, described by Bailey (1975). Modified versions of this model

5.2 Methods 123

have been used previously to analyse nosocomial transmission data (Pelu-

pessy et al., 2002; Cooper and Lipsitch, 2004; Forrester and Pettitt, 2005).

A schematic of the model is shown in Figure 5.2. The rate of cross-

transmission of VRE colonisation (per colonised per susceptible patient per

day) is denoted by β. It is assumed that the ward is of fixed size, N , hence the

number of uncolonised patients is N −C. Colonised patients are assumed to

remain colonised for their entire hospital stay, therefore transition from the

colonised to uncolonised compartments occurs via discharge of a colonised

patient and replacement with an uncolonised patient, which occurs at a rate

µC. Duration of stay of colonised patients was available from the dataset.

Acquisition of VRE that is transmitted is described by the mass-action

term, βC(N − C). VRE acquisition that is sporadic can arise through ward

admission of a colonised patient or any other process that is not related to

the number of colonised patients, and occurs at a rate, ν(N −C). Each of the

processes that lead to sporadic acquisition (for example prior colonisation

or colonisation from out-of-ward sources, endogenous gastrointestinal

colonisation) can reasonably be assumed to be independent of the number

of colonised patients in the ward.

CN-C

( ) ( )C N C N Cβ υ− + −

Cµ

Figure 5.2: The transmission of bacterial pathogens in the hospital ward.

The probability of a change in the number of colonised patients, C, in a short

time period, h, is given by

Pr[C(t + h) = i + 1|C(t) = i] = βi(N − i)h + ν (N − i)h + o(h),

P r[C(t + h) = i− 1|C(t) = i] = µih + o(h),

P r[C(t + h) = i|C(t) = i] = 1− βi(N − i)h− ν (N − i)h−µih + o(h),

P r[C(t + h) = j (j 6= i− 1, i, i + 1)|C(t) = i] = o(h). (5.1)


The number of colonised patients in the ward at time t, C(t), forms a Markov

process on state space 0, ..., N , where N is the number of patients on the

ward. Reflecting boundaries occur at states i = 0 and i = N , provided ν > 0,

otherwise 0 is an absorbing state, and provided µ > 0, otherwise N is an

absorbing state.

5.2.5 Hidden Markov model

We aim to estimate parameters associated with sporadic colonisation, ν, and

the colonisation caused by ward transmission, β, using the structured HMM

illustrated in Figure 5.3.

C1

Y1

C2

Y2

C4

Y4

C3

Y3

Figure 5.3: Hidden Markov model. Here C represents the number ofcolonised patients in the ward (detected or undetected), Y represents thenumber of patients detected at each time point. The horizontal arrows rep-resent the transition from one state to the next, and the vertical arrows rep-resent the relationship between the hidden state and the corresponding ob-servation.

Our hidden Markov model (HMM) consists of: observations, Y , the num-

ber of patients detected at each time point; underlying hidden states, C, the

number of colonised patients in the ward; a transition model linking each

hidden state with its adjacent states, represented by horizontal lines in Fig-

ure 5.3; an observation model linking the data with the hidden state, repre-

sented by the vertical lines in Figure 5.3. There is one hidden state for each

observation, denoted C1, C2, ..., Cn.

The full conditional probability of any node depends only on neighbouring

nodes to which it is connected directly. The observation component of the

HMM, denoted by Y , consists of 157 data inputs of weekly VRE prevalence

taken over 3 years and the vector of time points, t = t1, ..., tn, corresponding

to each observation time. The vector C consists of the n = 157 hidden states.

The transition probability matrix, giving the relationship between the hid-

den states, is described in Section 5.2.6. The observation model, giving the

relationship between the observed and hidden states, is described in Section

5.2.7.

5.2 Methods 125

The parameters used in the model are given in Table 5.1.

Parameter Symbol value sourceNumber of patients N 68 directly from data setRemoval rate of colonised pt µ 1/15 day−1 directly from data setTransmission rate β 1.0× 10−3 fitted using HMMSporadic acquisition rate ν 2.0× 10−4 fitted using HMMDetection probability d 0.58-0.97 literature review

Table 5.1: Parameters used in the model. Fitted values are discussed in sec-tion 5.3.

Model assumptions

The model makes the following assumptions

1. The ward is of fixed size, N .

2. The model parameters are time invariant (this assumption is relaxed

later in the study).

3. Each observation is conditionally independent given the corresponding

hidden state.

4. The hidden states follow a first order time homogenous Markov

process, that is Pr(C(tk)|C(t1), ..., C(tk−1)) = Pr(C(tk)|C(tk−1)) =

Pr(C(tk − tk−1)|C(0)).

5. Homogenous mixing of patients takes place.

These assumptions are discussed in Section 5.4.

5.2.6 Constructing a transition probability matrix

Following the theory of Cox and Miller (1965), we developed a transition

probability matrix, Γ(tk−tk−1). The ijth element of Γ(tk−tk−1) gives the proba-

bility of having j colonised patients on the ward at time tk, given that there

were i colonised patients on the ward at time tk−1.

To construct the transition probability matrix for an arbitrary time interval,

firstly we developed a discrete time transition probability matrix, A, for a

small time interval, h. Let A be the matrix in which the ijth element is given

by Pr(C(t + h) = j|C(t) = i). A is given using the system of equations 5.1.

Here, i and j are the number of patients colonised in the ward and can take

on values 0, ..., N .


Let p(t) be the (N +1) vector of probabilities of the number colonised at time

t. The generator matrix, G is a square, (N + 1)× (N + 1), matrix that has the

property that

dp(t)

dt= Gp(t). (5.2)

The ijth element of the generator matrix, G, is the instantaneous rate of

change of probability of being in state j, given a beginning in state i. Then G

is given by

G = limh→0

1

h(A− I). (5.3)

Following from Expression 5.2, we have

p(tk+1) = p(tk)e(tk+1−tk)G, (5.4)

in general. Specifically, after a time interval tk+−tk, the probability of being in

state j having begun in state i is the ijth element of the transition probability

matrix, given by

Γ(tk+1−tk)ij= Pr(Ck+1 = j|Ck = i) = (e(tk+1−tk)G) ij. (5.5)

Cox and Miller (1965, Chap 4.5) and MacDonald and Zucchini (1997) give an

expanded explanation. The matrix exponential e(tk−tk−1)G was calculated us-

ing the MatlabTM “expm” function.

5.2.7 Observation Model

The probability, d, of being known to be colonised (and therefore being in-

cluded in the prevalence data) given that a patient is colonised was unknown.

Literature sources regarding the sensitivity of rectal swabs in detecting VRE

were used to develop an expression for the uncertainty in this parameter. Es-

timates of the sensitivity range from 0.58 (D’Agata et al., 2002) to 0.97 (Reisner

et al., 2000) with values in between (Lemmen et al., 2001; Trick et al., 2004).

We allowed for the uncertainty regarding the probability of detection by as-

signing a uniform[0.58, 0.97] prior distribution to d.

The probability relationship between the states and the data is described

by the binomial distribution Yk ∼ Bin(Ck, d), where Yk is the kth observed

5.2 Methods 127

colonisation prevalence and Ck is the actual number of colonised patients,

the hidden state. This assumes that the probability, d, remains constant over

the study period (for each iteration) and the probability of detection of each

colonised patient is independent of the number of other colonised patients.

Alternative observation models with greater dispersion could have been

used. For example, the Poisson or negative binomial distribution could have

been chosen, had we been dealing with incidence rather than prevalence

data. We chose the Binomial distribution because it has a sound probabilis-

tic basis (assuming fixed detection) and, unlike the Poisson, ensures that

the hidden state (number colonised) is always larger than the observation

(number detected), a necessary result when using prevalence data.

5.2.8 Bayesian framework

The parameters for transmitted VRE, β, and sporadic VRE, ν, were estimated

using a Bayesian framework. Let θp = β, ν, d be the vector of model

parameters. Baum et al.’s recursion formula, summarised in Appendix 5.A,

was used to determine the likelihood of the data, L(Y |θp). Uniform U[0,

0.1] prior probability distributions were assigned to β and ν, because little

was known about these parameters other than that negative values or values

higher than 0.1 were completely implausible. The posterior probability

distribution is given by

p(θp|Y ) ∝ π(θp)L(Y |θp), (5.6)

where π(θp) is the prior probability distribution of θp. This was estimated us-

ing a Monte-Carlo Markov chain algorithm, described in Appendix 5.B.

The Bayesian framework can provide estimates (and full posterior probabil-

ity density) of any function of model parameters including functions which

depend upon knowledge of hidden states. Let θh be the vector of n inferred

hidden states C1, ..., Cn and let θ = θp, θh. The proportion of VRE acquisi-

tions due to ward transmission, f(θ), is given by:

f(θ) =

∑nk=1 βCk(N − Ck)∑n

k=1 βCk(N − Ck) + ν(N − Ck). (5.7)

We evaluate the expectation, E[f(θ)|Y ], by drawing samples θk, k = 1, ..., m


from p(θ|Y ) and using the approximation of Gilks et al. (1996, Chapt1)

E[f(θ)|Y ] ≈ 1

m

m∑

k=1

f(θk). (5.8)

The algorithm for this Monte Carlo integration is given in Appendix 5.B.

5.2.9 Comparison of cluster analysis results using genotyp-

ing with statistical analysis

A genotyping study was performed on the VRE isolates by Bartley et al. (2001).

Of the 49 isolates available for analysis, 44 were found to be E.fecium vanA

using glycopeptide resistance genotyping. The estimated number of isolates

having identical or closely related patterns on PFGE using the criteria of Ten-

over et al. (1995) was 41 of 49.

Cluster analysis based on genotypic relatedness

We compared the proportion of “identical isolates” (presumed to be part of

a cluster) with the estimated proportion of transmitted VRE derived from the

HMM and prevalence data. The posterior probability distribution of the pro-

portion of VRE cases that are identical can readily be derived using a Bayesian

framework and conjugate prior distribution (see Gelman et al., 2004). De-

note the parameter of interest, the proportion of VRE acquisitions that are

identical, by p. Assume the form Beta(1, 1) for the prior distribution for the

proportion; this is the same as the uniform[0,1] prior. The probability of the

data is given by given by the Binomial Bin(a; (a + b), p), where a is the num-

ber of identical isolates and b is the number of non-identical isolates, as de-

tected by the laboratory methods. The posterior probability density of p is

Beta(1 + a, 1 + b).

5.3 Results

5.3.1 Parameter estimation

The estimated value for the transmission coefficient, β was 10×10−4 (CI957.9×10−4, 13× 10−4) and the sporadic acquisition rate ν was 2.0× 10−4 (CI950.85×

5.3 Results 129

10−4, 3.8×10−4 ). The coefficient of correlation between β and ν was estimated

to be -0.24.

The basic reproduction ratio, R0, is “the average number of persons directly

infected by an infectious case during its entire infectious period, after enter-

ing a totally susceptible population” (Giesecke, 1994). In this model it can be

shown to be R0 = βNµ

. The basic reproduction ratio is estimated to be 1.07

(CI95 0.78-1.34).

The mean value for the estimated detection rate was 0.75 with a 95% credible

interval of 0.59 to 0.93.

5.3.2 Comparison of statistical model and genotyping data

The proportion of VRE acquisitions due to transmission, was estimated to be

89% (CI95=78-95%), using Bayesian inference applied to the hidden Markov

model structure. This compares with 84% (41/49) of isolates observed to be

identical or nearly identical using PFGE genotyping and 90% (44/49) using

glycopeptide resistance genotyping. The posterior distribution of the esti-

mated proportion of colonisations due to ward transmission compared with

those found to be identical by glycopeptide resistance genotype and PFGE

methods are displayed in Figure 5.4.

5.3.3 Model selection and validation

The values of the Deviance information criterion (DIC) were used to assess

the optimum model to fit the data (Gelman et al., 2004). Results are given in

Table 5.2.

Several models were explored. Setting either β or ν to zero led to much higher

values for the DIC, giving substantial statistical support to a mixed model, in

which VRE colonisation arose both from cross transmission in the ward and

sporadically. The model in which β changed after week 120 was a superior fit

to the model with time-invariant parameters. Allowing for a further change

in β after week 135 provided the best fit of those models investigated. The

effective number of parameters in a latent variable model depends on the

collinearity of the parameters and the influence of the latent variables.

Internal validation of the model was achieved using a parametric bootstrap

analysis. Data were simulated using the time-invariant model (one β one


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Proportion of colonisation due to cross−transmission

Pos

terio

r pr

obab

ility

den

sity

statistical resultPFGE dataglycopeptideresistance data

Figure 5.4: Posterior distribution of proportion of VRE acquisitions that aredue to ward transmission. The histogram gives the posterior distributionfrom the Bayesian analysis of the hidden Markov model, the solid curvegives the posterior distribution based on the observed proportion of identicalstrains using PFGE genotype data and the broken line gives the posterior dis-tribution based on observed proportion of identical strains using glycopep-tide resistance phenotype and genotype data (Bartley et al., 2001).

ν). The posterior distribution ν and β were estimated from simulated data,

demonstrating that the method achieves an unbiased estimate of the trans-

mission parameters.

5.4 Discussion

The aim of this study was to characterise transmission of VRE using sta-

tistical methods and simple serial surveillance data. We included a term

for sporadic colonisation because of evidence that new acquisitions of

VRE can occur through means other than within-ward patient to patient

cross-transmission. Sources of sporadic colonisation have been labelled in

the past as endogenous, spontaneous (Pelupessy et al., 2002) or background

(Forrester and Pettitt, 2005). Our statistical methods were designed to

distinguish between these two sources. Previous attempts have encountered

difficulties especially with identifiability of variables (Cooper and Lipsitch,

2004).

Full patient histories, PFGE and glycopeptide resistance genotype data were

used for validation but were not included in the statistical analysis in this

5.4 Discussion 131

Model Estimate ofβ (95%CI)× 10−4

Estimate ofν(95%CI)×10−4

DIC Pd

One value for ν and threevalues for β with changepoints at the end of week120 and 135

β13.4(0.28− 8.8)β215.3(13.5− 17.1)β310.9(7.1− 13.0)

2.2(0.96-4.0) 251 4.0

One value for ν and twovalues for β with changepoint at the end of week120

β13.4(0.28− 8.7)β211.9(10.2− 13.5)

2.2 (0.96-4.0) 253 2.3

One value for ν and onevalue for β

10(7.9-13) 2.0(0.85-3.8) 261 2.6

One value for ν and twovalues for β with changepoint at the end of week135

β111(7.6− 14.6)β29.6(7.9− 11.4)

2.0(0.88-3.7) 261 2.6

β = 0 and one value for ν 0 9.7(7.7-11.7) 393 1.2ν = 0 and one value for β 8.7(6.9-10.1) 0 531 1.5

Table 5.2: Comparison of different models using the Deviance InformationCriterion. Key Pd: effective number of parameters.

study. Estimates of the proportion of VRE resulting from cross-transmission

based on statistical methods (hidden Markov models) in this study were very

similar to those based on vancomycin resistance genotype data.

The proportion of identical isolates based on PFGE analysis was lower

than both the vancomycin resistance genotype data and the statistical

cluster analysis. This could be due to horizontal transfer of resistance

gene to new strains of enterococci, which has been reported previously

(Suppola et al., 1999; Weinstein, 2005). If horizontal transfer of resistance

genes occurs during an outbreak, cross-transmitted strains have identical

glycopeptide-resistance genotypes but different PFGE patterns, hence PFGE

under-estimates clustering.

Using a structured hidden Markov model, one can estimate the hidden

states behind the data, the number of patients colonised on the ward (both

detected and undetected). We used this to estimate the ward reproduction

ratio, which was 1.07. This value is just above the threshold value of unity,

which portends endemic VRE. We were able to make estimates of trans-

mission in the face of imperfect datasets in which transmission times and

patient histories were unknown and swab sensitivity was considerably less

than 100%.


For simplicity, this study assumed homogenous mixing of staff and patients.

Future studies could extend this model to include ward coupling, however

dividing the data to incorporate ward structure would lead to reduced preci-

sion in parameter estimates and increased model complexity.

The model presented in this study postulated that VRE acquisition arose

from both cross-transmission and sporadic sources. Model comparison

techniques found this model to be a far superior fit to the data compared

to models which relied on either cross-transmission or sporadic sources of

VRE acquisition alone, strongly supporting both modes of acquisition were

taking place.

We investigated changes in transmission over time using a structured epi-

demic model. Model comparison showed that there was evidence support-

ing the conclusion that there was an increase in cross-transmission just prior

to the outbreak. There was also evidence that the cross-transmission rate re-

duced after the epidemic peak at week 135, coinciding with the environmen-

tal cleaning intervention. Future studies using larger surveillance datasets

could extend the methodology presented to consider more models in which

parameters are time dependent. One approach to this would be to use the re-

versible jump Monte Carlo Markov chain method (Green, 1995) or the birth

death Markov process model (Stephens, 2000).

Inaccuracies in PFGE cluster analysis can arise from the horizontal transfer of

resistance genes. Glycopeptide resistance genotype analyses are not subject

to inaccuracies due to gene transfer but cannot distinguish different strains

that might all be of the same resistance genotype. Statistical methods are not

subject to these problems and have the additional advantage that they are

not resource intensive. They also have the potential to be used in real time,

within a control-chart outbreak alert system.

The model presented here can be applied to the surveillance of other

bacterial pathogens in small scale settings of healthcare institutions, such

as methicillin-resistant Staphylococcus aureus (MRSA), extended spectrum

beta-lactamase (ESBL) producing and other multi-resistant Gram-negative

pathogens.

Acknowledgements


Council Linkage Scheme (LP0347112) and NHMRC scholarship number

5.A Likelihood computation 133

290541. The authors would like to thank Dr Mike Whitby for providing data

and Dr Paul Bartley for helpful comments.

Appendix

5.A Likelihood computation

The probability of the full dataset and a particular sequence of hidden states,

C1, C2, ..., Cn is given by

Pr(Y1, ..., Yn, C1, ..., Cn| β, ν) = Pr(C1)Pr(Y1|C1)n∏

k=2

ΓCk−1 CkPr(Yk|Ck), (5.9)

with ΓCk−1 Ckas defined in Section 5.2.6.

The likelihood calculation of this single permutation of hidden states

requires 2n computations even after the matrix exponential has been

evaluated. The full likelihood of the data over all the states is

Pr(Y1, ..., Yn| β, ν) =N+1∑C1=1

, ...,

N+1∑Cn=1

Pr(Y1, ..., Yn, C1, ..., Cn| β, ν) (5.10)

which requires 2n(N + 1)n computations for one likelihood evaluation

(Le Strat and Carrat, 1999). This intractable calculation (with n = 157 and

N = 68) can be simplified using Baum’s recursion technique (Baum et al.,

1970), as shown below.

The forward recursion involves simplifying the likelihood computations by

considering a partial observation sequence and a single state sequence. Let

φk(i) be the probability of the partial observation sequence (Y1, Y2, ..., Yk) pro-

duced by all possible state sequences that end in state i. The probability is

given by

φk(i) = L(Y1, ..., Yk, Ck = i| ν, β), k ≤ n. (5.11)

Let δ be the (size N + 1) vector of probabilities of the first state, (δi = Pr(C1 =

i)). In the forward recursion method of likelihood computation, the value of

δ needs to be determined in the absence of data. The stationary distribution


of the transition matrix can be used for this (MacDonald and Zucchini, 1997).

The probability of the first state and first observation, Y1, is given by

φ1(i) = δi Pr(Y1|C1 = i). (5.12)

The forward recursion formula is then applied. We multiply every state prob-

ability, φk−1(i), by the transition probability Γij and by the probability of the

kth data point given the hidden state j. This results in a vector of probabilities

which is then summed to determine φk(j). Thus the probability of subse-

quent states is given by

φk(j) =

[N∑

i=0

φk−1(i)Γij

]Pr(Yk|Ck = j). (5.13)

At each step in the forward recursion, the procedure can be terminated and

the probability of the partial observation sequence determined by

Pr(Y1, ..., Yk| ν, β) = ΣNi=0φk(i). (5.14)

The likelihood of the data can then be determined by

Pr(Y1, ..., Yn|β, ν) = ΣNi=0φn(i). (5.15)

See Petrushin (2000) for a detailed discussion of the forward and backward

recursion formulae.

5.B Monte Carlo Markov chain algorithm

The algorithm for this Monte-Carlo integration used to estimate the propor-

tion of VRE acquisitions due to ward cross-transmission, f(θ), is given below.

The MCMC algorithm has the following steps:

1. Assume the prior probability for β and ν, to be (U[0, .1]). These priors

were used as little prior information was known except that negative

values and values greater than 0.1 are completely implausible.

2. Initialise β and ν and d′.

5.B Monte Carlo Markov chain algorithm 135

3. Assign the prior probability of the hidden states. A discrete uniform dis-

tribution on (0, ..., N) was used.

4. Initialise each hidden state using its corresponding observation and the

(binomial) observation model Yk ∼ Bin(Ck, d).

5. Determine the probability of the data and sequence of hidden states

using Equation 5.9.

6. Propose a new β′ using a simple random walk, the step size ∼N(0, 10−4).

7. Accept β′ using a Metropolis-Hastings step with the acceptance proba-

bility

a = min1, π(β′)Pr(Y , C|β′)q(β′ → β)

π(β)Pr(Y , C|β)q(β → β′), (5.16)

where q(β → β′) is the proposal probability for β′ from β which is the

normal density for β′ with mean β and variance 10−4.

8. Repeat for ν ′ and d′.

9. Update each hidden state using a Gibbs update, drawing from the dis-

tributions given by the conditional probability of the states, determined

by neighbouring states and observations, as described below.

10. Determine f(θ) for the particular sequence of hidden states and para-

meters β and ν using Expression (5.7).

11. Iterate by returning to step 4.

12. Burn in using 50 000 iterations. Use the following 50 000 updates to es-

timate the posterior probability distribution (using the ergodic average)

of the hidden states (C1, ..., Cn) and f(θ).

13. Repeat steps 2-12 to construct 10 such Markov chains each with differ-

ent initial values. Convergence tests showed that 50 000 updates were

sufficient to get precise estimates of the parameters (R = 1.02 for esti-

mates of logit(proportion)) (Gelman et al., 2004, Chapter 11.6).

14. Use 10× 50000 updates to determine the posterior probability densities

of the model parameters.

The Gibbs update involves determining the full conditional probability of the

hidden states (given everything else). The assumption that the hidden states


are a first order Markov process means that the conditional probability of the

hidden states is based only on neighbouring states and the corresponding da-

tum. The full conditional probability of the hidden state, Ck (k = 2, ..., n− 1),

is given by

Pr(Ck = i|C\k,y) ∝ Pr(Ck+1 = j|Ck = i)Pr(Ck = i|Ck−1 = h)Pr(Yk|Ck = i)

(5.17)

where C\k is the set of all states other than Ck and i is the proposed value of

the kth hidden state and h and j are the current values of the hidden states

k − 1 and k + 1, respectively.

The first and last state depend only on a single neighbour and the data asso-

ciated with that state. That is

Pr(C1 = i|C\1,Y ) ∝ Pr(C2 = j|C1 = i)Pr(Y1|C1 = i), (5.18)

and

Pr(Cn = i|C\n,Y ) ∝ Pr(Cn = i|Cn−1 = h)Pr(Yn|Cn = i). (5.19)

The conditional probability of the states can be determined and this becomes

the sampling distribution for the hidden state. Each of the n states can be up-

dated in a forward, backward or random manner. To estimate values of ν and

β, we do not need to infer hidden states. The simplified MCMC algorithm has

the following steps:

1. Assign the prior probability for β and ν using (U[0, .1]).

2. Initialise β and ν and d′.

3. Determine the likelihood of the data using Baum’s recursion formula.

4. Propose a new β′ using a simple random walk, the step size ∼N(0, .0001).

5.B Monte Carlo Markov chain algorithm 137

5. Accept β′ using a Metropolis-Hastings step with the acceptance proba-

bility

a = min1, π(β′)l(Y |β′)q(β′ → β)

π(β)l(Y |β)q(β → β′). (5.20)

6. Repeat for ν.

7. Iterate as above.

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CHAPTER 6

A Mathematical Model Investigating the

Impact of an Environmental Reservoir on

Prevalence and Control of

Vancomycin-Resistant Enterococci

This chapter consists of a publication in the form of a correspondence article which

was necessarily brief, followed by an elaboration of the model described in the pub-

lication.

Statement of joint authorshipEmma McBryde wrote the manuscript, developed the mathematical model,

wrote code for model extensions including interventions and acted as corre-

sponding author.

Sean McElwain assisted with the development of mathematical model and

proof read and critically reviewed the manuscript.

6.A Publication

To the Editor-

In an article recently published in Journal of Infectious Diseases, D’Agata

et al. (2005), present a mathematical model of transmission of vancomycin-

resistant enterococci (VRE). We developed an extension of that model that

incorporates an environmental reservoir for VRE. While our model (which

we call the environment model) supports many of the findings of D’Agata et

al., we predict different outcomes for some infection control interventions.

VRE is known to contaminate environmental surfaces (Bonten et al., 1996;

Smith et al., 1998) and case control studies suggest that this can contribute to

144 Chapter 6. Environmental reservoir model for VRE

VRE acquisition (Martinez et al., 2003). This has led many to speculate that

environment plays an important role in patient acquisition of VRE.

Mathematical models can give insight into the likely consequences of infec-

tion control practices such as hand hygiene, patient cohorting (Austin et al.,

1999a; Cooper et al., 1999), staff patient ratios (Austin et al., 1998) and an-

tibiotic restriction (D’Agata et al., 2005). However, mathematical models will

only deliver results based on the assumptions underlying them. All mod-

els published to date on nosocomial transmission of VRE have assumed that

there is no transmission due to environmental contamination. It is important

to estimate how environmental contamination could influence the outcomes

of infection control interventions.

The environment model uses the structure and assumptions of the model

presented by D’Agata et al. adding a new (environment) compartment. It

is assumed that the environment is saturable and that colonised patients

and healthcare workers contribute to environmental contamination. In

turn, the contaminated environment can cause contamination of healthcare

workers, indirectly leading to patient colonisation. The new model requires

the addition of three transmission parameters; βeh(0.15), the transmission

from healthcare workers to the environment, βe0(0.4) the transmission

from colonised patients not exposed to antibiotics to the environment and

βe1(4), the transmission from colonised, antibiotic exposed patients to the

environment. Parameters were chosen so that the rate of environmental

contamination was 25% that of patient or healthcare worker contamination.

Following the findings of Wendt et al. (1998) and Noskin et al. (2000), that

VRE persists in the environment for at least one week, we assumed that

VRE persists an average of 10 days in the environment (decontamination

parameter, κ=0.1). In order to make the equilibrium colonisation prevalence

the same as D’Agata et al., the “fitted” parameter, βp1, was 0.0074 in this

model. All other parameters followed D’Agata et al..

The system of ordinary differential equations describing the environment

model is:

6.A Publication 145

dPu0

dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0 (6.1)

dPu1

dt= Λu1 − (σu1 + γu1)Pu1 + τu0Pu0 − αp1βp1(1− η)Pu1Hc/Nh

dPc0

dt= Λc0 + σc1Pc1 − (τc0 + γc0)Pc0

dPc1

dt= Λc1 − (σc1 + γc1)Pc1 + τc0Pc0 + αp1βp1(1− η)Pu1Hc/Nh

dHc

dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np + αp1ρβh1E)(Nh −Hc)− µHc

dE

dt= (βe0

Pc0

Np

+ βe1Pc1

Np

+ βehHc

Nh

)(1− E)− κE

Please see D’Agata et al. for an explanation of parameters not given in the

text.

Our model predicted that, in the presence of an environmental reservoir, the

direction of the impact of infection control interventions is the same as the

predictions in the model by D’Agata et al. (which we call the original model)

but the magnitude is altered. The environment model predicts that improv-

ing hand hygiene compliance from 40% to 60% leads to a reduction in coloni-

sation prevalence by 17% for the environment model compared with 23% for

the original model. Increasing staff-patient ratios from 1:4 to 1:2 leads to a

reduction in colonisation prevalence by 24% in the environment model com-

pared with 32%, as predicted by the original model. Reducing the length of

stay of colonised patients from 28 days to 14 days led to a reduction in coloni-

sation prevalence of 51% in the environment model and 64% in the original

model.

Selective isolation of colonised patients (presuming 80% efficacy) led to

a predicted 44% reduction in colonisation prevalence in the environment

model compared with 42% reduction in the original model.

A significant prediction of the environment model is that even if colonised

patients are prevented from entering the ward, VRE remains endemic at 5.3%,

as illustrated in Figure 6.1. This differs from the conclusion by D’Agata et al..

The model predicts that the presence of an environmental reservoir reduces

the predicted efficacy of some interventions (compliance with hand hygiene,

increased staffing levels, reduced length of stay) yet marginally increases the

predicted efficacy of others (selective isolation). The results of the model with

regard to different effects due to interventions are not intuitively obvious. We


show that under certain circumstances, an environmental reservoir for VRE

could lead to endemic VRE transmission even if admission of VRE colonised

patients ceases.

These results suggests that, in the presence of an environmental reservoir,

VRE may be harder to eradicate, and infection control interventions less ef-

fective, with the exception of patient isolation which remains effective as pre-

dicted by this model.

0 100 200 300 400 500 600 7000

0.02

0.04

0.06

0.08

0.1

0.12

Time (days)

Pro

po

rtio

n c

olo

niz

ed

Figure 6.1: Model predictions of the prevalence of VRE over time. Both theenvironment and the original model begin with VRE at an endemic steady-state level of 12%. On day 200 further colonized patients are prevented fromentering the ward. In the environment model, a new equilibrium is estab-lished at 5.3%.

6.B Elaboration of Environment Model

6.B.1 Models

Standard Model

The model uses a four compartment model published previously (Austin and

Anderson, 1999; Austin et al., 1999b; Cooper et al., 1999). The compartments

in that model are; uncolonised patients, colonised patients, uncontaminated

healthcare workers and contaminated healthcare workers. The patient com-

partments are extended so that patients can be either antibiotic-exposed, or

not. This 6 compartment model is referred to as the “standard model” in this

6.B Elaboration of Environment Model 147

article and follows the model published by D’Agata et al., and adopts its as-

sumptions.

Environment Model

In the environment model, there is an additional compartment for the envi-

ronment. It is assumed that the environment can become contaminated by

both healthcare workers and patients and but that only healthcare workers

can acquire VRE from the environment. The interaction of these compart-

ments is shown in Figure 6.2.

Uncolonised patients

not receiving

antibiotics

Pu0

0uτ

1 1(1 ) /

p p c hH Nα β η ρ−

1cσ

Uncolonised patients

receiving antibiotics

Pu1

Colonised patients not


Pc0

Colonised patients


Pc1

0cτ

1uσ

1u∆0u∆

0c∆1c∆

0uγ

0cγ

1cγ

1uγ

0 0 0

1 1 1

/

( /

p h c p

p h c p

P N

P N

α β ρ

α β ρ

+

Uncontaminated HCW

Hu

Contaminated HCW

Hc

µ

)E+

1 1/e c pP Nβ

0 0/e c pP Nβ

/eh c h

H Nβ

κ

Environmental

contamination

E

Figure 6.2: Environmental model of VRE transmission in the hospital setting.The impact of the environment on colonisation of patients and contamina-tion of HCWs is indicated by E. The contamination of the environment arisesfrom colonised patients and contaminated HCWs.

Model assumptions

A summary of model assumptions and justification are given in Table 6.1. An-

tibiotic exposure in the colonised patient group is assumed to confer greater


likelihood of transmission of VRE to a healthcare worker per contact. This as-

sumption is based on the reduced VRE density in stool found following cessa-

tion of antibiotics in a mouse model (Donskey et al., 1999) and human cases

(Donskey et al., 2000). Additionally, only those not exposed to antibiotics are

able to revert to the uncolonised state. In the uncolonised group, it is as-

sumed that only those exposed to antibiotics are able to acquire colonisation.

Isolation can be applied to colonised patients, or to all patients. The effect of

each of these isolation strategies was examined separately.

6.B.2 Methods

Parameters

The parameters used in the standard model are those used by D’Agata

et al. Four additional model parameters were required to model environ-

mental contamination. These include transmission rate from patients and

healthcare workers to the environment, and transmission rate from the

environment to the hands of healthcare workers. At default values, environ-

mental contamination contributed to 25% of new VRE acquisitions, similar

to the proportion of VRE acquisitions in which environmental contamina-

tion was potentially implicated in the study by Bonten et al. (1996). In order

to begin with a prevalence level equal to that of the standard model (12%)

the transmission parameter, βp1, was refitted in the environmental model,

with the value 0.0074.

Equations governing models

The standard mathematical model taken from D’Agata et al. (2005) can be de-

scribed by a system of 5 ordinary differential equations. The compartments

include 4 patient compartments who can be either colonised or uncolonised,

(Pc, Pu) and exposed to antibiotics or not (Pc1, Pc0, Pu1, Pu0). The healthcare

workers also can be either colonised or uncolonised (Hc, Hu).

The nonlinear terms represent interactions between contaminated HCW and

uncolonised patient, αp1βp1(1− η)ρpu1Hc/Nh or colonised patient and uncon-

taminated HCW, (αp0βh0ρPc0/Np + αp1βh1ρPc1/Np)(Nh −Hc).

The environmental model incorporates a seventh compartment, E. The


environment is modelled as a saturable compartment taking values be-

tween zero and one. Additional terms in the environment model repre-

sent colonised patients’ contribution to environmental contamination,

(βe0Pc0

Np+ βe1

Pc1

Np)(1−E), HCW contribution to environmental contamination,

βehHc

Nh(1 − E) and the environmental contribution to HCW contamination,

αp1ρβh1E.

The system of ordinary differential equations that govern the model by

D’Agata et al. (2005) is given by

dPu0

dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0

dPu1


dPc0


dPc1


dHc

dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np)(Nh −Hc)− µHc. (6.2)

The system of ordinary differential equations that govern the extended model

is given by

dPu0

dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0

dPu1


dPc0


dPc1


dHc

dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np + αp1ρβh1E)(Nh −Hc)− µHc

dE

dt= (βe0

Pc0

Np

+ βe1Pc1

Np

+ βehHc

Nh

)(1− E)− κE. (6.3)

Simulations

Deterministic simulations were run using the MatlabT M ODE45 function.

Initial conditions that all patients were uncolonised and not exposed to an-

tibiotics (PU0 = Np). Interventions were predicted by incrementally changing


the value of one parameter while keeping each of the other parameters

constant at their default values.

The impact of restricting VRE admission was simulated using default para-

meters and initial conditions PU0 = Np for both models. In both models the

proportion of colonised patients reached equilibrium values of 12%. On day

200, we changed the conditions such that no colonised patients were admit-

ted to the ward (ΛC1 = 0, ΛC0 = 0).

6.B.3 Further discussion

This model is the first of its type, addressing the issue of environmental reser-

voirs for nosocomial pathogens. It was developed in response to the repeated

observation of antibiotic-resistant pathogens being retrieved from the envi-

ronment and anecdotal reports of failure to eliminate some pathogens fol-

lowing ward closure. Since the publication of the article (McBryde and McEl-

wain, 2006), an interrupted time series study has been published (Hayden

et al., 2006). In that study, environmental decontamination was associated

with a decline in VRE acquisition, lending further support for the role of envi-

ronment in transmission of VRE. Data from the study by Hayden et al. (2006)

could be incorporated into an environmental model to quantify further the

role of the environment in transmission.

The authors chose to allow 25% of transmission to be due to the environ-

ment at default values in this model to take a conservative approach. The

level could in fact be much higher. Environmental contamination was

assumed only to influence healthcare worker contamination not patient

colonisation directly. Such a model is appropriate for a setting where pa-

tients are relatively immobile and unlikely to make frequent contacts with

the environment, such as in an Intensive Care Unit. Additionally, many of

the objects on which pathogens have been found are equipment used by

healthcare workers, for example ward telephones or computer keyboards.

Future studies could incorporate direct environment to patient colonisation.


Assumption Support from literature Reference

Antibiotic exposureof colonised patientsincreases patient toHCW transmission

Increased stool density of VREin animal model followingantibiotic exposure Increasedstool density of VRE in humancases.

Donskey et al.(1999)Donskey et al.(2000)

Antibiotic exposurenecessary foruncolonisedpatients to acquireVRE.

Increased risk in case controlstudies for acquisition whenantibiotic exposed.

Carmeli et al.(2002)

This study incorporated thisassumption into the model toremain consistent withprevious modelling studies, forthe sake of comparison.

D’Agata et al.(2005)

Colonised patientscontribute toenvironmentalcontamination.

Rooms of colonised patientsmore likely to haveenvironmental contamination.

Trick et al.(2002)

Environmentalcontaminationcontributes tocolonisation.

23% of patients with roomspositive for VRE subsequentlyacquired colonisation.

Bonten et al.(1996)

Recurrence of VRE afterrecrudescence blamed onenvironmental point source.

Falk et al.(2000)

Patients colonised with VREmore likely to have beenexposed to contaminated roomin case-control study.

Martinez et al.(2003)

Patientdecolonisation doesnot occur inhospital.

Duration of colonisation ismuch longer (months) thanmean length of stay (weeks).

Lai et al. (1997);Byers et al.(2002)

Table 6.1: Model assumptions and their justifications.


Parameter Symbol Default ValuePatient→ environment transmission rate (per day)*

Antibiotic exposed βe1 4Unexposed βe0 0.4

Probability of patient colonisation per HCW contact βp1 0.0074HCW→ environment transmission rate* βeh 0.3Environment decontamination rate* κ 0.1Number of patients Np 400Ratio of patients/HCW ρ 4Uncolonised patients admitted (per day)

Antibiotic exposed Λu1 60Unexposed Λu0 3

Colonised patients admitted per dayAntibiotic exposed Λc1 0.4Unexposed Λc0 0.6

Length of hospital stayUncolonised patients

Unexposed to antibiotics 1γu0

14Antibiotic exposed 1

γu15

Colonized patientsUnexposed to antibiotics 1

γc028

Antibiotic exposed 1γc1

28HCW contact rate

Unexposed colonised patients αp0 8Antibiotic exposed colonised patients αp1 10

Probability of HCW contamination per contactUnexposed colonised patients βh0 0.05Antibiotic exposed colonised patients βh1 0.4

Antibiotics started per dayUncolonised patients τu0 0.15Colonised patients τc0 0.16

Antibiotics stopped per dayUncolonised patients σu1 0.15Colonised patients σc1 0.04

Hand hygiene compliance η 0.4Healthcare worker decontamination rate µ 93

Table 6.2: Table of parameters, their symbols and default values. The addi-tional parameters introduced for the environment model are indicated withan asterisk.

Bibliography

Austin, D. J., Anderson, R. M., 1999. Studies of antibiotic resistance within the pa-tient, hospitals and the community using simple mathematical models. PhilosTrans R Soc Lond B Biol Sci 354 (1384), 721–38.

Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999a.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.

Austin, D. J., Kristinsson, K. G., Anderson, R. M., 1999b. The relationship between thevolume of antimicrobial consumption in human communities and the frequencyof resistance. Proc Natl Acad Sci U S A 96 (3), 1152–6.

Austin, D. J., White, N. J., Anderson, R. M., 1998. The dynamics of drug action on thewithin-host population growth of infectious agents: melding pharmacokineticswith pathogen population dynamics. J Theor Biol 194 (3), 313–39.

Bonten, M. J., Hayden, M. K., Nathan, C., van Voorhis, J., Matushek, M., Slaughter,S., Rice, T., Weinstein, R. A., 1996. Epidemiology of colonisation of patients andenvironment with vancomycin-resistant enterococci. Lancet 348 (9042), 1615–9.

Byers, K. E., Anglim, A. M., Anneski, C. J., Farr, B. M., 2002. Duration of colonizationwith vancomycin-resistant enterococcus. Infect Control Hosp Epidemiol 23 (4),207–11.

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Donskey, C., Chowdhry, T., Hecker, M., Hoyen, C., Hanrahan, J., Hujer, A., Hutton-Thomas, R., Whalen, C., Bonomo, R., Rice, L., 2000. Effect of antibiotic therapy onthe density of vancomycin-resistant enterococci in the stool of colonized patients.N Engl J Med 343 (I), 1925–1932.

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Falk, P. S., Winnike, J., Woodmansee, C., Desai, M., Mayhall, C. G., 2000. Outbreak ofvancomycin-resistant enterococci in a burn unit. Infect Control Hosp Epidemiol21 (9), 575–82.

Hayden, M., Bonten, M., Blom, D., Lyle, E., van de Vijver, D., Weinstein, R., Jun 2006.Reduction in acquisition of vancomycin-resistant enterococcus after enforcementof routine environmental cleaning measures. Clinical Infectious Diseases 42 (11),1552–1560.

Lai, K. K., Fontecchio, S. A., Kelley, A. L., Melvin, Z. S., Baker, S., 1997. The epidemi-ology of fecal carriage of vancomycin-resistant enterococci. Infect Control HospEpidemiol 18 (11), 762–5.

Martinez, J. A., Ruthazer, R., Hansjosten, K., Barefoot, L., Snydman, D. R., 2003. Roleof environmental contamination as a risk factor for acquisition of vancomycin-resistant enterococci in patients treated in a medical intensive care unit. Arch In-tern Med 163 (16), 1905–12.

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Trick, W. E., Temple, R. S., Chen, D., Wright, M. O., Solomon, S. L., Peterson, L. R.,2002. Patient colonization and environmental contamination by vancomycin-resistant enterococci in a rehabilitation facility. Arch Phys Med Rehabil 83 (7), 899–902.

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CHAPTER 7

Bayesian Modelling of an epidemic of

Severe Acute Respiratory Syndrome

Statement of joint authorshipEmma McBryde wrote the manuscript, constructed the dataset, developed

the mathematical model, developed the code for the data analysis and acted

as corresponding author.

Gavin Gibson assisted with the analysis of data, Bayesian inference and

piecewise hazard model and proof read and critically reviewed the manu-

script.

Tony Pettitt assisted with the analysis of data, Bayesian inference and piece-

wise hazard model and proof read and critically reviewed the manuscript.

Y.Zhang and B.Zhao initiated the concept for the manuscript, constructed

the dataset and reviewed the manuscript.

Sean McElwain assisted with the development of mathematical model and

analysis and proof read and critically reviewed the manuscript.

156 Chapter 7. Bayesian modelling of an epidemic of SARS

Abstract

This paper analyses data arising from a SARS epidemic in Shanxi Province of

China involving a total of 354 people infected with SARS-CoV between late

February and late May, 2003. Using Bayesian inference, we have estimated

critical epidemiological determinants. The estimated mean incubation pe-

riod was 5.3 days (95%CI 4.2-6.8 days), mean time to hospitalisation was 3.5

days (95%CI 2.8-3.6 days), mean time from symptom onset to recovery was

26 days (95%CI 25-27 days) and mean time from symptom onset to death was

21 days (95%CI 16-26 days). The reproduction ratio was estimated to be 4.8

(95%CI 2.2-8.8) in the early part of the epidemic (February and March, 2003)

reducing to 0.75 (95%CI 0.65-0.85) in the later part of the epidemic (April and

May, 2003). The infectivity of symptomatic SARS cases in hospital and in the

community was estimated. Community SARS cases caused transmission to

others at an estimated rate of 0.4 per infective per day during the early part

of the epidemic, reducing to 0.2 in the later part of the epidemic. For hos-

pitalised patients, the daily infectivity was approximately 0.15 early in the

epidemic, but this fell to 0.0006 in the later part of the epidemic. Despite

the lower daily infectivity level for hospitalised patients, the long duration of

the hospitalisation led to a greater number of transmissions within hospitals

compared with the community in the early part of the epidemic, as estimated

by this study. This study investigated the individual infectivity profile dur-

ing the symptomatic period, with an estimated peak infectivity on the ninth

symptomatic day.

7.1 Introduction

Severe acute respiratory syndrome, (SARS), caused a perplexing epidemic

with propensity for hospital transmission, rapid worldwide spread and

markedly different epidemic curves in different countries (Wallinga and

Teunis, 2004). Beginning in November 2002 in the Guangdong province of

China, the SARS epidemic spread to Hong Kong, Viet Nam and Singapore by

March, 2003 and eventually to 29 countries around the world (Poon et al.,

2004). The World Health Organisation (WHO) issued a global alert on March

12, 2003 regarding a cluster of cases of severe atypical pneumonia and 3

days later gave a case definition and name to the condition (WHO, 2003c). A

novel coronavirus, named SARS-CoV, was identified as the infectious agent

responsible for SARS in April 2003 (Peiris et al., 2003b; Drosten et al., 2003;

7.1 Introduction 157

Ksiazek et al., 2003). In total, 8098 SARS infections and 774 deaths were

reported in the 2002/2003 epidemic of SARS (Gumel et al., 2004). The largest

outbreaks occurred in mainland China, in which there were 5327 infections

and 349 deaths reported (WHO, 2003a). Despite the initial worldwide spread

and early predictions of high case numbers, the 2003 SARS epidemic was

contained relatively rapidly with no further spread reported after July, 2003

(Donnelly et al., 2004).

SARS-CoV is likely to have an animal reservoir, possibly the palm civet cat,

Paguma lavata, (Guan et al., 2003; Webster, 2004), and further epidemics

are anticipated. Laboratory associated infections in Singapore (Lim et al.,

2004), Taiwan (Orellana, 2004) and China (WHO, 2004), the latter involving

onward transmission (Normille, 2004), remind us that further outbreaks of

SARS could occur. To help contain future epidemics of SARS, it is essential to

have an understanding of the infectivity, incubation period and likely course

of the illness.

Nosocomial transmission was a prominent feature of SARS epidemiology.

Early in the SARS pandemic, a majority of cases arose from hospital trans-

mission in many places, including Toronto (Booth et al., 2003), Hong Kong

(Wong et al., 2004; Riley et al., 2003) and Singapore (Gopalakrishna et al.,

2004). Later in the course of the epidemic, hospitals were effective sites

of containment of SARS (Gopalakrishna et al., 2004). Factors believed to

be important in reducing nosocomial transmission of SARS include hand

washing and wearing of masks, while contact with respiratory secretions is

highly correlated with SARS transmission (Teleman et al., 2004). Thorough

contact tracing and quarantine of exposed cases led to reduced transmission

in Singapore (Gopalakrishna et al., 2004). In this study, we compare the

estimated infectivity of SARS cases in the community and in hospitals. We

also examine how this changes over time.

Mathematical models of the SARS epidemic have the potential to give

insights into the disease process, to estimate critical epidemiological deter-

minants and ultimately to predict outcomes of public health interventions.

Models of SARS transmission published to date have already been useful

tools for designing control strategies; estimating the incubation period

(Donnelly et al., 2003), the infectivity (Lipsitch et al., 2003; Riley et al., 2003;

Wallinga and Teunis, 2004) and the potential impact of interventions (Riley

et al., 2003). Models have been used to predict the effect of public health

measures on the SARS epidemic in many countries including Canada (Choi

and Pak, 2003; Chowell et al., 2003), Hong Kong (Chowell et al., 2003; Lee


et al., 2003; Riley et al., 2003), Singapore (Chowell et al., 2003; Lipsitch et al.,

2003), Taiwan (Hsieh et al., 2004), and mainland China (Wang and Ruan,

2004).

For transmission models to be realistic and predictive, accurate measures of

the various transition times are required, including the incubation period,

and the time from symptom onset to removal (isolation, recovery or death).

Estimates of infectivity, particularly those based on the early behaviour of an

epidemic, are sensitive to the estimate of the incubation period. Models are

also sensitive to the full distribution of the transition periods (Lloyd, 2001),

such that summary measures (mean, median) alone are often inadequate in

modelling the behaviour of the epidemic.

In order to design effective and safe interventions, public health practitioners

also need an accurate estimate of the incubation period. Decisions regard-

ing quarantine time require estimates of the mean incubation period and the

probability of outliers. Therefore, the full probability distributions of the in-

cubation and symptomatic periods are required.

The general aims of this study are to estimate accurately the full distribution

of the transition times; the incubation period, time from symptom onset to

hospitalisation, recovery and death, to determine the infectivity of SARS in-

cluding the relative infectivity of symptomatic SARS cases in and out of hos-

pital and early and late in the epidemic, and to estimate the individual infec-

tivity profile over the course of SARS infection.

This study makes some unique contributions to the study of SARS transmis-

sion. Firstly, it uses a Bayesian framework to infer transmission times and

calculate the incubation period. In doing so, it investigates three different

models of viral transmission. Secondly, it compares the infectiousness of

SARS cases in the community and in hospital and during different times

of the epidemic. Thirdly, this study considers three different models for

individual infectivity profiles over time, using model selection criteria to

determine the optimal model. The current study investigates a database

from mainland China which has not been published previously.

7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model 159

S E RI

Figure 7.1: The schematic of the SEIR model.

7.2 Susceptible-Exposed-Infectious-Removed

(SEIR) model

The model used in this study is an extension of the stochastic version of the

compartmental Susceptible-Exposed-Infectious-Removed (SEIR) model (see

Figure 7.1) used extensively in infectious disease modelling literature (see for

example, Kermack and McKendrick (1927)). In the SEIR model individuals in

a population begin as susceptible (S) and move to the exposed (E) state fol-

lowing transmission of a contagion. This occurs at a rate that is proportional

to the number of infectious (I) and the proportion of susceptible people in

the community, SN

, (the mass action effect) so that in a small time interval, dt,

the probability of a transmission occurring is given by

Pr(S(t + dt) = i− 1, E(t + dt) = j + 1 | S(t) = i, E(t) = j) =β S(t) I(t) dt

N(t), (7.1)

where β is a constant. In the simplest version of the SEIR model, transition

between subsequent model compartments occurs at a constant rate, becom-

ing infectious as they move into the I compartment and being neither infec-

tious nor susceptible after being Removed (see Figure 7.1). This leads to

Pr(E(t + dt) = j− 1, I(t + dt) = k + 1 |E(t) = j, I(t) = k) = δ E(t) dt (7.2)

Pr(I(t + dt) = k− 1 | I(t) = k) = γ I(t) dt, (7.3)

where δ and γ are constants.

The assumption of a constant transition rate in the basic SEIR model,

adopted for ease of calculation, leads to an exponential distribution of the

probability density function for the time to transition. Other distributions,

parametric or non-parametric, could be used to model sojourn times (Diek-

mann and Heesterbeek, 2000). In the case of SARS, the incubation period,

time to hospitalisation and time from hospital admission to discharge have

been shown not to be exponentially distributed (Donnelly et al., 2003).


Assuming an exponentially distributed incubation period, with a mode of

zero, (when in fact the mode of the incubation period is considerably greater

than zero) leads to under-estimation of infectivity inferred from the early

epidemic growth curve.

In the current study, we implemented an alternative parameterisation of the

transition times. Following Donnelly et al. (2003), the Gamma distribution

was used. Other distributions could also be utilised to approximate the incu-

bation period, such as the Weibull distribution, used by Lipsitch et al. (2003).

In this study we use Gamma(α, β) notation, where α is the shape parameter

and β is the reciprocal of the scale parameter, such that

p(x) =βαxα−1e−βx

Γ(α)(x > 0, α > 0, β > 0)

and

Γ(α) =

∫ ∞

0

tα−1e−t dt.

S E H

R

D

I

Figure 7.2: The schematic of the extended SEIHRD model used in this study.The heavy arrows represent the transitions that were observed or inferred inthe current study. The thin arrows represent events that probably occur, butwith a low frequency relative to other transitions and therefore are not con-sidered in the current study.

The current study extends the SEIR model by considering two infectious

groups and two removed groups. As shown in Figure 7.2, in this model the

patients can either be infectious and in the community, I, or infectious and

hospitalised H. Removal can represent either recovery, R, or death, D. This

model, similar to that used by Riley et al. (2003) and Lipsitch et al. (2003), will

be referred to as the SEIHRD model.

In addition to dividing the infectious compartments into 2 groups, commu-

nity and hospitalised, the study also examines infectivity early and late in the

7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model 161

epidemic. Hence there are 4 infectious groups to consider (a) early commu-

nity (b) early hospitalised (c) late community (d) late hospitalised.

In this study, three different models of individual infectivity profiles are con-

sidered:

Uniform transmission model: Constant infectivity within each of the 4

groups of patients (a)-(d), but different between groups.

Model with transmission proportional to viral load: Infectivity is modelled

as a triangular distribution, with zero infectivity on day 0 and day 20

and a peak at day 10, following the viral load as described by Peiris et al.

(2003a). This is also influenced by the group (a)-(d) into which the pa-

tient falls.

Model with transmission given by a Gamma distribution: Infectivity takes

on values given by the Gamma distribution, the shape and scale para-

meters of which are inferred. Again this is modified by the co-efficient

of infectivity based on the group into which the patient falls (a)-(d).

This study assumes that the proportion of the population who are suscepti-

ble, S/N , remains at unity throughout the epidemic. The authors justify this

by the large number of people in the region investigated in this study, with

the largest city in Shanxi Province having a population of around 3 million

people, compared with the small number (354) of SARS cases observed in the

epidemic in the region. The full description of the database used in this paper

is given in the next section.

Other assumptions implicit in the current model are that there is homoge-

neous mixing of the population and that SARS cases were only infectious dur-

ing the symptomatic period. Early contact tracing studies suggest that infec-

tivity is indeed low during the incubation period (Poutanen et al., 2003). The

current study also assumes that sub-clinical SARS cases (not recorded in the

database) did not contribute significantly to the epidemic. This assumption

is supported by the finding of a very low proportion of asymptomatic con-

tacts who were SARS antibody positive (0.2%)(Leung et al., 2004).


7.3 Severe Acute Respiratory Syndrome Data from

Shanxi Province

The data used in this study come from Shanxi province in China. On 23rd

April, the WHO travel warning to China was extended to include Beijing

and Shanxi province (WHO, 2003d). The travel warning was removed on

13th June, 2003, after it was concluded no further chains of transmission

were occurring (WHO, 2003b). The Shanxi province epidemic began when

a person returned to the province while incubating SARS after visiting

Beijing in February, 2003. There were 354 reported cases of SARS during the

epidemic which began in late February 2003 and ended late-May 2003.

Appendix 7.A gives the full Gantt chart of the epidemic in Shanxi province.

Figure 7.3 shows the daily number of hospital admissions of SARS cases in

the Shanxi province. It can be seen that the peak incidence of SARS cases

admitted to hospital in Shanxi province was in mid to late April, 2003.

0

5

10

15

20

25

2-M

ar

9-M

ar

16-M

ar

23-M

ar

30-M

ar

6-A

pr

13-A

pr

20-A

pr

27-A

pr

4-M

ay

11-M

ay

Figure 7.3: Histogram of daily admissions to hospital.

Data recording the duration of exposure to another person with SARS were

available in 85 cases. Exposure-time, recorded by calendar day, ranged from

zero to a maximum of 26 days as shown in Figure 7.4. The mean time from

the day of first known exposure to the day of symptom onset (inclusive) was

7.3 SARS Data from Shanxi Province 163

8.5 days using the discrete dataset. The time from the end of exposure to the

symptomatic period had a mean of 2.9 days. This places an upper and lower

limit on estimates of the mean incubation period.

0 5 10 15 20 25 300

2

4

6

8

10

12

14

Time from first exposure to symptom onset (days)

Nu

mb

er o

f ca

ses

Figure 7.4: Histogram of time from first exposure to another SARS case tosymptom onset.

The time from symptom onset to hospitalisation was recorded in 351 of the

354 cases. In two cases, the recorded hospital admission day preceded the

recorded time of symptom onset. This was due to quarantining of exposed

individuals during the incubation period. These patients were excluded from

the analysis of time from symptom onset to hospitalisation, leaving 349 avail-

able patient records. Figure 7.5 shows a histogram of the time from symptom

onset to hospitalisation. It is an approximately exponential distribution and

the majority of SARS cases reached hospital within 4 days. It is widely dis-

persed, however, with some people taking more than 10 days to reach hospi-

tal. There is a clear outlier among these data with one SARS case reporting 44

days of symptoms prior to hospitalisation. This is also evident on the Gantt

chart, shown in Figure 7.16. It seems most likely that the date of onset of

symptoms is erroneous and this case has been excluded from the remainder

of the analysis.

Of the 354 cases in the epidemic, 344 had a recorded outcome (recovery or

death), of whom 20 died and the remainder were discharged from hospital

following recovery. The time from symptom onset to recovery was available

in all 324 cases and the time from symptom onset to death was available in 18

of the 20 cases. The distributions of symptom onset to recovery and symptom


0 5 10 15 20 25 30 35 40 450

10

20

30

40

50

60

70

80

90

100

Time from symptom onset to hospital admission (days)

Nu

mb

er o

f ca

ses

Figure 7.5: Recorded time interval from symptom onset to hospitalisation.

onset to death are shown in Figures 7.6 and 7.7, respectively.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

Time from symptom onset to recovery (days)

Nu

mb

er o

f ca

ses

Figure 7.6: Recorded time interval from symptom onset to recovery.

7.3 SARS Data from Shanxi Province 165

0 20 40 60 800

1

2

3

4

Time from symptom onset to death (days)

Nu

mb

er o

f ca

ses

Figure 7.7: Recorded time interval from symptom onset to death.


7.4 Challenges and specific aims of the study

A major challenge of the study was to estimate the distribution of the incuba-

tion period of SARS. The time of transmission of SARS is unobservable, such

that estimates of the incubation period are necessarily based on inference.

A Bayesian inference framework was used in this study as described in Sec-

tion 7.5. Only a limited number of cases have recorded known symptomatic

SARS contacts and these are used to infer transmission times and thereby es-

timate the incubation period. The cases with the shortest contact periods are

most informative. Section 7.6 describes the methodology used to parame-

terise the distributions of time to hospitalisation, recovery and death. This is

more straightforward as the times are observed and recorded.

In Section 7.7 we estimate the infectivity of the two compartments assumed

to be infectious, the symptomatic patients in the community and in hospi-

tal. This requires inference regarding missing data and transmission times.

Extending the SEIR model to include two infectious compartments, allows

us to estimate the relative impact of hospitalised and community SARS cases

on the epidemiology. Additionally, we can compare how infectivity changed

over time in each group, reflecting the effects of interventions. In this sec-

tion we also estimate the changepoint; the date that marked the transition

from high to relatively low infectivity. Finally, this study explores individual

infectivity profiles over the course of SARS illness, in Section 7.8.

7.5 Estimation of time to transmission and incu-

bation period

The incubation period was estimated only from those cases who had known

contact with another SARS case, and when there was a single contact of

known duration. In the Shanxi database this included 85 cases. It was

assumed that transmission occurred from the known contact during the

contact period and that the rate of transmission, given the contact was

independent of the state of the epidemic. The required times of exposure for

transmission to occur for the 85 cases under consideration was assumed to

be a set of independent random variables. Incubation periods of the SARS

cases are also assumed to be independent.

The model assumes that during periods of exposure to symptomatic SARS

cases, susceptible individuals acquire the disease at a fixed daily hazard rate,

7.5 Estimation of time to transmission and incubation period 167

λ. This constant hazard model is compared with two other models, a model

assuming immediate transmission and a model in which the probability

of transmission is uniform across the contact period. Following transmis-

sion, there is an incubation period that occurs before patients become

symptomatic. This period is assumed to be drawn from a Gamma(αL, βL)

distribution.

7.5.1 Bayesian approach to estimating incubation period

A Bayesian approach was used to estimate the incubation period:

π(λ, αL, βL|data) ∝ π(λ, αL, βL)L(data|λ, αL, βL), (7.4)

where π(λ, αL, βL) is the prior probability of the parameters, L(data|λ, αL, βL)

is the likelihood of the data given the parameters and π(λ, αL, βL|data) is

the posterior probability distribution of the parameters. Explanation of

the choice of prior probability distributions for the parameters, use of

augmented data and determination of likelihood of the data are given in this

section. Details of computations are given in Appendix 7.B.

Choice of prior probability distributions

Gamma priors were chosen for the three parameters. Vague prior distrib-

utions, Gamma(0.001, 0.001), were chosen for λ, αL, and βL because little is

known about the transmission rate.

Likelihood of the data given the parameters

The data used for estimation of the incubation period are the durations of ex-

posure to another SARS case, denoted vi for each individual, i, and the time

from first exposure to onset of symptoms, denoted si for each individual i. If

N is the total number of cases, the vector of the N exposure times is denoted

by v and the vector of N times to symptom onset is denoted by s.

The time that each individual in the dataset acquired SARS-CoV is not known.

It is assumed to be during the period of exposure to another symptomatic

SARS case. The time to transmission, denoted by ui, was estimated and in-

cluded in the model as an auxiliary variable. The remaining time to onset of

symptoms (si − ui) is the incubation period.


In the dataset available, all patients developed SARS, so we are considering

the probability density of ui conditional on transmission having occurred

(therefore ui < vi). Assuming a constant hazard of transmission throughout

the contact period, the conditional probability density of ui is a truncated

exponential distribution given by

fi(ui, λ) =λe−λui

1− e−λvi(0 < ui < vi ). (7.5)

The likelihood of u is also dependent on the probability density of the incu-

bation period, si − ui. The distribution, g, of the incubation period, given ui,

is determined by the Gamma(αL, βL) distribution, so that

g((si − ui)|αL, βL) ∼ Gamma(αL, βL). (7.6)

Assuming the observations are independent, the likelihood of the augmented

data (observations plus auxiliary variables, u) is given by

L(u, s|λ, αL, βL) =N∏

i=1

fi(ui|λ) g((si − ui)|αL, βL). (7.7)

The likelihood of the full set of N observations is given by

L(s |λ, αL, βL) =N∏

i=1

∫ vi

0

fi(ui|λ)g((si − ui)|αL, βL) dui.

(7.8)

Because the integral (7.8) is not straight forward to compute, a Markov chain

Monte Carlo (MCMC) algorithm, given in Appendix 7.B, was used to deter-

mine the posterior probability distributions of the parameters.

7.5.2 Results: Time to transmission and incubation period

The posterior distribution of the hazard of transmission, λ, had a maxi-

mum density close to zero and a mean of 0.18 per day, see Figure 7.8. The

inferred mean time from exposure to transmission was 2.5 days (95%CI

0.19-4.4). The estimated incubation period is shown in Figure 7.9. It follows

a Gamma(1.4, 0.26) distribution. The standard deviation for the incubation

period was 4.5 days (95%CI 3.4-5.9 days) and mean was 5.3 days (95% CI

7.5 Estimation of time to transmission and incubation period 169

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

λ

Po

ster

ior

pro

bab

ility

den

sity

Figure 7.8: Posterior distribution for the hazard of transmission, λ.

4.2-6.8 days). The median is 4.2 days, shorter than that reported by Lee et al.

(2003), 6 days, but similar to that reported by Donnelly et al. (2003), 3.8 and

Meltzer (2004), 4 days.

Appendix 7.C.1 compares the sensitivity of the results for the incubation pe-

riod to the value of λ and to model choice, showing that the conclusions re-

garding the incubation period are robust to these.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Incubation period (days)

Pro

abili

ty d

ensi

ty f

un

ctio

n

Figure 7.9: Estimated distribution of the incubation period based on max-imum posterior probability density estimates for the shape and scale para-meters of the Gamma distribution.


7.5.3 Discussion: Time to transmission and incubation pe-

riod

Estimation of the incubation period for SARS-CoV has proven to be a consid-

erable challenge. Numerous studies have attempted to make estimates (see

Donnelly et al. (2004) for a review). In papers in which interval censoring

methodology is outlined, a common strategy to deal with censored data is

to assume a uniform probability of transmission across the exposure period,

(see, for example Donnelly et al. (2003) and Meltzer (2004)). An alternative is

to assume immediate transmission upon exposure to a known symptomatic

SARS case, (see, for example Lee et al. (2003)).

The methodology used to estimate the incubation period in the current study

was to assume a constant hazard of transmission within the contact period.

The estimated incubation period, for a given dataset, using this model would

be expected to be longer than the estimations using the uniform probability

model, but shorter than the estimates based on the assumption of immediate

transmission.

The constant hazard model has the advantage that it has a biologically plau-

sible basis. However, because the estimated value of the hazard of transmis-

sion, λ, had a large probability mass near zero in this study, it would be rea-

sonable to use a uniform probability density function for time to transmis-

sion as an approximation. Figure 7.18 illustrates the estimated incubation

period based on the two different models. There is little difference between

the result of the incubation period assuming a constant hazard and assuming

uniform probability of infection during the exposure period, and the subse-

quent conclusions of the model are robust to the estimates of λ. Figure 7.18

also gives the expected value of the incubation period assuming instanta-

neous transmission at the time of contact, which is considerably longer than

the estimated incubation period in the constant hazard or uniform transmis-

sion models.

Determining the incubation period following point exposure avoids the as-

sumptions required to infer transmission times. Olsen et al. (2003) investi-

gated cases following a 3 hour in-flight exposure to a symptomatic SARS case

and found an incubation period of 4 (2-8) days. The numbers in that study

were small (22 cases), and the rapid transmission may reflect a large inocu-

lum which could impact on incubation period. Studies using larger datasets

of fully observed exposure times would be useful.

7.6 Estimation of other transition periods 171

A deficiency in the current study is that there is only weak information on

hazard of transmission, λ, since only those known to be infected with SARS-

CoV are included in the dataset. This leads to the posterior probability den-

sity for λ taking on values similar to the prior probability. In future studies,

more informative estimates of λ could be obtained by incorporating knowl-

edge about those who had exposure to a SARS case but did not become in-

fected. Alternatively, the number of contacts per infectious patient per day

could be incorporated into the model. This would provide a direct relation-

ship between the daily hazard of transmission for a single contact and the

infectivity per patient per day, which is estimated from the large scale behav-

iour of the epidemic (see Section 7.7).

7.6 Estimation of other transition periods

A Bayesian framework was also used to estimate the other transition periods

in the SEIHRD model: time from symptom onset to hospitalisation, time

from hospital admission to recovery and time from hospital admission to

death. The transition periods were assumed to be drawn from Gamma(α, β)

distributions. The parameters of the Gamma distributions were given vague

prior probability densities (π(α, β) ∼ Gamma(0.001, 0.001)). All observations

for transition periods were assumed to be independent. The posterior

probability densities of the Gamma distribution parameters (α, β) were

determined for each of the transition periods using

π(α, β|z) ∝ π(α, β)L(z|α, β), (7.9)

where z is the vector of observations for each of the transition period and

L(z|α, β) is the likelihood given by

L(z|α, β) =N∏

i=1

g(zi|α, β), (7.10)

where N is the number of observations. The calculations were performed

using Metropolis-Hastings steps in a manner similar to that described in Ap-

pendix 7.B.


7.6.1 Results: Estimation of other transition periods

Figure 7.10 gives the parameterised posterior probability distribution of the

time interval from symptom onset to hospitalisation, with the recorded dis-

crete data in the background. The distribution is approximately exponential,

with a mean of 3.5 days and a median of 2.9 days.

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

Pro

bab

ility

den

sity

fu

nct

ion

Time from symptom onset to hospital admission (days)

Figure 7.10: Estimated best fit Gamma distribution for time from symptomonset to hospitalisation, based on maximum posterior probability density es-timates for the shape and scale parameters. A histogram of recorded discretetimes is also shown.

Figure 7.11 shows the parameterised distribution of the time from symptom

onset to recovery. The mean time from symptom onset to death was 26 days

with a standard deviation of 11 days. Figure 7.12 shows the parameterised

distribution of the time from symptom onset to death. The distribution is

widely dispersed, with a mean of 21 days and standard deviation of 9.4 days.

Table 7.1 gives the means and standard deviation for the duration of each of

the stages of infection. Appendix 7.D gives the estimated values of the shape

and scale parameters of the inferred Gamma distributions.

7.7 Model for estimating coefficients of infectivity

The extended SEIHRD model was used to estimate the infectivity of SARS

cases. Coefficients of infectivity were defined in this study as the expected

number of new transmissions per infectious case per day. The infectious

group was divided into community, I, and hospitalised, H symptomatic

7.7 Model for estimating coefficients of infectivity 173

0 10 20 30 40 50 60 700

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time from symptom onset to recovery (days)

Pro

bab

ility

den

sity

fu

nct

ion

Figure 7.11: Estimated best fit Gamma distribution for time from symptomonset to recovery, based on maximum posterior probability density estimatesfor the shape and scale parameters. A histogram of recorded discrete times isalso shown.

0 10 20 30 40 50 60 70 800

0.01

0.02

0.03

0.04

0.05

0.06

Time from symptom onset to death (days)

Pro

bab

ility

den

sity

fu

nct

ion

Figure 7.12: Estimated best fit Gamma distribution for time from symptomonset to death, based on maximum posterior probability density estimatesfor the shape and scale parameters. A histogram of recorded discrete times isalso shown.

SARS cases. The epidemic was assumed to begin on February 28, 2003 when

the first introduced SARS case became symptomatic. Following the SEIHRD

model outlined in Section 7.2, the rate of new transmissions was assumed to

be proportional to the number of infectious patients at that time and their

infectivity.


Mean 95% CI Standard Deviation 95% CISymptoms to hospitalisation 3.5 3.2-3.9 3.2 2.8-3.6Symptoms to recovery 26 25-27 11 10-12Symptoms to death 21 15-29 15 9.7-23Hospitalisation to recovery 23 21-24 11 10-12Hospitalisation to death 17 11-28 16 6.9-31

Table 7.1: The posterior mean and standard deviation (in days) of the timesto hospitalisation, hospital discharge and death.

Two different states, community and hospitalised, and two different time

periods, early and late in the epidemic, were investigated. The time of

change from high to low infectivity was also estimated. The change-point

was considered an additional parameter, and its posterior probability was

investigated.

7.7.1 Bayesian approach to estimation of the transmission

coefficients

The parameters of interest in this part of the model are the coefficients of

infectivity of symptomatic community SARS cases (prior to hospitalisation)

early and late in the epidemic, denoted by x1, and x2, respectively, and the co-

efficients of infectivity of hospitalised patients early and late in the epidemic,

denoted by y1, and y2, respectively. Also of interest is the change-point, de-

noted by C.

We did not assume a fixed change-point because there were a number

of stages of intervention in the Shanxi epidemic. Firstly, the global alert

occurred on March 12, 2003. This was followed by a concerted public

health campaign in early April. It was not until April 23, 2003 that WHO

included Shanxi province on its travel warning. We therefore included the

change-point as an unknown parameter, requiring estimation.

7.7.2 Prior specification

Gamma(0.001, 0.001), were used for the four coefficients of infectivity. A dis-

crete, uniform U [1, n] distribution was used as the prior for the change-point,

where n is the number of days of the epidemic.


7.7.3 Likelihood estimation

Following the SEIHRD model and assuming constant infectivity within each

of the 4 groups of symptomatic SARS cases, the transmission pressure, ρj , on

day, j, is given by

ρj = xiI(j) + yiH(j), (7.11)

where i = 1 (j < C) and i = 2 (j ≥ C). I(j) is the number of symptomatic

community patients and H(j) is the number of symptomatic hospitalised pa-

tients.

The likelihood of Tj transmissions occurring on day j is assumed to be drawn

from the Poisson distribution:

k(Tj, Hj, Ij|x, y) ∼ Poisson(ρj). (7.12)

In a small scale epidemic, if the number of susceptibles were known, the

Binomial probability distribution could be used. In this epidemic in which

there are approximately 3 million susceptibles, the Poisson approximation

is reasonable, although it may underestimate the dispersion of the offspring

distribution (as would the binomial distribution), particularly if there is

marked heterogeneity of spreading (for example super-spreaders). An

alternative parameterisation with higher dispersion would be the Negative

Binomial distribution.

With all data included it is straightforward to find the full likelihood of the

data given the parameters:

L(T,H, I|x, y, C) =C∏

j=1

k(Tj, Hj, Ij|x1, y1)n∏

j=C+1

k(Tj, Hj, Ij|x2, y2), (7.13)

where n is the number of days of the epidemic and T ,H and I, represent

the vectors of n values of daily transmissions, community case numbers and

hospitalised case numbers, respectively.

Because the times of transmission are unknown, and there are some missing

values in the hospitalisation and recovery and death times, missing data and

unobserved data need to be inferred. The simulated data are drawn from

the distributions of the incubation period, time to hospitalisation and time

to recovery and discharge estimated in the first part of the study. The likeli-

hood estimation of the day of transmission for each individual was based on

the parameterised incubation period and for cases with known contacts the


truncated exponential distribution given in Expression (7.5). The state of the

epidemic was not considered in the likelihood computation. The techniques

used for data augmentation and computation are given in Appendix 7.E.

7.7.4 Results: Change point Estimation

The epidemic was measured from the day of symptom onset of patient 1,

which was February 28, 2003. Figure 7.13 shows the posterior distribution

for the estimated time of the change in infectivity (change-point). The maxi-

mum density is taken to be the end of day 29 of the epidemic, corresponding

to the beginning of March 29, 2003. Following this, the estimates of the co-

efficients of infectivity were performed assuming a change point at midnight

March 28/29.

20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Days from start of epidemic

Pos

terio

r pr

obab

ility

den

sity

Figure 7.13: Posterior distribution for change-point.

Figure 7.13 demonstrates that there is considerable uncertainty with this

estimate, with the posterior probability also giving some support to an

earlier change point time. The posterior weight rapidly declines for times

after March 29th (day 30 from the start of the epidemic), suggesting later

times are unlikely.

7.7.5 Results: Coefficients of Infectivity

The estimated means of the four coefficients of SARS-CoV transmission,

representing the mean number of new infections per infectious case per


day, are given in Table 7.2. The relative infectivity of community compared

to hospitalised SARS cases increases markedly after the change point, with

x1/y1 = 5.1 (95% CI 0.8-17), and x2/y2 = 350 (95% CI 95-1400), where xi

refers to symptomatic community SARS cases (prior to hospitalisation) and

yi refers to hospitalised patients.

Parameter Mean 95% CredibleInterval

x1 0.41 0.24-0.59y1 0.15 0.023-0.34x2 0.21 0.18-0.24y2 0.0006 0.000018-

0.0022Ra 4.8 2.2-8.8Rb 0.75 0.65-0.85

Table 7.2: Table of transmission coefficients (mean number of transmissions perinfective per day) x1: symptomatic community cases before March 29, x2: symp-tomatic community cases after March 29 y1: symptomatic hospitalised cases beforeMarch 29, y2: symptomatic hospitalised cases after March 29. March 29 was thechange-point with the highest posterior probability density.


7.7.6 Results: Reproduction ratio

The basic reproduction ratio, R0, is defined as the expected number of

secondary cases per primary case in a fully susceptible population (Ander-

son and May, 1991; Diekmann and Heesterbeek, 2000). As the epidemic

progresses, the reproduction ratio could be modified both by a decrease in

the number of susceptible cases or a change in infectivity (for example, due

to infection control interventions). In this study, we estimated the effective

reproduction ratio before and after the change point.

The effective reproduction ratio can be deduced from the inferred coeffi-

cients and the known data. The mean time from symptom onset to hospital

admission is 3.5 days and the mean time from hospital admission to either

recovery or death is 22.2 days. The posterior probability distribution of the

effective reproduction ratio can be calculated using

Ra = x1X + y1Y , (7.14)

where Ra is the reproduction ratio prior to the change point, X is the mean

duration of symptoms prior to hospitalisation and Y is the mean duration of

symptoms in hospital. Similarly Rb, the reproduction ratio after the change

point can be calculated using

Rb = x2X + y2Y . (7.15)

Ra is estimated to be 4.8 (95%CI 2.2-8.8 )and Rb is estimated to be 0.75 (95%CI

0.65-0.85). The distributions for Ra and Rb are displayed in Figure 7.14. The

greatest impact on the reproduction ratio was the change in infectivity of the

hospitalised group.

During the first part of the epidemic prior to March 29, the expected number

of transmissions resulting from each symptomatic SARS case is 1.4 during

the community period, and 3.4 during the hospitalised period. For the SARS

cases from March 29 onwards, the expected number of transmissions result-

ing from each symptomatic SARS case is 0.73 during the community period,

and 0.013 during the hospitalised symptomatic period. The ratio of infec-

tivity in the community to infectivity following hospitalisation is 5.1, simi-

lar to Riley et al. (2003)’s estimate of 5. After March 29 however, this figure

was much higher, owing to very much reduced estimated infectivity in hos-

pitalised patients.

7.8 Individual Infectivity profiles 179

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

Reproduction ratio

Po

ster

ior

pro

bab

ility

den

sity

Figure 7.14: Posterior distribution for the reproduction ratios prior to (white)and after (black) March 29, 2003.

7.8 Individual Infectivity profiles

A preliminary analysis compares three models of individual infectivity over

the course of SARS-CoV infection. The first is the uniform transmission model

in which infectiousness within the 4 groups (out of hospital early, in hospi-

tal early, out of hospital late, in hospital late) is uniform over the course of

illness. In the second model is the model with transmission proportional to

viral load. In this model infectivity takes on a triangular distribution peaking

on day 10, following the results for viral load described by Peiris et al. (2003a).

In the third model, the model with transmission given by a Gamma distribu-

tion; shape and scale parameters were inferred.

Using the Akaike information criterion (AIC) (Akaike, 1974), the model with

transmission given by a Gamma distribution is superior (AIC=320) to the uni-

form transmission model (AIC=328). The model model with transmission pro-

portional to viral load performs the worst (AIC=356). With reference to the

model with transmission given by a Gamma distribution, the inferred Gamma

distribution of the infectivity profile is shown in Figure 7.15. The peak infec-

tivity is estimated to be on the 9th day following symptom onset in the Gamma

model. The infectivity follows the Gamma(3.9, 0.36) distribution.

This finding is based on an initial exploration of the dataset and the analysis

can be extended. In particular, the infectivity profiles could inform the trans-

mission times. In this study, as a simplification, the unobserved transmission


0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Day since symptom onset

Est

imat

ed in

fect

ivit

y

Figure 7.15: Infectivity profile versus time since symptom onset.

times were inferred using the uniform transmission model only.

7.9 Discussion and Conclusions

Important conclusions regarding the infectivity of SARS-CoV can be drawn

from this analysis. The estimated daily infectivity of the hospitalised patients

was lower than for community patients. Despite this, it was estimated that

early in the epidemic, a larger number of secondary cases resulted from hos-

pitalised patients because people remained in this stage for a longer time (an

average of 22.2 days symptomatic in hospital compared with 3.5 days prior to

hospitalisation). Later in the epidemic, the transmission rate of symptomatic

community SARS cases decreased to around 50% of previous levels, whereas

the decline in transmission rate for those SARS patients in hospital reduced

more dramatically to around 0.4% of previous levels.

These results support the conclusion that interventions were effective at con-

trolling the SARS epidemic in Shanxi province, particularly those interven-

tions directed at hospital isolation. However, other possible causes for these

results need to be considered. The relatively high infectivity of the commu-

nity SARS cases could be due to their earlier stage in the course of SARS-CoV

infection. Much of the time spent in hospital is associated with the conva-

lescent stages of the illness and it could be argued that SARS patients would

be less infectious during this period. On the other hand, Peiris et al. (2003a)

showed that viral shedding peaks around day 10, suggesting that for many

people the most infectious stage of the illness occurs following hospitalisa-

tion.

7.9 Discussion and Conclusions 181

The reduction of infectivity over time could be partly explained by the pro-

portion of contacts who are susceptible decreasing as an epidemic proceeds.

While depletion of susceptibles undoubtedly occurs in widespread viral epi-

demics, the authors believe that the transmission of SARS-CoV to 354 people

in a population of over 3 million would not account for a significant drop in

the proportion of contacts who are susceptible, assuming homogenous pop-

ulation mixing. In the hospital setting and in families, in which contacts tend

to cluster, depletion of susceptibles may account for some of the change in

the reproduction ratio. This could be further explored using a network or

household model.

Another reason for the difference in infectivity before and after the March 29

could be seasonal. It is possible that SARS-CoV , like many other respiratory

viruses, is transmitted more efficiently in winter. However, this would result

in a general decline in infectivity, which does not explain the much greater re-

duction in infectivity of hospitalised SARS cases compared with community

SARS cases, observed in this study.

The estimated date on which the infectivity of SARS declined (the change

point) predated the peak incidence of admission of SARS cases to hospital.

Both the incubation period and the delay between symptom onset and hos-

pitalisation contributed to this lag. It is a lesson for future epidemics, that

even after appropriate interventions are successful in reducing transmission,

we can expect a further increase in infection notifications.

The reproduction ratio late in the Shanxi epidemic is very similar to those

estimated by Wallinga and Teunis (2004) in Singapore and Hong Kong, both

estimated to be 0.7. Wallinga and Teunis (2004) studied 4 countries (Singa-

pore, Viet Nam, Hong Kong and Canada) and found that although the epi-

demic curves initially were markedly different, following interventions, the

estimated reproduction ratio was very similar in 3 of the 4 countries exam-

ined in that study. Although it is reassuring that in most cases (all except

Canada), a reproduction ratio of less than one was achieved, it was only fol-

lowing the implementation of stringent control measures. It could be pre-

dicted that if complacency occurs in future epidemics, it may be difficult to

achieve a reproduction ratio of less than one for SARS.

Three different models of infectivity profiles over the course of SARS-CoV in-

fection were considered in this study. The model considering a Gamma shape

for infectivity appeared statistically slightly superior to the model assuming

uniform infectivity. Of interest is that the estimated peak infectivity occurs


on the ninth day following symptom onset. This is consistent with specimen

positivity in the lower and upper respiratory tract and gut reported by Cheng

et al. (2004). Additionally, (Peiris et al., 2003a) measured nasopharyngeal as-

pirate viral loads of 14 SARS cases on day 5, 10 and 15 following symptom

onset and found that day 10 was consistently the highest of these measure-

ments. The concordance between viral load data and infectivity inferred in

this study warrants further investigation. A larger dataset in which contact

times are fully observed would be useful in elucidating infectivity profile.

There are several ways in which the current model can be extended. This

study assumed Gamma distributions for transition times. Other distributions

could be considered including the Weibull and non-parametric approaches.

A mixture model may be particularly useful for estimating susceptibility,

infectiousness and duration of infectivity. The possibility of more than one

change point or a gradual transition could also be explored. Reversible

jump MCMC would be a useful tool in determining this (Green, 1995). SARS

models to date including the current study have assumed zero infectivity

during the incubation period. Infectivity of SARS cases during the incubation

period could be estimated by extending the Bayesian inference model. While

there were no clearly identified super-spreaders in the Shanxi epidemic,

heterogeneity of infectivity was a major feature of the epidemiology of

SARS in Singapore and Hong Kong (Li et al., 2004). This could be further

investigated using the current dataset, however a dataset containing detailed

information on transmission trees would be more informative.

Appendix

7.A Gantt chart of Shanxi epidemic

Figure 7.16 displays a visual depiction of the epidemic. Each individual is

represented as a horizontal line, with the colour code indicating the stage of

SARS-CoV infection for that individual. It can be seen from Figure 7.16 that

in mid to late April, the daily number of new cases began to decline.

7.B Computations for time to transmission and incubation period183

Figure 7.16: Gantt chart of epidemic. The time of exposure to another SARScase is mid-blue, the time that a patient is asymptomatic following exposureis light blue, the time of symptoms prior to hospitalisation is yellow, the timeof hospitalisation is orange and the time of discharge or death, maroon. Pa-tients are ordered according to hospital admission date.

7.B Computations for time to transmission and

incubation period

Computations were performed using a Markov chain Monte Carlo (MCMC)

algorithm.

1. Initialise the parameters λ, αL and βL.

2. For each patient i, propose a new ui by drawing u′i randomly from the

distribution described in Expression (7.5)

3. Accept u′i using the acceptance probability,

Pacc = min

1,

g((si − u′i)|αL, βL)

g((si − ui)|αL, βL)

(7.16)

4. Propose λ′ using a simple random walk step such that λ′ = λ + ε, where


ε is drawn from the N(0, 100) distribution. In this paper, we follow the

Bayesian notation where N(0, 100) is used for a normal distribution with

a mean of zero and a precision of 100 and hence a variance of 0.01. The

precision of the proposal distribution was chosen as a balance of the need

to have rapid mixing and the desire to improve acceptance probability

5. Accept λ′ with a probability Pacc given by

Pacc = min

1,

∏Ni=1 f(ui|λ′)π(λ′)∏Ni=1 f(ui|λ)π(λ)

, (7.17)

6. Update αL, proposing a new value α′L using a simple random walk, each

step is drawn from a random normal distribution N(0, 100). Accept α′Lwith probability Pacc given by

Pacc = min

1,

∏Ni=1 g((si − ui), α

′L, βL)π(α′L)∏N

i=1 g((si − ui), αL, βL)π(αL)

(7.18)

7. Update βL using a Gibbs step. A conjugate prior, π(βL) ∼ Gamma(l,m),

is assigned to βL, making the full conditional posterior for βL

βL|(s− u), αL ∼ Gamma(l + αLN,m +N∑

i=1

(si − ui)), (7.19)

which enables a Gibbs update of βL by drawing a value randomly from

this distribution.

Steps 2 to 7 constitute a single iteration of the algorithm.

The “burn-in” period was 10 000 iterations. The posterior probability distri-

butions of u, λ, αL, βL were determined by taking the next 90 000 iterations.

7.C Diagnostics: Convergence and Sensitivity

analysis

Visual inspection of the trace plots showed that the chains for all parameters

appeared to converge within 1000 iterations. A number of different initial

values were considered for the parameters and the results were essentially

unchanged. Figure 7.17, for example, shows values of αL plotted against iter-

ation number for 6 different initial values. The plots show that the estimates

7.C Diagnostics: Convergence and Sensitivity analysis 185

of αL settle down well before the end of the 10 000 iteration burn-in.

500 1000 1500 2000 2500 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Iterations

α

Figure 7.17: Output from one series of Markov chains. Six different initial val-ues of the shape parameter of the incubation period, αL all settle down to thesame distribution after a few hundred iterations.

7.C.1 Sensitivity of estimate of incubation period to model

choice and hazard of transmission parameter

Sensitivity analysis was performed on the choice of model used to estimate

the incubation period. The current study assumed that during a contact the

hazard of transmission remained constant, leading to an exponential proba-

bility density function for time to transmission. Two alternative approaches

would be

1. to assume that the probability of transmission was constant throughout

the contact period, a uniform probability density for time to transmis-

sion, effectively putting λ = 0

2. to assume transmission coincides with onset of infection challenge, ef-

fectively putting λ = ∞.

The posterior probability density of the incubation period was estimated us-

ing these models and compared with the estimation in the current study as

summarised in Table 7.3 and illustrated in Figure 7.18.

In the model used in the current study (the assumption of constant hazard),

the maximum posterior density for the daily hazard of transmission, λ was


Incubation period Mean 95% CI Standard Deviation 95 % CIConstant hazard 5.3 4.2-6.8 4.5 3.4-5.9Uniform 5.1 4.1-6.3 4.4 3.4-5.6Immediate Transmission 7.9 6.9-9.0 4.9 4.1-5.9

Table 7.3: The estimated mean and standard deviation (in days) of the in-cubation period comparing the estimates using the assumption of constanthazard, used by the current study, and the assumptions of uniform probabil-ity and immediate transmission.

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Incubation period (days)

Pro

bab

ility

den

sity

fu

nct

ion

constant hazarduniform probabilityimmediate transmission

Figure 7.18: Comparison of the incubation period as estimated in the currentstudy using the constant hazard model, the uniform probability model andthe immediate transmission model. The constant hazard model used in thisstudy leads to a similar result to the uniform probability model.

close to zero. Little information was available in the data-set regarding λ,

therefore λ took on a distribution similar to its prior probability, with a large

probability mass near zero and a long tail. This effectively makes the model

in which a constant hazard is assumed equivalent to the model of uniform

probability, the model that suggests infection is equally likely at any stage

during the exposure period. Even at the extreme values of λ, the effects of the

estimate of λ on incubation period shown in Figure 7.18 are relatively small.

Therefore the conclusions of the subsequent components of the model are

robust to the choice of model for transmission and the value of λ.

7.D Estimated values of shape and scale parameters for the Gammadistributions 187

7.D Estimated values of shape and scale parame-

ters for the Gamma distributions

Table 7.4 gives the estimated values of the parameters of the Gamma distrib-

utions applied in the model. These values can be used along with the coef-

ficients of infectivity to reconstruct the epidemic and explore the large scale

effect of interventions, including reduced time to isolation, quarantine, and

more effective isolation.

shape parameter scale−1 parameterIncubation period 1.4 0.26Symptom onset to hospitalisation 1.3 0.37Symptom onset to recovery 5.6 0.22Symptom onset to death 2.1 0.11Hospital admission to recovery 4.1 0.18Hospital admission to death 1.2 0.068Individual infectivity 3.9 0.36

Table 7.4: Estimated values (based on maximum posterior density) for theshape and scale−1 parameters of the Gamma distributions fitted to the data.

7.E Techniques used for data augmentation and

computation to determine coefficients of in-

fectivity and change point

7.E.1 Augmented data

There were missing values for the time of symptom onset hospitalisation

times and time to recovery (1, 2 and 10 missing values respectively out of

the 354 SARS cases in the database). Missing data were simulated for each

iteration of the Markov chain using the inferred distributions of transition

times. The likelihood of the data given the parameters is given by

L(d|θ) =

∫L(d, s|θ) ds, (7.20)

where d is the known data and s is the simulated data.

Because the integral above is not straightforward, L(d|θ) was inferred by

drawing s using the known times and the parameterised distributions,

estimated in Section 7.6. For example, where recovery times were missing,


these were inferred from the hospitalisation date and the parameterised time

to recovery distribution.

The date of each individual’s transmission of SARS-CoV also became an aux-

iliary variable in the model. The times were inferred from

1. the known date of onset of symptoms (taken directly from the database)

2. the parameterised incubation period,

so that transmission date = date of symptom onset - incubation period, where

the incubation period was drawn randomly from the Gamma(αL, βL) distrib-

ution. If the time of exposure to another SARS case was known, the proposed

transmission time (ti) was drawn from a distribution based on the joint prob-

ability of (a) time to transmission, calculated using Expression 7.5 and (b) the

incubation period, with Gamma(αL, βL) distribution.

7.E.2 Computations to determine posterior distributions of

the coefficients of infectivity and the change point

For each iteration of the model, the auxiliary variables were firstly de-

termined using Gibbs sampling of the parameterised distributions. The

likelihood of the augmented data was calculated using Expression (7.13).

Coefficients of infectivity were proposed and accepted according to:

Pacc = min

1,

C∏i=1

k(Tj, Hj, Ij, x′1, y1)p(x′1)prop(x′1 → x1)

k(Tj, Hj, Ij, x1, y1)p(x1)prop(x1 → x′1)

, (7.21)

where C is the date of the change point and prop(x′1 → x1) is the proposal

probability of x1 from x′1. Similarly, x′2 is updated by:

Pacc = min

1,

n∏i=C+1

k(Tj, Hj, Ij, x′2, y2)p(x′2)prop(x′2 → x2)

k(Tj, Hj, Ij, x2, y2)p(x2)prop(x2 → x′2)

, (7.22)

where n is the number of days of the epidemic. Acceptance equations were

similarly constructed for y1 and y2.

The change-point day was updated as follows:

1. For each iteration a new change-point day was proposed drawn as an

integer from the U [1, n] distribution, where the epidemic begins on day

1 and ends on day n

7.E Statistical inference used to estimate infectivity and change points189

2. The change-point day was updated using a Metropolis step based on

the full likelihood given by:

Pacc = min

1,

p(C ′)∏C′

j=1 k(Tj, x1, y1)∏n

j=C′+1 k(Tj, x2, y2)

p(C)∏C

j=1 k(Tj, x1, y1)∏n

j=C+1 k(Tj, x2, y2)

. (7.23)

The process was iterated 100 000 times and the first 10 000 iterations were

used as a burn-in period. The following 90 000 updates of x1, x2, y1, y2 and C

were used to determine the posterior distribution.

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CHAPTER 8

Conclusions and suggestions for future

work

Mathematical models add value to the study of infectious diseases. Models are

required both to form a basis of valid statistical inference and, in the absence of ade-

quate epidemiological data, produce some evidence for the efficacy of interventions

through simulation. Knowledge of the transmission characteristics of contagions

can be incorporated into statistical models to improve statistical inference. Such

structured models have an advantage over standard statistical techniques in that

they give a meaningful interpretation to the value of the estimated parameters. Epi-

demic models, founded on biologically plausible assumptions, can be an extremely

useful predictive tool. One can explore some “what if?” scenarios, most importantly

the predicted effect of infection control interventions. Such models are crucial in the

study of infectious diseases epidemics because the standard frequentist statistical

analysis involving repeated trials is infeasible in outbreak settings and randomised

controlled trials of infection control interventions for bacterial pathogens are

logistically challenging. Additionally cost-effectiveness studies can be teamed with

such models to ensure optimal resource utilisation.

The Bayesian approach was adopted in this thesis because the main questions posed

by the studies were How does the information, provided in this single dataset, mod-

ify my belief regarding the transmission of the organism? Such a question does not

have meaning in a frequentist context. In the case of a single pandemic, such as

SARS, there is no opportunity for repeated measurements. Additionally, the Markov

chain Monte-Carlo algorithm is a very convenient tool for numerical integration of

the complex expressions derived from incorporation of latent variables into trans-

mission models. Bayesian inference also allows a researcher to incorporate prior

information into models. In models developed in this thesis, priors probabilities

were vague. Mostly, this was because little was known about the model parameters.

In the cases where a small amount of independent data were available, these were

used to independently validate model conclusions (in Chapters 5 and 7) rather than

incorporated as priors. The results of the studies could be used to develop prior

196 Chapter 8. Conclusions and suggestions for future work

probabilities for subsequent studies.

8.1 What has been achieved?

This thesis used previously unpublished datasets to develop pathogen-specific mod-

els of infectious disease transmission. Using these models, the studies in this thesis

have quantified the cross-transmission rate of three pathogens and explored poten-

tial alternative sources of pathogen acquisition. By allowing the transmission para-

meters to be time-dependent, the studies in Chapters 5 and 7 assessed the impact of

infection control interventions that took place during the data collection period. For

the other studies, the potential outcome of infection control measures were mod-

elled and predictions were made regarding their effect on transmission.

8.1.1 Estimation of basic reproduction ratio and cross-

transmission rates

In Chapters 3, 5 and 7, statistical inference was used to estimate infectivity and the

basic reproduction ratio. Time dependence in the parameters for infectivity was in-

corporated into the study in Chapters 5 and 7, to assess the impact of interventions

that occurred during the data collection period.

The study in Chapter 3 estimated the reproduction ratio to be below unity, and finds

that methicillin-resistant Staphylococcus aureus (MRSA) was endemic to the ward

due to continued importation of new cases through admissions of patients already

colonised. The study in Chapter 5 found that the majority of VRE acquisition in

the institute occurred through cross-transmission on the ward. The study also

concluded that there is some evidence that the infectivity changed just prior to the

hospital outbreak and following infection control interventions.

The study in Chapter 7 estimated the reproduction ratio for SARS-CoV before and

after infection control interventions. It concluded that people were less infectious in

hospital than in the community in Shanxi province, and that the difference increased

after infection control interventions were put in place.

8.1.2 Development of new models

Chapter 6 developed and explored a model that has not been described previously.

The study examined the hypothesis that an environmental reservoir would affect

the impact of infection control interventions and the endemic level of colonisation.

Chapter 7 considered models both for the incubation period of SARS and infectivity

profile over time that gave insight into the transmission dynamic of SARS.

8.1 What has been achieved? 197

8.1.3 Using of models to inform health policy

Chapter 3 examined the predicted impact of a number of infection control interven-

tions. Hand hygiene was predicted to be the most effective intervention. Increasing

staff/patient ratio was predicted to increase MRSA transmission if no cohorting took

place, a finding that contradicts a number of other studies. The findings suggest that

caution should be taken when a ward implements a policy to increase staffing lev-

els. Increasing the number of staff in the setting of cohorting is predicted to reduce

transmission. This study also demonstrates that stochastic model predictions are of-

ten different from deterministic model predictions on the scale of the hospital ward

setting. The changes in ward size leading to differences in transmission cannot be

predicted using deterministic models. Some of the model predictions are not intu-

itive, for example, that very small increments in hand hygiene lead to large reduc-

tions in transmission, and that increasing staff could lead to increased transmission.

Chapter 6 suggests that an environmental reservoir should be considered for

pathogens known to survive in the environment as such a reservoir is predicted to

reduce the efficacy of many infection control interventions.

8.1.4 Methodological framework for future studies

Methods that allowed for serial dependence in infection control data are used

throughout this thesis, namely structured models that allow for changes in coloni-

sation or infection pressure. Censored transmission data are accounted for by

inferring transmission times using latent variables in a Bayesian framework. Like-

lihood estimates are based on piecewise constant hazard formulae. Monte-Carlo

Markov chain integration is used to simplify the intractable integrals that result from

the latent variable models.

The study in Chapter 5 employed a method that could be applied to simple serial

surveillance data with no information on event histories. A hidden Markov model

is used in which the transition component is based on a structured Susceptible-

Infectious (SI) model. By following this methodology, we are able to estimate the

transmission characteristics of VRE without assuming full or immediate detection

of transmission events. This model could be applied to a number of datasets in the

future. Serial surveillance is a common way of measuring the status of hospitals

with regard to nosocomial pathogen containment. These datasets are, therefore,

available for analysis in many hospitals.


Models developed in this thesis can form the foundation of future statistical mod-

els of nosocomial transmission. Models of pathogens need to consider both cross-

transmission and independent sources of colonisation. Such a model could be in-

corporated into conventional statistical approaches using a Cox proportional haz-

ards model with colonisation pressure included in the model as a time dependent

covariate, for example. The possibility of an environmental reservoir could be in-

corporated into future analyses of transmission of agents such as Acinetobacter spp.,

other Gram negative bacteria and norovirus.

Chapter 7 uses a Bayesian framework to estimate incubation period and infectious-

ness of SARS in mainland China. There are many emerging threats including the

H5N1 strain of influenza to which similar methodology could be applied.

8.1.5 Model comparison

Several different models are compared using the Deviance Information Criterion

(DIC) in Chapter 5. The model that suggested VRE colonisation arose both from

cross-transmission and sporadically was superior to the models that included only

one of these. Additionally, comparison of models with a time dependent cross-

transmission parameter suggested there is some evidence that the transmission

changed at the time of the interventions. Chapter 7 compares different individual

infectivity profile models for SARS using the Akaike Information Criterion (AIC).

8.1.6 Model diagnostics

The study described in Chapters 3 and 5 used parametric bootstrap technique to test

the model. The data were simulated using estimated model parameters and the pre-

cision with which the model was able to estimate parameters was measured. Chap-

ter 5 also used genotyping data as an external comparison with model results. The

study described in Chapter 7 compared the individual infectivity profile estimated by

the study with virological data, finding a close relationship with this external source

of information.

8.2 Limitations of the approach adopted in this

thesis and opportunities for extensions

Chapter 3

Chapter 3 used a four compartment model to quantify the transmission of MRSA in

the hospital intensive care unit. The model can readily be adapted to incorporate

8.2 Limitations and opportunities for extensions 199

covariates in future studies. Time-dependence could be incorporated into parame-

ters; for example, in the context of a planned interrupted time series studies with the

aim of investigating the impact of infection control interventions.

A major draw-back of the model described in Chapter 3 is that it assumed only one

mode of transmission of contagion, namely, via the hands of healthcare workers.

There was no exploration of alternative models. Additionally, the inclusion of four

compartments led to a number of parameters, to which the model outcomes are

often highly sensitive, as explored in Chapter 3. A more parsimonious model con-

sisting of only two compartments, was developed and presented in Chapter 5. The

study in Chapter 3 also assumes perfect swab sensitivity. This is addressed in Chap-

ter 5.

Chapter 5

The model in Chapter 5 investigated two possible sources of vancomycin-resistant

enterococci (VRE); acquisition arising from cross-transmission in the hospital en-

vironment and acquisition which is sporadic. Other possibilities could be consid-

ered. For example, one could model a hyper-endemic period that might arise from a

point-source outbreak, by including a parameter and a time-dependent indicator.

The model described in Chapter 5 used serial prevalence data. Often, however, lon-

gitudinal datasets involve incidence data. An obvious extension of this model is to

use incidence data in a HMM framework. One could use an approximate relation-

ship between underlying hidden state (prevalence of colonised patients) and obser-

vations (incidence of detection/infection), as was done in the study by Cooper and

Lipsitch (2004). An alternative method would be to derive a direct relationship be-

tween the incidence of transmission and the colonisation pressure over a time in-

terval. Colonisation pressure over time could be estimated by integrating across the

hidden states.

Chapter 5 assumes that there was homogenous mixing of patients and staff and that

the rate of cross-transmission was density dependent (Reed-Frost assumption). Al-

ternative models, based on the Greenwood assumption (Becker, 1989), for example,

could be explored. Here, the maximum risk of cross-transmission is achieved by a

single colonised patient (a saturation effect), with no further risk as more colonised

patients are in the ward. Larger databases could give information on networks or

mixing across wards and relax the assumption of homogenous mixing.

A weak prior probability distribution was used for the parameter values estimated in

Chapter 5. We could have incorporated the data from the genotype study as a more

informative prior probability distribution. Instead these data were used to validate

the model.


There was some evidence of a change in transmissibility in this study. The cause of

this change is unclear, but given the mounting evidence for environmental contam-

ination and its contribution to transmission, a model incorporating an environmen-

tal reservoir needs to be considered. This model was developed in Chapter 6.

Chapter 6

The model presented in Chapter 6 is not based on a data series and has not been

validated against other methods. It is therefore a theoretical model with the aim of

generating hypotheses. Future studies may compare models with an environmental

compartment with those without and use model comparison techniques to estab-

lish the evidence for this reservoir compartment. Alternatively, intervention studies

that quantify environmental contamination, compare environmental strains with

human pathogens and aim to decontaminate the environment could be used to

validate the model.

Chapter 7

This study relaxed the assumption of homogeneity of transmission by sub-dividing

the infectious compartment (early and late, hospitalised and community). However,

within each compartment homogeneity was assumed. While there were no clearly

identified super-spreaders in the Shanxi epidemic, heterogeneity of infectivity was

a major feature of the epidemiology of SARS in Singapore and Hong Kong (Li et al.,

2004). This was not explored in the study as detailed chains of transmission were not

available. Additionally it was assumed that all people were equally susceptible.

This study assumed Gamma distributions for sojourn times. Other distributions

could be considered including the Weibull, lognormal and non-parametric ap-

proaches. The study investigates the time of change from a rising epidemic to a

declining epidemic. The possibility of more than one change point or a gradual

transition could also be explored. Reversible jump Markov chain Monte Carlo would

be a useful tool in determining this. SARS models to date, including the current

study, have assumed zero infectivity during the incubation period. The possibility

of infectivity of SARS cases during the incubation period could be investigated by

extending the Bayesian inference model.

8.3 Future work

The motivation for this work is the high mortality, morbidity and cost of healthcare

associated infection. The reduction of healthcare associated infections is a complex

8.3 Future work 201

process, dependent on human perceptions, motivations and ultimately behaviour.

For example, while hand hygiene is widely acknowledged to be a critical factor in

transmission of diseases in hospitals, hand hygiene compliance continues to be poor

(Whitby et al., 2006). In addition to knowledge; motivation, intervention, measure-

ment and feedback are required to change behaviour.

This thesis has focussed principally on valid measurement of the rate of healthcare

associated transmission of infectious agents, and the manner in which these organ-

isms are acquired and estimating the effect of interventions through measurement

and simulation. Methods described in this thesis could be incorporated into con-

trol charts to deliver real-time response to infection control interventions, providing

greater incentive for behaviour change.

The models developed in Chapters 3, 5 and 7 have application to other emerging

threats such as antibiotic resistant Gram-negative bacteria, some of which are found

in the environment, for example carbapenem resistant Acinetobacter baumanii. The

models in Chapter 7 have application to emerging community infectious diseases

such as H5N1 influenza.

Further adaptation and utilisation of models can be made as technology and knowl-

edge improves. The model in Chapter 3 could be readily adapted to investigate

the effect of patient isolation and early patient detection when technology for

more rapid and sensitive detection of colonised patients is achieved and when the

protective effect of isolation is known. As well as predicting the impact of emerg-

ing infection control initiatives, the models help structure study design. Models

developed in this thesis could be components of economic models for cost-utility

analysis.

This thesis confined itself to available hospital datasets and as such did not consider

issues of control of hospital pathogens at a regional level. Chapter 3 showed that

MRSA persisted in the Intensive Care Ward despite a reproduction ratio within the

ward of well below unity. The cause for this was introduction of MRSA via patients

colonised on admission. Cooper et al. (2004) showed that readmission of patients

from the community with MRSA may lead to an effective reproduction ratio greater

than unity despite a ward reproduction ratio less than unity.

For pathogens that are carried long term and have reached significant levels in the

community, infection control must be conducted by a higher stratum than the hos-

pital ward and models comparing regional control strategies are essential. Network

models and structured community models could be used to predict large scale epi-

demic behaviour at this level.

Multi-resistant bacteria, previously the domain of hospitals, especially intensive care

units, are now seen increasingly in the community. Two examples are the emergence


of community acquired MRSA and the spread of VRE in the community in Europe,

linked to the use of avoparcin prior to its removal from the market in 1997 (Ridwan

et al., 2002). Future models of multi-resistant pathogens will need to address the

complex interaction between community, long term healthcare facilities and hospi-

tals.

Bibliography


Cooper, B., Lipsitch, M., 2004. The analysis of hospital infection data using hiddenMarkov models. Biostatistics 5 (2), 223–37.


Li, Y., Yu, I., Xu, P., Lee, J., Wong, T., Ooi, P., Sleigh, A., 2004. Predicting super spread-ing events during the 2003 severe acute respiratory syndrome epidemics in HongKong and Singapore. Am J Epi 160 (8), 719–728.

Ridwan, B., Mascini, E., van der Reijden, N., Verhoef, J., Bonten, M., 16 March 2002.What action should be taken to prevent spread of vancomycin resistant entero-cocci in European hospitals? Brit Med J 324, 666–668.

Whitby, M., McLaws, M., Ross, M., 2006. Why healthcare workers don’t wash theirhands: a behavioral explanation. Infect Control Hosp Epidemiol. 27 (5), 484–492.

mathematical and statistical modelling of infectious ... · emma mcbryde mbbs (honours) university...

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