Modeling with Data

Jon ZelnerPostdoctoral Fellow, Ecology & Evolutionary Biology @

Princeton University

Postdoctoral Fellow, NIH Fogarty International Center Research and Policy for Infectious Disease Dynamics (RAPIDD) Program

Data in the Modeling Process

Model

Agents

Environment Simulated Behavior

Observed Behavior

Pattern Oriented Modeling (POM)

� Term coined by Grimm et al. in 2005 Science paper

� Modeling process should be guided by patterns of interest� Can use patterns @ multiple levels:

� Individual agents� Environment� Aggregate agent behavior

� Patterns should be used both to guide model development and to calibrate and validate models.

Types of data for modelers� Counts/Proportions:

� Infections� Occupied patches

� Distributions� Age� Lifespan� Duration of infection

� Rates� Birthrates� Transmission rate

� Time Series:� Evolution of outbreak in

time� Timeline of conflict� Number of firms over

time

� Qualitative:� ‘Norovirus-like’

outbreaks� Size and shape of forest

patches� Clusters of settlements

Pattern Oriented Modeling

Pattern Oriented Modeling: Kayenta Anasazi (Axtell et al. 2001)� Trying to understand

population growth and collapse among the Kayenta Anasazi in U.S. Southwest

� Many factors in this:� Weather� Farming� Kinship

� Optimize models by explaining multiple patterns @ one time.

Anasazi (cont’d)

Inference for POM

� Bayesian/Qualitative� Use some kind of quality function to score goodness of runs

and optimize by minimizing distance between model output and optimum quality and/or data.� Number of occupied patches� Size of elephant herds

� Frequentist/Likelihood-based� Define a likelihood function for Data | Model� Simulate runs from the model and evaluate likelihood of

data as (# runs == Data) / # runs

How Infections Propagate After Point-Source Events

An Analysis of Secondary Norovirus Transmission

Jon Zelnera,b, Aaron A. Kinga,c, Christine Moee & Joseph N.S. Eisenberga,d

University of Michigana Center for the Study of Complex Systems, b Sociology & Public Policy,

c Ecology & Evolutionary Biology, d School of Public Health

Emory Universitye Rollins School of Public Health

Norovirus (NoV) Epidemiology� Most common cause of non-bacterial

gastroenteritis in the US and worldwide.� Est. 90 million cases in 2007� Explosive diarrhea & projectile vomiting

in symptomatic cases.

� Single-stranded, non-enveloped RNA virus� Member of family Caliciviridae

� Often transmitted via food� Salad greens� Shellfish

� Most person-to-person transmission is via the environment and fomites.

Why model transmission after point-source events?� Typical analysis of point-source events

focuses on primary, one-to-many risk:� How many cases are created by an

infectious food handler?� How many people infected after water

treatment failure?

� However, actual size of point source events is underestimated without including secondary transmission risk.

� Within-household transmission is an important bridge between point-source events.� So, even if within-household Ro < 1,

household cases have important dynamic consequences at the community level.

H IS

S

IA

NoV Transmission Dynamics

� Norovirus transmission dynamics tend to be locally unstable but globally persistent.� E.g., small, explosive outbreaks in Mercer

County, but no local NoV epidemic� Multiple reported NoV outbreaks

throughout New Jersey every week.� Stochasticity operates at multiple levels.

� Disease/Contact

NoV Transmission Dynamics

Exponential Growth, Global Invasion

(e.g.,Pandemic Flu)

Short, Explosive & Limited

(Typical of NoV outbreaks)

Outbreak Data

� Gotz et al. (2001) observed 500+ households exposed to NoV after a point-source outbreak in a network of daycare centers in Stockholm, Sweden.� Traceable to salad prepared by a

food handler who was shedding post-symptoms.

� Followed 153 of these households� Eliminating those with only one

person.� 49 had secondary cases� 104 have no secondary cases

Deterministic SEIR model� Infinite population

� Mass-action mixing

� Frequency-dependent transmission

� When I > 0, a fraction of the susceptible population is infected at every instant� Constant average rate of recovery� Doesn’t matter who is infected � ‘Nano-fox’ problem !

€

dSdt

= −βSI

dEdt

= βSI −εE

dIdt

= εE − γI

dRdt

= γI

Why use a stochastic model?� Deterministic models work well

when assumptions are plausible, but are less useful when:� Populations are small:

� e.g.,Household outbreak

� Global contact patterns deviate from homogeneous mixing:� Social networks� Realistic behavior

� Disease natural history is not memoryless:� Recovery period is not

exponentially or gamma distributed

� Lots of variability in individual infectiousness

Exponential RV

Lognormal RV

Progression of NoV Infection

� Short incubation period (~1.5 days)

� Typical symptom duration around 1.5 days.� Exceptional cases up to a year have been reported.

� Most people shed asymptomatically after recovery of symptoms:� Typically for several days� Not uncommon for shedding to last > 1 month, year or more� 15-50% of all infections may be totally asymptomatic

Basic NoV Transmission Model for Household Outbreaks

� SEIR Transmission Model� Individuals may be in one of four states:

� Susceptible� Exposed/Incubating� Infectious� Recovered/Immune

� Multiple boxes in E & I states correspond to shape parameter of gamma distributed waiting times.

� Background infection parameter, α. (Fixed to 0.001/day)

� Although NoV immunity tends to be partial and short-lived, this model is adequate for analyzing short-lived outbreaks.

Analysis Objectives� Estimate daily person-to-person rate of infection (β).

� Estimate average effective duration of infection (1/γ) and shape parameter of gamma-distributed infectiousness duration.

� Effect of missing household sizes on results.

� Effect of asymptomatic infection.

� 0.14/infections per day

� 1.2 days; γs = 1

� Minimal

� .035 increase in β for each 10% increase in proportion of individuals who are asymptomatically infectious

What makes these data challenging to work with?� We want to understand:

� Daily person-to-person rate of infection (β).� Average effective duration of infection (1/γ).� Variability in 1/γ.� Generation of asymptomatic infections.

� But household data are noisy and only partially observed:� We know time of symptom onset but are missing:

� Time of infection� Time of recovery� Firm estimate of asymptomatic ratio & infectiousness� Household Sizes (!)

� Strength of these data are that we can treat each household as an independent trial of a random infection process.

Likelihood Function for Fully Observed Household Outbreaks

€

λ(Sij ,I ij ,β ,α) = Sij βI ij +α( )Force of Infection @ t

€

i, a = exp −λ(Sij ,I ij ,β ,α)(tj+1− tj )( )j=0

NQ −1

∏Likelihood of no infections over all infection-free intervals

Probability of all infections

€

i,b = λ(Sik,Iik,β,α)k=1

NK

∏

x = infection; = symptom onset; = recovery

Likelihood Function for Fully Observed Household Outbreaks

Likelihood of a household observation

€

i = i,a × i,b !

€

O = ii∈H∏Likelihood of all household

observations

x = infection; = symptom onset; = recovery

Likelihood Function for Fully Observed Household Outbreaks

€

λ(Sij ,I ij ,β ,α) = Sij βI ij +α( )Force of Infection @ t

€

i, a = exp −λ(Sij ,I ij ,β ,α)(tj+1− tj )( )j=0

NQ −1

∏Likelihood of no infections over all infection-free intervals

Probability of all infections

€

i,b = λ(Sik,Iik,β,α)k=1

NK

∏

Likelihood of a household observation

€

i = i,a × i,b !

€

O = ii∈H∏Likelihood of all household

observations

Unobserved Infection States

+ 104 Households w/ No Secondary Cases

Unobserved Infection States

� Use data augmentation to generate complete observations.

� For each symptom onset event (q):

� Draw incubation time, k, from distribution� Infection time, a = q – k� If you draw any a < 0, whole sample has

likelihood = 0.

� Draw recovery time, r, from symptom duration distribution.� If r > observation period, w:

� r = w� For right-censoring in data.

� Repeat for many (1K+) samples

Unobserved Infection States

x = infection; = symptom onset; = recovery

� Evaluate likelihood w/respect to β and α for each sample.� E(L) is estimated likelihood of data.

Unobserved Household Sizes

� Sizes of households in Stockholm outbreak are unknown.

� Expected number of cases is:� S(βI + a)Δt

� Missing S!

� Solution:� Assume exposed households are sampled at random from

the whole population.� For each augmented household time series, sample household

size from Swedish census distribution.� Save samples by setting a lower bound:

� Likelihood of outbreak with have fewer individuals than observed infections = 0, so don’t sample these.

Results: MLE Parameter Values and 95% Confidence Intervals

1/γ limited to values >= 1 day; infectiousness duration < 1 day not plausible

Results: Likelihood Surface

� Contour plot shows likelihood for combinations of β and 1/γ for γs = 1.

� Triangle is location of MLE; Dashed oval 95% confidence bounds

� Parameter space isn’t very large, optimize using brute force.

Goodness of fit� Simulate from SEIR model using fitted parameters

and same demographics as outbreak.

If :

!

" = #SI Draw number of new infections, x, from

!

Binomial(S,") S = S – x

E = E + x Draw symptom onset times from for all new infections.

t = t + dt At end of step:

Transition from

!

E "I those who have infectiousness onset time <= t.

Transition

!

I"R those who have recovery time <= t Else: STOP

If :

!

" = #SI Draw number of new infections, x, from

!

Binomial(S,") Draw number never symptomatic, a, from

!

Binomial(x,") S = S – (x-a)

E = E + (x-a) R = R + a Draw symptom onset times from for all new infections.

t = t + dt At end of step:

Transition from

!

E "I those who have infectiousness onset time <= t.

Transition

!

I"R those who have recovery time <= t Else: STOP

Goodness of fit� Simulate from SEIR model using

fitted parameters and same demographics as outbreak.

� Quantify model performance based on closeness to outbreak characteristics� Average number of infections in

households with secondary cases.� Simulated: 1.9, SD = 0.2� Stockholm: 1.6

� Average number of households with no secondary cases.� Simulated: 110.5, SD = 5.5� Stockholm: 104

!# of infections in households w/ 2-ary transmission

!# of households with zero secondary cases

Sensitivity Analysis:Household Sizes

� Want to understand the extent to which using sampled household sizes biases results.

� Simulate outbreaks with household sizes drawn from Swedish census distribution.� Estimate parameters using:

� Sampled household sizes� Known sizes from simulation

� Compare results.

Results: Sensitivity Analysis� Estimate parameters for outbreak with β = 0.14/day

and 1/γ = 1.2 days

� Dashed lines show fit when household sizes are known, solid are unknown.

� Results almost exactly the same.

Asymptomatic Infections� Problem: Only observed symptomatic infections

� Asymptomatics likely don’t contribute much to outbreaks in households with symptomatic cases, but can be infected during these outbreaks.� Are very important for seeding new outbreaks:

� Stockholm outbreak started by post-symptomatic food-handler� Afternoon Delight outbreak in Ann Arbor� Subway outbreaks in Kent County, MI

� Full analysis of asymptomatic infections requires active surveillance� e.g., Stool and environmental samples.

� Solution: Estimate parameters for outbreaks with varying levels of asymptomatic infection using simulated data.

Modeling asymptomatic infection

� π is proportion of new infections that are asymptomatic.� Assume asymptomatic infections are non-infectious during household

outbreak.

� Sample 20 outbreaks each for combinations of:� Β = {0.075,0.085,…,.2} � π = {0, .1,…,.5}

Modeling asymptomatic infection

If :

!

" = #SI Draw number of new infections, x, from

!

Binomial(S,") S = S – x

E = E + x Draw symptom onset times from for all new infections.

t = t + dt At end of step:

Transition from

!

E "I those who have infectiousness onset time <= t.

Transition

!

I"R those who have recovery time <= t Else: STOP

If :

!

" = #SI Draw number of new infections, x, from

!

Binomial(S,") Draw number never symptomatic, a, from

!

Binomial(x,") S = S – (x-a)

E = E + (x-a) R = R + a Draw symptom onset times from for all new infections.

t = t + dt At end of step:

Transition from

!

E "I those who have infectiousness onset time <= t.

Transition

!

I"R those who have recovery time <= t Else: STOP

Modeling asymptomatic infection

� π is proportion of new infections that are asymptomatic.� Assume asymptomatic infections are non-infectious during household

outbreak.

� Sample 20 outbreaks each for combinations of:� Β = {0.075,0.085,…,.2} � π = {0, .1,…,.5}

� Estimate parameters using data augmentation method.� Assume π = 0, as when fitting Stockholm data.

� Find expected value of β for each tau when estimated β = 0.14.

Modeling asymptomatic infection

Norovirus outbreaks in realistic communities� Norovirus has interesting qualitative outbreak dynamics in the

community.� Outbreaks are explosive but typically limited.� Multiple levels of transmission:

� Can embed findings about household transmission.

� Community rate of transmission is unknown.

� Data on community and region-level Norovirus outbreaks are rare.

� Take a pattern-oriented approach to building community-level models of NoV transmission.

� Build a model based on observed patterns and data that can recreate outbreaks with NoV-like characteristics.

Detailed Transmission Model

S E IS

IA1

R

IA2

(βIS*IS) + (βIA*IA)

NoV transmission is marked by heterogeneous asymptomatic infectious periods.

~5% of the population will shed for 100+ days.

Existing theory predicts that increasing variability in individual infectiousness makes outbreaks less predictable, but smaller on average.

Want to understand how this heterogeneity impacts outbreak dynamics in the context of heterogeneous contact structure.

Contact structure

� Household sizes:� Assume a representative community, i.e., household sizes are a

random sample from the census distribution of household sizes.

� Contacts in the community:� Individuals separated into compartments:

� School, work, etc

� Social network:� How do we choose a network topology that is useful and informative?

� Food handlers:� About 1% of U.S. adults are food handlers� Average norovirus point-source outbreak size is about 40

Empirical contact networks

� Many empirical community contact networks have an exponentially distributed degree.� Moderate

heterogeneity in contact

Outbreak Realizations

� = Household Transmission

� = Community Transmission

� = Point Source Event