Disease spread in small-size directed networks

Marco Pautasso, Mathieu Moslonka-Lefebvre, & Mike Jeger - Imperial College London, Silwood Park

Bath University, 2nd July 2009

Outline of the talk

1. why small-size networks?

2. case study: Phytophthora ramorum

3. simulations of disease spread in small-size directed networks

4. conclusions

Hufnagel et al. (2004) Forecast and control of epidemics in a globalized world. PNAS

number of passengers per day

Disease spread in a globalized world

Matisoo-Smith et al. (1998) Patterns of prehistoric human mobility in Polynesia indicated by mtDNA from the Pacific rat. PNAS

Understanding human mobility patterns

Vendramin et al. (2008) Genetically depauperate but widespread: the case of an emblematic Mediterranean pine. Evolution

Understanding plant mobility patterns

Dunne et al. (2002) Food-web structure and network theory:the role of connectance and size. PNAS

Food webs: an example of small-size networks

Outline of the talk

1. why small size-networks?

2. case study: Phytophthora ramorum

3. simulations of disease spread in small-size directed networks

4. conclusions

from: Rizzo et al. (2005) Annual Reviews of Phytopathology, Photo: Susan Frankel

P. ramorum in Monterey County, California

P. ramorumconfirmations on

the US West Coast vs. national risk

Map from www.suddenoakdeath.orgKelly, UC-Berkeley

Hazard map: Koch & Smith,

3rd SOD Science Symposium (2007)

from: McKelvey et al. (2007) SOD Science Symposium III

nurseries& garden

centres

gardens/woodlands

Phytophthora ramorum in England & Wales (2003-2008)

Outbreak maps courtesy of David Slawson, PHSI, DEFRA, UK

Climatic match courtesy of Richard Baker, CSL, UK

Outline of the talk

1. why small-size networks?

2. case study: Phytophthora ramorum

3. simulations of disease spread in small-size directed networks

4. conclusions

step 1

step 2

step 3

step n

…

Simple model of infection spread (e.g. P. ramorum) in a network

pt probability of infection transmission

pp probability of infection persistence

… 100node 1 2 3 4 5 6 7 8

1. spread in theornamental plant trade

(asymmetric)

Features of the P. ramorum pathosystem → model

2. garden centres/plant nurseries are not just either

susceptible or infected

3. nurseries at risk even after eradication

if still trading susceptible spp

asymmetry in the adjacency matrices (directed networks)

0 < pi < 1 (continuum model)

absence of removal/immunization

(SIS model)

The four basic types of network structure used

local

random

small-world

scale-free

SIS Model, 100 Nodes, directed networks, P [i (x, t)] = Σ {p [s] * P [i (y, t-1)] + p [p] * P [i (x, t-1)]}

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 26 51 760

10

20

30

40

50

60

70

80

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 26 51 760

5

10

15

20

25

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 26 51 760

10

20

30

40

50

60

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1 51 101 151 2010

5

10

15

20

25

30

35

40

Examples of epidemic development in four kinds of directed networks of small size (at threshold conditions)

local

sum

pro

babi

lity

of in

fect

ion

acro

ss a

ll no

des

randomscale-free

% n

odes

with

pro

babi

lity

of in

fect

ion

> 0.

01

from: Pautasso & Jeger (2008) Ecological Complexity

small-world

0.00

0.25

0.50

0.75

1.00

0.00 0.25 0.50 0.75 1.00

probability of transmission

prob

abili

ty o

f per

sist

ence

localrandomsmall-worldscale-free (two-way)scale-free (uncorrelated)scale-free (one way)

Lower epidemic threshold for scale-free networks with positive correlation between in- and out-degree

modified from: Pautasso & Jeger (2008) Ecological Complexity

Epidemic does not develop Epidemic develops

Lower epidemic threshold for two-way scale-free networks (unless networks are sparsely connected)

N replicates = 100; error bars are St. Dev.; different letters show sign. different means

at p < 0.05

from: Moslonka-Lefebvre et al. (in press) Journal of Theoretical Biology

0.0

0.2

0.4

0.6

0.8

1.0

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

local random

small-world scale-free 2

scale-free 0 scale-free 1

thre

shol

d pr

obab

ility

of t

rans

mis

sion

correlation coefficient between in- and out-degree

(100) (200 links)

(400) (1000 links)

from: Moslonka-Lefebvre et al. (in press) Journal of Theoretical Biology

0

25

50

75

100

0 25 50 75 1000

25

50

75

100

0 25 50 75 100

0

25

50

75

100

0 25 50 75 100

epid

emic

fina

l siz

e (N

of n

odes

with

infe

ctio

n st

atus

> 0

.01)

0

2 5

5 0

7 5

1 0 0

0 2 5 5 0 7 5 1 0 0

(local) (sw)

(rand) (sf2)

0

2 5

5 0

7 5

1 0 0

0 2 5 5 0 7 5 1 0 00

25

50

75

100

0 25 50 75 100

(sf0) (sf1)

starting node of the epidemic

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8

-1 .0

0 .0

1 .0

-1 0 1 2 3

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

0 1 2 3 4 5 6

sum

at e

quili

briu

m o

f inf

ectio

n st

atus

ac

ross

all

node

s (+

0.01

for s

fnet

wor

ks)

local

rand sf2 (log-log)

n of links from starting node n of links from starting node

sw

sf0 (log-log) sf1 (log-log)

Correlation of epidemic final size with out-degree of starting node increases with network connectivity

N replicates = 100; error bars are St. Dev.; different letters show sign. different means at p < 0.05

Conclusions

1. lower epidemic threshold for two-way scale-free networks

2. importance of the in-out correlation

3. out-degree as a predictor of epidemic final size

4. implications for biological invasions

Contemporary ornamental

trade patterns

From International Statistics Flower and Plants 2004, Institut

fuer Gartenbau-oekonomie der

UniversitaetHannover, Germany

NATURAL

TECHNOLOGICAL SOCIAL

food webs

airport networks

cell metabolism

neural networks

railway networks

ant nests

WWWInternet

electrical power grids

software mapscomputing

gridsE-mail

patterns

innovation flows

telephone calls

co-authorship nets

family networks

committees

sexual partnerships DISEASE

SPREAD

Food web of Little Rock Lake, Wisconsin, US

Internet structure

Network pictures from: Newman (2003) SIAM Review

HIV spread

network

Epidemiology is just one of the many applications of network theory

urban road networks

modified from: Jeger et al. (2007) New Phytologist

Acknowledgements

Ottmar Holdenrieder,

ETHZ, CH

Mike Shaw, University of

Reading

Alan Inman,

DEFRA

Joan Webber, Forest Research,

Farnham

Tom Harwood,

CEP, Imperial College

Jennifer Parke, Univ. of Oregon

Xiangming Xu, East Malling

Research

Richard Baker, CSL

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