Networks and Epidemiology

Mike Jeger & Marco Pautasso,Division of Biology, Imperial College London,

Wye Campus, Kent, UK

APS, CPS & MSA Joint Meeting,Quebec City, Jul 31, 2006

Networks and Epidemiology

1. Introduction: interconnected world, growing interest in network theory and disease spread in networks

2. Examples of recent work modelling disease (i) spread and (ii) control in networks of various kind

4. Conclusion: call for enhanced use of network theory in plant pathology

3. Case study: Phytophthora ramorum and epidemiological simulations in networks of small size

Networks are formed by:

• physical structures

• associations/relationships

• processes/flows on a structure

Armillaria rhizomorph network near Wageningen, Netherlands

From: Lamour et al. (submitted to FEMS Microbiology Ecology)

from Lazaro et al. 2005, Bird-made fruit orchards in northern Europe: nestedness and network properties. Oikos 110: 321-329

Plant-frugivores network in a Denmark forest

From: Hufnagel et al. (2004) Forecast and control of epidemics in a globalized world. PNAS 101: 15124-15129

number of passengers per day

From: Pautasso & Jeger (submitted)

Epidemic spread of studies applying network theory

2001

2004

2002

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20052006

2005

200520052003

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20052004

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20062005

2005 2005

200520052005

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NATURAL

TECHNOLOGICAL SOCIAL

food webs

airport networks

cell metabolism

neural networks

railway networks

ant nests

WWWInternetelectrical

power grids

software mapscomputing

grids

E-mail patterns

innovation flows

telephone callsco-authorship

nets

family networks

committees

sexual partnerships DISEASE

SPREAD

Food web of Little Rock Lake, Wisconsin, US

Internet structure

Network pictures from: Newman (2003) The structure and function of complex networks. SIAM Review 45, 2: 167-256

HIV spread

network

Epidemiology just one of the many applications of network theory

urban road networks

Networks and Epidemiology

1. Introduction: interconnected world, growing interest in network theory and disease spread in networks

2. Examples of recent work modellingdisease (i) spread and (ii) control in networks of various kinds

4. Conclusion: call for enhanced use of network theory in plant pathology

3. Case study: Phytophthora ramorum and epidemiological simulations in networks of small size

Different types of networks

Modified from: Keeling & Eames (2005) Networks and epidemic models. Interface 2: 295-307

random scale-free

local small-world

Epidemic development in different types of networks

scale-freerandom2-D lattice rewired2-D lattice1-D lattice rewired1-D lattice

From: Shirley & Rushton (2005) The impacts of network topology on disease spread. Ecological Complexity 2: 287-299

N of nodes of networks = 500;p of infection = 0.1;

latent period = 2 time steps;infectious period = 10 time steps

Clustering vs. path length

Modified from: Roy & Pascual (2006) On representing network heterogeneities in the incidence rate of simple epidemic models. Ecological Complexity 3, 1: 80-90

randomlocal small-world

local small-world random

From: Keeling (2005) The implications of network structure for epidemic dynamics. Theoretical Population Biology 67: 1-8

Simulations of a wide variety of networks with

average of 10 contacts

per individuals

Initial R0

Asymptotic R0

Reproductive ratio R0 in networks of differing degree of clustering

random local(C/Cmax)

From: Kiss et al. (2005) Disease contact tracing in random and clustered networks. Proceedings of the Royal Society B, 272: 1407-1414

(a) low clustering

Epidemic control in networks with low vs. high clustering

(b) high clustering

average number of connections per node = 10

From: Shirley & Rushton (2005) Where diseases and networks collide: lessons to be learnt from a study of the 2001 foot-and-mouth disease

epidemic. Epidemiology & Infection 133: 1023-1032

Super-connected individuals in scale-free networks

A reconstruction of the recent UK foot-and-mouth disease

epidemic (20 Feb–15 Mar 2001).

Vertices marked with a label are livestock markets,

unmarked vertices are farms.

Only confirmed infected premises are included.

Arrows indicate route of infection.

From: Shirley & Rushton (2005) Where diseases and networks collide: lessons to be learnt from a study of the 2001 foot-and-mouth disease

epidemic. Epidemiology & Infection 133: 1023-1032

Degree distribution of nodes in a scale-free network

The degree distribution of a reconstruction of the

UK foot-and mouth disease network.

Fitted line: y= 118.5x -1.6,

R2 = 0.87

From: May (2006) Network structure and the biology of populations. Trends in Ecology & Evolution, in press

uniform degree distribution

scale-free network with P(i) ≈ i-3

Fraction of population infected (l) as a function of ρ0

ρ0 is coincident with R0

for a uniform degree distribution;

for a scale-free network, theory says that

R0 = ρ0 + [1 + (CV)2], where CV is the

coefficient of variation of the degree distribution

From: Eames & Keeling (2003) Contact tracing and disease control. Proceedings of the Royal Society B 270: 2565-2571

Critical tracing efficiency to control an SIS-type epidemic in a network with uniform degree distribution

Connectivity loss in the North American power grid due to the removal of transmission substations

From: Albert et al. (2004) Structural vulnerability of the North American power grid. Physical Review E 69, 025103

transmission nodes removed (%)

Networks and Epidemiology

1. Introduction: interconnected world, growing interest in network theory and disease spread in networks

2. Examples of recent work modelling disease (i) spread and (ii) control in networks of various kinds

4. Conclusion: call for enhanced use of network theory in plant pathology

3. Case study: Phytophthora ramorumand epidemiological simulations in networks of small size

Photo: Marin County Fire DepartmentMarin County, CA, US (north of San Francisco)

Sudden Oak Death in California

Source: United States Department of Agriculture, Animal and Plant Health Inspection Service, Plant Protection and Quarantine

Trace forward/back zipcode

Positive (Phytophthora ramorum) site

Hold released

Trace-forwards and positive detections across the USA, July 2004

European garden & nursery finds

Photos: Hans DeGruyter, Netherlands Plant Protection InstitutePhytophthora ramorum infection on Rhododendron in Europe

UK: records positive to Phytophthora ramorum;

n = 2788

Jan 2003-Dec 2005

Data source: Department for Environment, Food and Rural Affairs, UK

UK, 2003-2005; n = 2788

Records positive to P. ramorum

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50

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Jan-03

Apr-03

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Oct-05

n of

reco

rds

unclear which

estates/environment

nurseries/gardencentres

Data source: Department for Environment, Food and Rural Affairs, UK

Own epidemiological investigations in four basic types of directed networks of small size

SIS-model;N nodes = 100; n links = 369; directed networks;

probability of infection for the node x at time t+1 = Σ px,y iy where px,y is the probability of connection between node x and y, and iy is the infection status of the node y at time t;

20 replicates for each type of network

(a) local; (b) small world;

(c) random; (d) scale-free

(a) (b)

(c) (d)

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

random network nr 8;starting node = nr 80

scale-free network nr 2; starting node = nr 11

local network nr 6; starting node = nr 100

small-world network nr 4;starting node = nr 14

sum

pro

babi

lity

of in

fect

ion

acro

ss a

ll no

des

iteration iteration

% n

odes

with

pro

babi

lity

of in

fect

ion

> 0.

01

0.00

0.25

0.50

0.75

1.00

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

probability of transmission

prob

abili

ty o

f per

sist

ence

localsmall-worldrandomscale-free

epidemic develops

no epidemic

Linear epidemic threshold on a graph of the probability of persistence and of transmission

0.000

0.100

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0.300

0.400

0.500

-0.500 0.000 0.500 1.000

correlation coefficient between number of links to and links from nodes

thre

shol

d (p

of t

rans

mis

sion

bet

wee

n no

des)

localsmall worldrandomscale-free (one way)scale-free (two ways)

probability of persistence = 0

Lower epidemic threshold for higher correlation coefficient between links to and links from nodes

scale-free network nr 8

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local network nr 2

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starting node

% n

odes

at e

quili

briu

m w

ith p

roba

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y of

infe

ctio

n >

0.01

starting node

random network nr 9

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small world network nr 6

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Marked variations in the final size of the epidemic at threshold conditions depending on the starting point

a b

dc

Further developments of these simulations?

• effect on these relationships of number of links/size of networks?

• integration in simulations of different sizes of nodes and of a dynamic contact structure?

• migration of network theory into GIS with spatially explicit network modellingof epidemics?

• applications in the control of Phytophthora ramorum spread?

Spatially-explicit modelling framework

UK- distribution centres of tree nurseries from

Hort Week suppliers guide 2003; n = 476

kindly provided by Tom Harwood

Sites Distribution Centres Incoming material Outgoing material

Further developments of these simulations?

• effect on these relationships of number of links/size of networks?

• integration in simulations of different sizes of nodes and of a dynamic contact structure?

• migration of network theory into GIS with spatially explicit network modellingof epidemics?

• applications in the control of Phytophthora ramorum spread?

Scale-free properties in the database of sites tested positive to Phytophthora ramorum, UK (2002-2005)

0.0

0.5

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1-4 5-49 50-284

n of positive P. ramorum records in database

log 1

0 num

ber o

f sit

es

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1-4 5-49 50-499 500-4999 5000-

total amount plants affected by P. ramorum

log 1

0 of n

of r

ecor

dsScale-free properties in the database of sites tested positive to Phytophthora ramorum, UK (2002-2005)

Networks and Epidemiology

1. Introduction: interconnected world, growing interest in network theory and disease spread in networks

2. Examples of recent work modelling disease spread and control in networks of various kinds

4. Conclusion: call for enhanced use of network theory in plant pathology

3. Case study: Phytophthora ramorum and epidemiological investigations in networks of small size

PLANT

HUMAN ANIMAL

HIVMycoplasmapneumoniae

avian fluDengueSARS

foot and mouth disease

fish diseases

bovine tuberculosisRotavirus

Where are the applications to plant pathology?

Neisseriagonorrhoeae

raccoon rabies

computer viruses

(rumor propagation)

(plant-pollinator

interactions)

(plant meta-populations)

(plant-frugivore

interactions)(bats in networks of

hollow trees)

(mycorrhiza) (plant metabolomics –

cellular pathways)

[plant-vector interactions e.g. viruses]

[nursery networks]

[quarantine][epiphytoticsmanagement

& control]

[recreation/ amenities landscape]

LEGEND:

no brackets = application existing

(…) = application existing, but not strictly involving disease

[…] = would involve plant pathology, but application of network theory lacking

Possible reasons for delay in the application of network thinking to plant pathology

• Lack of data on network structure in plant epidemics relative to human and animal ones?

• Homogeneous mid-field conditions more than adequate for plant diseases?

• Just lagging behind? Clustering effects may have slowed down the spread of the concept into this meta-population?

AcknowledgementsMike Shaw & Tom Harwood, Univ. of Reading, UK

Xiangming Xu, East Malling Research, UK

Ottmar Holdenrieder, ETHZ, CH

Sandra Denman, Forest Research, Alice Holt, UK

Judith Turner, Central Science Laboratory, York, UK

Department for Environment, Food and Rural Affairs, UK

ReferencesDehnen-Schmutz K, Holdenrieder O, Jeger MJ & Pautasso M (2010) Structural change in the international horticultural industry: some implications for plant health. Scientia Horticulturae 125: 1-15Harwood TD, Xu XM, Pautasso M, Jeger MJ & Shaw M (2009) Epidemiological risk assessment using linked network and grid based modelling: Phytophthora ramorum and P. kernoviae in the UK. Ecological Modelling 220: 3353-3361 Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126Jeger MJ, Pautasso M, Holdenrieder O & Shaw MW (2007) Modelling disease spread and control in networks: implications for plant sciences. New Phytologist 174: 179-197 Lonsdale D, Pautasso M & Holdenrieder O (2008) Wood-decaying fungi in the forest: conservation needs and management options. European Journal of Forest Research 127: 1-22 MacLeod A, Pautasso M, Jeger MJ & Haines-Young R (2010) Evolution of the international regulation of plant pests and challenges for future plant health. Food Security 2: 49-70 Moslonka-Lefebvre M, Pautasso M & Jeger MJ (2009) Disease spread in small-size directed networks: epidemic threshold, correlation between links to and from nodes, and clustering. J Theor Biol 260: 402-411Moslonka-Lefebvre M, Finley A, Dorigatti I, Dehnen-Schmutz K, Harwood T, Jeger MJ, Xu XM, Holdenrieder O & Pautasso M (2011) Networks in plant epidemiology: from genes to landscapes, countries and continents. Phytopathology 101: 392-403Pautasso M (2009) Geographical genetics and the conservation of forest trees. Perspectives in Plant Ecology, Systematics & Evolution 11: 157-189Pautasso M (2010) Worsening file-drawer problem in the abstracts of natural, medical and social science databases. Scientometrics 85: 193-202Pautasso M & Jeger MJ (2008) Epidemic threshold and network structure: the interplay of probability of transmission and of persistence in directed networks. Ecological Complexity 5: 1-8Pautasso M et al (2010) Plant health and global change – some implications for landscape management. Biological Reviews 85: 729-755Pautasso M, Moslonka-Lefebvre M & Jeger MJ (2010) The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks. Ecological Complexity 7: 424-432 Pautasso M, Xu XM, Jeger MJ, Harwood T, Moslonka-Lefebvre M & Pellis L (2010) Disease spread in small-size directed trade networks: the role of hierarchical categories. Journal of Applied Ecology 47: 1300-1309Xu XM, Harwood TD, Pautasso M & Jeger MJ (2009) Spatio-temporal analysis of an invasive plant pathogen (Phytophthora ramorum) in England and Wales. Ecography 32: 504-516