Photo: Marin County Fire Department, CA, USA

Models of disease spread and establishment in small-size

directed networks

Mathieu Moslonka-Lefebvre, Marco Pautasso & Mike Jeger

Imperial College London, Silwood Park, UK

Rutgers University, March 2009

From: Hufnagel, Brockmann & Geisel (2004) Forecast and control of epidemics in a globalized world. PNAS 101: 15124-15129

number of passengers per day

Disease spread in a globalized world

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, Pautasso, Holdenrieder & Shaw (2007) New Phytologist

P. ramorumconfirmations on

the US West Coast vs. national risk

Map from www.suddenoakdeath.orgKelly, UC-Berkeley

Hazard map: Frank Koch & Bill Smith, 3rd SOD Science

Symposium (2007)

from: McKelvey, Koch & Smith (2007) SOD Science Symposium III

168 historic gardens/ woodlands

Phytophthora ramorum in England & Wales (2003-2006)

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

Climatic match courtesy of Richard Baker, CSL, UK

85

426

46

122

2003-Jun 2008

511 nurseries/ garden centres

2003-Jun 2008

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

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)]}

Epidemic threshold and network structure

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 0.05 0.10 0.15 0.20 0.25 0.30

probability of transmission

prob

abili

ty o

f per

sist

ence local

small-world

random

scale-free

Lower epidemic threshold for scale-free networks

from: Pautasso & Jeger (2008) Ecological Complexity

Epidemic does not develop

Epidemic develops

Connectance, in-out correlations

and clustering

Correlation of number of links in and number of links out for wholesalers/retailers

Courtesy of Tom Harwood

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, Pautasso & Jeger (submitted)

(a) (b)

(c) (d)

from: Moslonka-Lefebvre et al. (submitted)

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. (submitted)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

local random

small-world scale-free 2

scale-free 0 scale-free 1

thre

shol

d pr

obab

ility

of t

rans

mis

sion

clustering coefficient

(100 links) (200)

(400) (1000)

from: Moslonka-Lefebvre et al. (submitted)

Starting node and epidemic final size

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 epidemicfrom: Pautasso, Moslonka-Lefebvre & Jeger (submitted)

-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

from: Pautasso et al. (submitted)

0.0

0.2

0.4

0.6

0.8

1.0

100 200 400 1000links

corr

elat

ion

coef

fici

ent b

etw

een

epid

emic

fin

al s

ize

(0.0

1) a

nd o

ut-

degr

ee o

f st

artin

g no

de localrandomswsf2sf0sf1

A

B B

CED

A A

BC DE

A BC

DDE E

C CA

E

BD

from: Pautasso et al. (submitted)

links

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

local

random sw sf2 sf0 sf1

corr

elat

ion

betw

een

epid

emic

fi

nal s

ize

(sum

) and

in-d

egre

e of

the

star

ting

node

1002004001000

A

D CB

ABBB

A

A

DB

C

B

CD

A

DC B

D

C

A

B

from: Pautasso et al. (submitted)

-0.80-0.60-0.40-0.200.000.200.400.600.801.00

100 200 400 1000

links

corr

elat

ion

coef

fici

ent b

etw

een

epid

emic

fin

al s

ize

(0.0

1) a

nd

in-d

egre

e of

sta

rtin

g no

de

localrandomswsf2sf0sf1

A

B CDE E D

BC

A

E E DBC

A

EF CB B

A

E D

from: Pautasso et al. (submitted)

Main results

1. lower epidemic threshold for scale-free networks

2. in-out correlation more important than clustering

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

4. implications for the horticultural trade

Photo: Marin County Fire Department

ReferencesChiari C, Dinetti M, Licciardello C, Licitra G & Pautasso M (2010) Urbanization and the more-individuals hypothesis. Journal of Animal Ecology 79: 366-371Dehnen-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 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 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: 1300-1309Pecher C, Fritz S, Marini L, Fontaneto D & Pautasso M (2010) Scale-dependence of the correlation between human population and the species richness of stream macroinvertebrates. Basic Applied Ecology 11: 272-280Xu 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