Epidemiology of complex networks

Marco Pautasso,Division of Biology,

Imperial College London, Wye Campus, Kent, UK

Universität Bayreuth,25 Jan 2007

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

Phytophthora alni along water courses in Bayern

Modified from: Holdenrieder, Pautasso, Weisberg & Lonsdale (2004) Tree diseases and landscape processes: the challenge of landscape pathology. Trends in Ecology & Evolution 19, 8: 446-452

From: Jung & Blaschke (2004) Phytophthora root and collar rot of alders in Bavaria: distribution, modes of spread and possible management strategies. Plant Pathology 53: 197–208

10 km

Web of susceptible genera connected by Phytophthora ramorum (based on genus co-existence in 2788 positive findings in England & Wales, 2003-2005)

Rhodo-dendron

Magnolia

Fagus

Castanea Taxus

Festuca

Laurus

Umbellularia

Drimys

Leucothoe

Kalmia

Parrotia

Syringa

Hamamelis

CamelliaViburnum

Pieris

Quercus

From: Pautasso, Harwood, Shaw, Xu & Jeger (2007) Epidemiological modeling of Phytophthora ramorum: network properties of susceptible plant genera movements in the UK nursery sector. Accepted for the Sudden Oak Death Science Symposium III, Santa Rosa, CA, US

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: 167-256

HIV spread

network

Epidemiology is just one of the many applications of network theory

urban road networks

Modified from: Jeger, Pautasso, Holdenrieder & Shaw (in press) Modelling disease spread and control in complex networks: implications for plant sciences. New Phytologist

Epidemic spread of studies applying network theory

2001

2004

2002

2004

2005

20052006

2005

200520052003

2004

2003

2003

2006

20052004

2005

20062005

2005 2005

200520052005

2004

2005

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: further potential work applying network theory in biogeographic modelling

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

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

based on 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 21, 7: 394-399

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

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: further potential work applying network theory in biogeographic modelling

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

Map courtesy of Ross Meentemeyer

Sudden Oak Death ground survey, Northern California, 2004

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

Vascular plant species richness as a function of human population size in US counties

From: Pautasso & McKinney (in review) The botanist effect revisited: plant species richness, county area, and human population size in the United States. Conservation Biology

P. ramorum: an aggressive AND generalist pathogen

Modified from: Pautasso, Holdenrieder & Stenlid (2005) Susceptibility to fungal pathogens of forests differing in tree diversity. Scherer-Lorenzen, Körner & Schulze (eds)

Forest Diversity and Function: Temperate and Boreal Systems. Ecological Studies, 176: 263-289

Acer macrophyllum, Aesculuscalifornica, Lithocarpus densiflorus, Quercus agrifolia, Quercus kelloggii, Quercus chrysolepis, Quercus parvula,

Pseudotsuga menziesii, Sequoia sempervirens

England and Wales: records positive to Phytophthora ramorum

n = 2788

Jan 2003-Dec 2005

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

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

SIS-modelN nodes = 100 constant n of linksdirected 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

local small-world

random scale-free

from: Pautasso & Jeger (in review) Epidemic threshold and network structure: the interplay of probability of transmission and of persistence. Ecological Complexity

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)

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

from: Pautasso & Jeger (in review) Ecological Complexity

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

from: Pautasso & Jeger (in review) Ecological Complexity

0.000

0.100

0.200

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

from: Pautasso & Jeger (in preparation) Proceedings Royal Society B

scale-free network nr 8

0

25

50

75

100

0 25 50 75 100

local network nr 2

0

25

50

75

100

0 25 50 75 100

% n

odes

at e

quili

briu

m w

ith p

roba

bilit

y of

infe

ctio

n >

0.01

random network nr 9

0

25

50

75

100

0 25 50 75 100

small world network nr 6

0

25

50

75

100

0 25 50 75 100

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

a b

dc

from: Pautasso & Jeger (in preparation) Proceedings Royal Society B

starting node starting node

Temporal development; England & Wales, 2003-2005; n = 2788

R ecords positive to P. ram orum

0

50

100

150

200

250

Jan-03Apr-0

3Ju

l-03

Oct-03

Jan-04Apr-0

4Ju

l-04

Oct-04

Jan-05Apr-0

5Ju

l-05

Oct-05

n of

reco

rds

unclear which

estates/environm ent

nurseries/gardencentres

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

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 modelling of epidemics

Local Trade

Heathland

Woodland

Spatially-explicit modelling framework

Long-distance tradeClimate suitability

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: further potential work applying network theory in biogeography

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

Further potential work applying network theory in biogeographic modelling

• conservation biology (e.g. meta-populations, reserve networks, botanical gardens)

• invasion ecology (for exotic organisms particularly when spread by the nursery trade)

• plenty of open questions of mathematical interest, to be addressed using theoretical analyses, but also numerical simulations

Acknowledgements

Mike Jeger, Imperial College,

Wye, UKMike Shaw,

Univ. of Reading, UK

Kevin Gaston, Univ. of

Sheffield, UK

Ottmar Holdenrieder,

ETHZ, CH

Emanuele Della Valle, Politecnico di

Milano, ItalyKatrin

Boehning-Gaese,

Univ. Mainz

Peter Weisberg, Univ. of Nevada,

Reno, US

Mike McKinney, Univ. of Tennessee, US

Chris Gilligan, Univ. of Cambridge, UK

ReferencesJokimäki J, Kaisanlahti-Jokimäki M-L, Suhonen J, Clergeau P, Pautasso M & Fernández-Juricic E (2011) Merging wildlife community ecology and animal behavioral ecology for a better urban landscape planning. Landscape & Urban Planning 100: 383-385Moslonka-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, Böhning-Gaese K, Clergeau P, Cueto VR, Dinetti M, Fernandez-Juricic E, Kaisanlahti-Jokimäki ML, Jokimäki J, McKinney ML, Sodhi NS, Storch D, Tomialojc L, Weisberg PJ, Woinarski J, Fuller RA & Cantarello E (2011) Global macroecology of bird assemblages in urbanized and semi-natural ecosystems. Global Ecology & Biogeography 20: 426-436Barbosa AM, Fontaneto D, Marini L & Pautasso M (2010) Is the human population a large-scale indicator of the species richness of ground beetles? Anim Cons 13: 432-441Barbosa AM, Fontaneto D, Marini L & Pautasso M (2010) Positive regional species–people correlations: a sampling artefact or a key issue for sustainable development? Animal Conservation 13: 446-447Cantarello E, Steck CE, Fontana P, Fontaneto D, Marini L & Pautasso M (2010) A multi-scale study of Orthoptera species richness and human population size controlling for sampling effort. Naturwissenschaften 97: 265-271Chiari 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. ScientiaHorticulturae 125: 1-15Golding J, Güsewell S, Kreft H, Kuzevanov VY, Lehvävirta S, Parmentier I & Pautasso M (2010) Species-richness patterns of the living collections of the world's botanic gardens: a matter of socio-economics? Annals of Botany 105: 689-696MacLeod A, Pautasso M, Jeger M & Haines-Young R (2010) Evolution of the international regulation of plant pests & challenges for future plant health. Food Security 2: 49-70 Pautasso M (2010) Worsening file-drawer problem in the abstracts of natural, medical and social science databases. Scientometrics 85: 193-202Pautasso M & Pautasso C (2010) Peer reviewing interdisciplinary papers. European Review 18: 227-237Pautasso M & Schäfer H (2010) Peer review delay and selectivity in ecology journals. Scientometrics 84: 307-315Pautasso M, Dehnen-Schmutz K, Holdenrieder O, Pietravalle S, Salama N, Jeger MJ, Lange E & Hehl-Lange S (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: 272-280Harwood 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 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. Journal of Theoretical Biology 260: 402-411

References (bis)Pautasso M (2009) Geographical genetics and the conservation of forest trees. Perspectives in Plant Ecology, Systematics and Evolution 11: 157-189Pautasso M & Dinetti M (2009) Avian species richness, human population and protected areas across Italy’s regions. Environmental Conservation 36: 22-31Pautasso M & Powell G (2009) Aphid biodiversity is correlated with human population in European countries. Oecologia 160: 839-846Pautasso M & Zotti M (2009) Macrofungal taxa and human population in Italy's regions. Biodiversity & Conservation 18: 473-485Xu XM, Harwood TD, Pautasso M & Jeger MJ (2009) Spatio-temporal analysis of an invasive plant pathogen (Phytophthora ramorum) in England and Wales. Ecography 32: 504-516Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126Jeger MJ & Pautasso M (2008) Plant disease and global change – the importance of long-term data sets. New Phytologist 177: 8-11Lonsdale 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 Pautasso M & Chiarucci A (2008) A test of the scale-dependence of the species abundance-people correlation for veteran trees in Italy. Annals of Botany 101: 709-715 Pautasso M & Fontaneto D (2008) A test of the species-people correlation for stream macro-invertebrates in European countries. Ecological Applications 18: 1842-1849Pautasso 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 & Weisberg PJ (2008) Density-area relationships: the importance of the zeros. Global Ecology and Biogeography 17: 203-210Schlick-Steiner B, Steiner F & Pautasso M (2008) Ants and people: a test of two mechanisms behind the large-scale human-biodiversity correlation for Formicidae in Europe. J of Biogeography 35: 2195-2206Steck CE & Pautasso M (2008) Human population, grasshopper and plant species richness in European countries. Acta Oecologica 34: 303-310Jeger MJ, Pautasso M, Holdenrieder O & Shaw MW (2007) Modelling disease spread and control in networks: implications for plant sciences. New Phytologist 174: 179-197 Pautasso M (2007) Scale-dependence of the correlation between human presence and plant/vertebrate species richness. Ecology Letters 10: 16-24 Pautasso M & McKinney ML (2007) The botanist effect revisited: plant species richness, county area and human population size in the US. Conservation Biology 21, 5: 1333-1340 Pautasso M & Parmentier I (2007) Are the living collections of the world’s botanical gardens following species-richness patterns observed in natural ecosystems? BotanicaHelvetica 117: 15-28 Pautasso M & Gaston KJ (2006) A test of the mechanisms behind avian generalized individuals-area relationships. Global Ecology and Biogeography 15: 303-317 Pautasso M & Gaston KJ (2005) Resources and global avian assemblage structure in forests. Ecology Letters 8: 282-289Pautasso M, Holdenrieder O & Stenlid J (2005) Susceptibility to fungal pathogens of forests differing in tree diversity. In: Forest Diversity and Function (Scherer-Lorenzen M, Koerner Ch & Schulze D, eds.). Ecol. Studies Vol. 176. Springer, Berlin, pp. 263-289 Holdenrieder O, Pautasso M, Weisberg PJ & Lonsdale D (2004) Tree diseases and landscape processes: the challenge of landscape pathology. Trends in Ecology and Evolution 19, 8: 446-452

Networks and Epidemiology

Marco Pautasso,Division of Biology,

Imperial College London, Wye Campus, Kent, UK

Universität Bayreuth,25 Jan 2007

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

path length

clustering

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, Green & Kao (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: 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, Albert & Nakarado (2004) Structural vulnerability of the North American power grid. Physical Review E 69, 025103

transmission nodes removed (%)