This document discusses networks and epidemiology, with a focus on applying network theory to model disease spread. It provides examples of modeling disease spread in different types of networks, including local, small-world, random and scale-free networks. It also presents a case study on modeling the spread of Phytophthora ramorum in directed networks of small size. The conclusion discusses potential future work applying network theory and spatially-explicit modeling to problems in conservation biology, invasion ecology, and other areas of biogeography.

1. Epidemiology
of complex networks
Marco Pautasso,
Division of Biology,
Imperial College London,
Wye Campus, Kent, UK
Universität Bayreuth,
25 Jan 2007

2. Disease spread in
a globalized world
number of passengers per day
From: Hufnagel, Brockmann & Geisel (2004) Forecast and control
of epidemics in a globalized world. PNAS 101: 15124-15129

3. Phytophthora alni along water courses in Bayern
10 km
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
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

4. Web of susceptible genera connected by Phytophthora ramorum (based on
genus co-existence in 2788 positive findings in England & Wales, 2003-2005)
Viburnum
Camellia Umbellularia
Castanea Taxus
Syringa
Drimys
Fagus Rhodo-
dendron
Festuca
Hamamelis Quercus
Kalmia Pieris
Laurus Magnolia Parrotia
Leucothoe
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

5. Epidemiology is just one of the
many applications of network theory
Network pictures from:
Newman (2003) NATURAL
The structure and function
of complex networks. food webs
SIAM Review 45: 167-256
cell
metabolism
neural
networks Food web of Little Rock
ant nests Lake, Wisconsin, US
sexual
DISEASE partnerships
SPREAD
family
innovation networks
flows
Internet co-authorship HIV
structure railway nets spread
telephone calls
networks urban road network
electrical networks E-mail
committees
power grids airport Internet WWW patterns
computing networks
grids software maps
TECHNOLOGICAL SOCIAL

6. Epidemic spread of studies applying network theory
2005
2005
2005
2005 2005
2004
2005 2006
2004
2004
2001
2005
2002 2006
2004
2003 2005
2004
2005
2003 2005
2005
2005
2006
2005
2003
2005
Modified from: Jeger, Pautasso, Holdenrieder & Shaw (in press) Modelling disease spread
and control in complex networks: implications for plant sciences. New Phytologist

7. 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
3. Case study: Phytophthora ramorum and
epidemiological simulations in networks of small size
4. Conclusion: further potential work applying
network theory in biogeographic modelling

8. Different types of networks
local small-world
random scale-free
Modified from: Keeling & Eames (2005) Networks and epidemic models. Interface 2: 295-307

9. Epidemic development in different types of networks
scale-free
random
2-D lattice rewired
2-D lattice
1-D lattice rewired
1-D lattice
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) The impacts of network topology on disease spread.
Ecological Complexity 2: 287-299

10. 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

11. 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: 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

12. Fraction of population infected (l) as a function of ρ0
uniform degree
distribution
scale-free network
with P(i) ≈ i-3
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: May (2006) Network structure and the biology of populations.
Trends in Ecology & Evolution 21, 7: 394-399

13. 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
3. Case study: Phytophthora ramorum
and epidemiological simulations
in networks of small size
4. Conclusion: further potential work applying
network theory in biogeographic modelling

14. Sudden Oak Death
Marin County, CA, US
Photo: Marin County Fire Department (north of San Francisco)

15. Sudden Oak Death ground survey,
Northern California, 2004
Map courtesy
of Ross Meentemeyer

16. Trace-forwards and positive detections across the USA, July 2004
Trace forward/back zipcode
Positive (Phytophthora ramorum) site
Hold released
Source: United States Department of Agriculture,
Animal and Plant Health Inspection Service, Plant Protection and Quarantine

17. 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

18. P. ramorum: an aggressive AND generalist pathogen
Acer macrophyllum, Aesculus
californica, Lithocarpus densiflorus,
Quercus agrifolia, Quercus kelloggii,
Quercus chrysolepis, Quercus parvula,
Pseudotsuga menziesii,
Sequoia sempervirens
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

19. England and Wales: records positive
to Phytophthora ramorum
n = 2788
Jan 2003-Dec 2005
Data source: Department for Environment, Food and Rural Affairs, UK

20. Own epidemiological investigations in four
basic types of directed networks of small size
local small-
world SIS-model
N nodes = 100
constant n of links
directed networks
probability of infection
for the node x at time
random scale- t+1 = Σ px,y iy where
free 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
from: Pautasso & Jeger (in review) Epidemic threshold and network structure:
the interplay of probability of transmission and of persistence. Ecological Complexity

21. Examples of epidemic development in four kinds of
directed networks of small size (at threshold conditions)
sum probability of infection across all nodes
1.2 40 1.2 25
35
% nodes with probability of infection > 0.01
1.0 1.0
20
small-world network nr 4;
30
0.8 0.8
25
starting node = nr 14 15
0.6 20 0.6
10
15
0.4 0.4
local network nr 6; 10
5
starting node = nr 100
0.2 0.2
5
0.0 0 0.0 0
1 51 101 151 201 1 26 51 76
1.6 iteration 60
1.2 iteration 80
1.4
1.0
scale-free network nr 2; 70
starting node = nr 11
50
1.2 60
40 0.8
1.0 50
0.8 30 0.6 40
0.6
random network nr 8; 30
0.4
starting node = nr 80 20 0.4
20
10 0.2
0.2 10
0.0 0 0.0 0
1 26 51 76 1 26 51 76
from: Pautasso & Jeger (in review) Ecological Complexity

22. Linear epidemic threshold on a graph of the
probability of persistence and of transmission
1.00
local
epidemic
develops small-world
probability of persistence
0.75 random
scale-free
0.50
0.25
no
epidemic
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
probability of transmission
from: Pautasso & Jeger (in review) Ecological Complexity

23. Lower epidemic threshold for higher correlation
coefficient between links to and links from nodes
0.500
probability of
threshold (p of transmission between nodes)
0.400 persistence = 0
0.300
0.200
local
small world
0.100 random
scale-free (one way)
scale-free (two ways)
0.000
-0.500 0.000 0.500 1.000
correlation coefficient between number of links to and links from nodes
from: Pautasso & Jeger (in preparation) Proceedings Royal Society B

24. Marked variations in the final size of the epidemic at
threshold conditions depending on the starting point
100 100
local network nr 2
% nodes at equilibrium with probability of infection > 0.01
a b small world network nr 6
75
75
50
50
25
25
0
0
0 25 50 75 100
starting node 0 25 50
starting node
75 100
100 100
random network nr 9
c d scale-free network nr 8
75 75
50 50
25 25
0 0
0 25 50 75 100 0 25 50 75 100
from: Pautasso & Jeger (in preparation) Proceedings Royal Society B

25. Temporal development; England & Wales, 2003-2005; n = 2788
250
R ecords positive to P . ram orum
unclea r w hic h
200
n of records
esta tes/env ironm ent
150
nurseries/ga rden
centres
100
50
0
3
4
5
03
3
04
4
05
5
3
4
5
-0
-0
-0
l-0
l-0
l-0
-0
-0
-0
n-
n-
n-
pr
pr
pr
ct
ct
ct
Ju
Ju
Ju
Ja
Ja
Ja
O
O
O
A
A
A
Data source: Department for Environment, Food and Rural Affairs, UK

26. 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

27. Spatially-explicit
modelling framework
Climate Long-distance trade
suitability
Local Trade
Heathland
Woodland

28. 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
3. Case study: Phytophthora ramorum and
epidemiological investigations in networks of small size
4. Conclusion: further potential work
applying network theory in biogeography

29. 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

30. Acknowledgements
Peter Weisberg, Chris Gilligan, Univ.
Univ. of Nevada, of Cambridge, UK
Reno, US Mike Jeger,
Ottmar Imperial College, Mike Shaw,
Holdenrieder, Wye, UK Univ. of
ETHZ, CH Reading, UK
Kevin
Gaston,
Univ. of
Mike Sheffield, Emanuele Della
McKinney, UK Katrin Valle, Politecnico di
Univ. of Boehning Milano, Italy
Tennessee, -Gaese,
US Univ. Mainz

31. References
Jokimäki 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: 383-385
Moslonka-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: 392-403
Pautasso 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: 426-436
Barbosa 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: 432-441
Barbosa 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: 446-447
Cantarello 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: 265-271
Chiari C, Dinetti M, Licciardello C, Licitra G & Pautasso M (2010) Urbanization and the more-individuals hypothesis. Journal of Animal Ecology 79: 366-371
Dehnen-Schmutz K, Holdenrieder O, Jeger MJ & Pautasso M (2010) Structural change in the international horticultural industry: some implications for plant health. Scientia
Horticulturae 125: 1-15
Golding 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: 689-696
MacLeod 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: 193-202
Pautasso M & Pautasso C (2010) Peer reviewing interdisciplinary papers. European Review 18: 227-237
Pautasso M & Schäfer H (2010) Peer review delay and selectivity in ecology journals. Scientometrics 84: 307-315
Pautasso 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: 729-755
Pautasso 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: 1300-1309
Pecher 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: 272-280
Harwood 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

32. References (bis)
Pautasso M (2009) Geographical genetics and the conservation of forest trees. Perspectives in Plant Ecology, Systematics and Evolution 11: 157-189
Pautasso M & Dinetti M (2009) Avian species richness, human population and protected areas across Italy’s regions. Environmental Conservation 36: 22-31
Pautasso M & Powell G (2009) Aphid biodiversity is correlated with human population in European countries. Oecologia 160: 839-846
Pautasso M & Zotti M (2009) Macrofungal taxa and human population in Italy's regions. Biodiversity & Conservation 18: 473-485
Xu 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
Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126
Jeger MJ & Pautasso M (2008) Plant disease and global change – the importance of long-term data sets. New Phytologist 177: 8-11
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
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: 1842-1849
Pautasso M & Jeger MJ (2008) Epidemic threshold and network structure: the interplay of probability of transmission and of persistence in directed networks. Ecological
Complexity 5: 1-8
Pautasso M & Weisberg PJ (2008) Density-area relationships: the importance of the zeros. Global Ecology and Biogeography 17: 203-210
Schlick-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: 2195-2206
Steck CE & Pautasso M (2008) Human population, grasshopper and plant species richness in European countries. Acta Oecologica 34: 303-310
Jeger 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? Botanica
Helvetica 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: 282-289
Pautasso 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

33. Networks and
Epidemiology
Marco Pautasso,
Division of Biology,
Imperial College London,
Wye Campus, Kent, UK
Universität Bayreuth,
25 Jan 2007

34. Clustering vs. path length
local small-world random
clustering
path
length
local small-world random
Modified from: Roy & Pascual (2006) On representing network heterogeneities
in the incidence rate of simple epidemic models. Ecological Complexity 3, 1: 80-90

35. Reproductive ratio R0 in networks
of differing degree of clustering
Initial R0
Asymptotic R0
Simulations of a
wide variety of
networks with
average of
10 contacts
per individuals
random (C/Cmax) local
From: Keeling (2005) The implications of network structure for epidemic dynamics.
Theoretical Population Biology 67: 1-8

36. Epidemic control in networks with low vs. high clustering
(a) low clustering (b) high clustering
average number of connections per node = 10
From: Kiss, Green & Kao (2005) Disease contact tracing in random and
clustered networks. Proceedings of the Royal Society B, 272: 1407-1414

37. Critical tracing efficiency to control an SIS-type epidemic
in a network with uniform degree distribution
From: Eames & Keeling (2003) Contact tracing and disease control.
Proceedings of the Royal Society B 270: 2565-2571

38. Connectivity loss in the North American power grid
due to the removal of transmission substations
transmission nodes removed (%)
From: Albert, Albert & Nakarado (2004) Structural vulnerability of the
North American power grid. Physical Review E 69, 025103