network epidemiology, Phytophthora ramorum, network theory, plant pathology, epidemic spread, clustering, small-world, random, scale-free. Introduction: interconnected world, growing interest in network theory and disease spread in networks. Examples of recent work modelling disease (i) spread and (ii) control in networks of various kinds

1. 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

2. 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
3. Case study: Phytophthora ramorum and
epidemiological simulations in networks of small size
4. Conclusion: call for enhanced use
of network theory in plant pathology

3. Networks are formed by:
• physical structures
• associations/relationships
• processes/flows on a structure

4. Armillaria rhizomorph network near Wageningen, Netherlands
From: Lamour et al. (submitted to FEMS Microbiology Ecology)

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

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

7. 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
2003 2005
2005
From: Pautasso & Jeger (submitted)

8. Epidemiology 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, 2: 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

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

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

11. 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

12. Clustering vs. path length
local small-world random
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

13. 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

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

15. 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

16. 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: 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

17. 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, in press

18. 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

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

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

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

22. 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

23. European garden & nursery finds
Phytophthora ramorum infection on Rhododendron in Europe
Photos: Hans DeGruyter, Netherlands Plant Protection Institute

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

25. UK, 2003-2005; n = 2788
250
Records positive to P. ramorum
unclear which
200
n of records
estates/environment
150
nurseries/garden
centres
100
50
0
O 3
O 4
O 5
A 03
A 04
A 05
Ja 3
Ja 4
5
Ju 3
Ju 4
Ju 5
l-0
l-0
l-0
-0
-0
-0
-0
-0
-0
n-
n-
n-
pr
pr
pr
ct
ct
ct
Ja
Data source: Department for Environment, Food and Rural Affairs, UK

26. Own epidemiological investigations in four
basic types of directed networks of small size
(a) (b) SIS-model;
N nodes = 100;
n links = 369;
directed networks;
probability of infection
for the node x at time
(c) (d) 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
(a) local; (b) small world;
type of network
(c) random; (d) scale-free

27. 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
1.2 80
1.6 60
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
iteration iteration

28. Linear epidemic threshold on a graph of the
probability of persistence and of transmission
1.00
epidemic
local
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

29. Lower epidemic threshold for higher correlation
coefficient between links to and links from nodes
0.500
probability of
threshold (p of transmission between nodes)
persistence = 0
0.400
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

30. 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 0 25 50 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
starting node starting node

31. 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?
• applications in the control of
Phytophthora ramorum spread?

32. Spatially-explicit modelling framework

34. UK- distribution centres of
tree nurseries from
Hort Week suppliers guide
2003; n = 476
kindly provided
by Tom Harwood

35. Sites Distribution Centres Incoming material Outgoing material

36. 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?
• applications in the control of
Phytophthora ramorum spread?

37. Scale-free properties in the database of sites tested
positive to Phytophthora ramorum, UK (2002-2005)
3.0
log10 number of sites
2.5
2.0
1.5
1.0
0.5
0.0
1-4 5-49 50-284
n of positive P. ramorum records in database

38. Scale-free properties in the database of sites tested
positive to Phytophthora ramorum, UK (2002-2005)
4.0
log10 of n of records
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1-4 5-49 50-499 500-4999 5000-
total amount plants affected by P. ramorum

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

40. Where are the applications to plant pathology?
LEGEND:
PLANT
no brackets = (plant
application existing (mycorrhiza) metabolomics –
(plant meta- cellular pathways)
(…) = application
existing, but not populations)
strictly involving
disease [nursery
networks]
[…] = would involve
plant pathology, but [quarantine] [plant-vector
application of network interactions
theory lacking [epiphytotics e.g. viruses]
management
& control]
(plant-
[recreation/ pollinator
amenities interactions)
(plant-
landscape] frugivore
(bats in
networks of interactions)
computer hollow trees)
viruses
Neisseria foot and fish diseases
(rumor gonorrhoeae mouth disease
propagation) Mycoplasma HIV Dengue avian flu bovine
pneumoniae Rotavirus SARS raccoon rabies
tuberculosis
HUMAN ANIMAL

41. Possible reasons for delay in the application
of network thinking to plant pathology
• Homogeneous mid-field conditions
more than adequate for plant diseases?
• Lack of data on network structure in plant
epidemics relative to human and animal ones?
• Just lagging behind? Clustering effects may
have slowed down the spread of the concept
into this meta-population?

42. Acknowledgements
Mike 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

43. References
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
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
Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126
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
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: 402-411
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 (2009) Geographical genetics and the conservation of forest trees. Perspectives in Plant Ecology, Systematics & Evolution 11: 157-189
Pautasso M (2010) Worsening file-drawer problem in the abstracts of natural, medical and social science databases. Scientometrics 85: 193-202
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 et al (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
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