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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
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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
2004
2005
20052006
2005
200520052003
2004
2003
2003
2006
20052004
2005
20062005
2005 2005
200520052005
2004
2005
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
0
50
100
150
200
250
Jan-03
Apr-03
Jul-03
Oct-03
Jan-04
Apr-04
Jul-04
Oct-04
Jan-05
Apr-05
Jul-05
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)
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
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
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
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
starting node
% n
odes
at e
quili
briu
m w
ith p
roba
bilit
y of
infe
ctio
n >
0.01
starting node
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
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
1.0
1.5
2.0
2.5
3.0
1-4 5-49 50-284
n of positive P. ramorum records in database
log 1
0 num
ber o
f sit
es
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
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
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