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Why small-size networks? They are good models for regional horticultural networks spreading plant diseases such as Phytophthora ramorum. Main result: Lower epidemic threshold for scale-free networks with positive correlation between in- and out-degree
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Disease spread in small-size directed networks
Marco Pautasso, Mathieu Moslonka-Lefebvre, & Mike Jeger - Imperial College London, Silwood Park
Bath University, 2nd July 2009
Outline of the talk
1. why small-size networks?
2. case study: Phytophthora ramorum
3. simulations of disease spread in small-size directed networks
4. conclusions
Hufnagel et al. (2004) Forecast and control of epidemics in a globalized world. PNAS
number of passengers per day
Disease spread in a globalized world
Matisoo-Smith et al. (1998) Patterns of prehistoric human mobility in Polynesia indicated by mtDNA from the Pacific rat. PNAS
Understanding human mobility patterns
Vendramin et al. (2008) Genetically depauperate but widespread: the case of an emblematic Mediterranean pine. Evolution
Understanding plant mobility patterns
Dunne et al. (2002) Food-web structure and network theory:the role of connectance and size. PNAS
Food webs: an example of small-size networks
Outline of the talk
1. why small size-networks?
2. case study: Phytophthora ramorum
3. simulations of disease spread in small-size directed networks
4. conclusions
from: Rizzo et al. (2005) Annual Reviews of Phytopathology, Photo: Susan Frankel
P. ramorum in Monterey County, California
P. ramorumconfirmations on
the US West Coast vs. national risk
Map from www.suddenoakdeath.orgKelly, UC-Berkeley
Hazard map: Koch & Smith,
3rd SOD Science Symposium (2007)
from: McKelvey et al. (2007) SOD Science Symposium III
nurseries& garden
centres
gardens/woodlands
Phytophthora ramorum in England & Wales (2003-2008)
Outbreak maps courtesy of David Slawson, PHSI, DEFRA, UK
Climatic match courtesy of Richard Baker, CSL, UK
Outline of the talk
1. why small-size networks?
2. case study: Phytophthora ramorum
3. simulations of disease spread in small-size directed networks
4. conclusions
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
1. spread in theornamental plant trade
(asymmetric)
Features of the P. ramorum pathosystem → model
2. garden centres/plant nurseries are not just either
susceptible or infected
3. nurseries at risk even after eradication
if still trading susceptible spp
asymmetry in the adjacency matrices (directed networks)
0 < pi < 1 (continuum model)
absence of removal/immunization
(SIS model)
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)]}
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 0.25 0.50 0.75 1.00
probability of transmission
prob
abili
ty o
f per
sist
ence
localrandomsmall-worldscale-free (two-way)scale-free (uncorrelated)scale-free (one way)
Lower epidemic threshold for scale-free networks with positive correlation between in- and out-degree
modified from: Pautasso & Jeger (2008) Ecological Complexity
Epidemic does not develop Epidemic develops
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 et al. (in press) Journal of Theoretical Biology
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. (in press) Journal of Theoretical Biology
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 epidemic
-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
Conclusions
1. lower epidemic threshold for two-way scale-free networks
2. importance of the in-out correlation
3. out-degree as a predictor of epidemic final size
4. implications for biological invasions
Contemporary ornamental
trade patterns
From International Statistics Flower and Plants 2004, Institut
fuer Gartenbau-oekonomie der
UniversitaetHannover, Germany
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 et al. (2007) New Phytologist
Acknowledgements
Ottmar Holdenrieder,
ETHZ, CH
Mike Shaw, University of
Reading
Alan Inman,
DEFRA
Joan Webber, Forest Research,
Farnham
Tom Harwood,
CEP, Imperial College
Jennifer Parke, Univ. of Oregon
Xiangming Xu, East Malling
Research
Richard Baker, CSL
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: 3353-3361MacLeod 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. Journal of Theoretical Biology 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 and Evolution 11: 157-189Pautasso 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, 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: 1300-1309Xu 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