Models of disease spread in small-size directed networks
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Transcript of Models of disease spread in small-size directed networks
Photo: Ottmar Holdenrieder
Plant disease spread and establishment in small-size
directed networks
Mathieu Moslonka-Lefebvre, Marco Pautasso & Mike Jeger
Imperial College London, Silwood Park
IEW 10, Geneva, NY - 10 June 2009
Outline of the talk
1. The relevance of networks for disease epidemiology
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
Brenn et al. (2008) Community structure of Phialocephala fortinii s. lat. in European tree nurseries, and assessment of the potential of the seedlings as dissemination vehicles. Mycological Research
100 km
Plant nurseriesas hubs
2000-2004
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
Outline of the talk
1. The relevance of networks for disease epidemiology
2. Case study: Phytophthora ramorum
3. Simulations of disease spread in small-size directed networks
4. Conclusions
from: McKelvey et al. (2007) SOD Science Symposium III
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)
168 historic gardens/ woodlands
Phytophthora ramorum in England & Wales (2003-2006)
Outbreak maps courtesy of David Slawson, PHSI, DEFRA, UK
Climatic match courtesy of Richard Baker, CSL, UK
85
426
46
122
2003-Jun 2008
511 nurseries/ garden centres
2003-Jun 2008
Outline of the talk
1. The relevance of networks for disease epidemiology
2. Case study: Phytophthora ramorum
3. Simulations of disease spread in small-size directed networks
4. Conclusions
May (2006) Network structure and the biology of populations. Trends in Ecology & Evolution
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
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
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
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1.0
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1 26 51 760
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40
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0.0
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25
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1 51 101 151 2010
5
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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. (submitted)
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
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1.0
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0
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0.8
1.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
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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. (submitted)
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 the horticultural trade
A very short history of ornamental gardens
Thebes, ~1500 BCE Florence, 16th century Gardens of Heligan, 17-18th century
California, ramorum-affected nursery, 2004
California, ramorum-affected urban setting, 2000
Contemporary ornamental
trade patterns
From International Statistics Flower and Plants 2004, Institut
fuer Gartenbau-oekonomie der
UniversitaetHannover, Germany
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: 3353-3361 Jeger MJ & Pautasso M (2008) Comparative epidemiology of zoosporic plant pathogens. European Journal of Plant Pathology 122: 111-126MacLeod 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, 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