models of disease spread and establishment in small-size directed networks
DESCRIPTION
disease, globalized world, epidemiology, network theory, epidemic threshold, starting node, clustering, final size. Main results 1. lower epidemic threshold for scale-free networks 2. in-out correlation more important than clustering 3. out-degree as a predictor of epidemic final size 4. implications for the horticultural tradeTRANSCRIPT
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Photo: Marin County Fire Department, CA, USA
Models of disease spread and establishment in small-size
directed networks
Mathieu Moslonka-Lefebvre, Marco Pautasso & Mike Jeger
Imperial College London, Silwood Park, UK
Rutgers University, March 2009
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From: Hufnagel, Brockmann & Geisel (2004) Forecast and control of epidemics in a globalized world. PNAS 101: 15124-15129
number of passengers per day
Disease spread in a globalized world
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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, Pautasso, Holdenrieder & Shaw (2007) New Phytologist
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P. ramorumconfirmations on
the US West Coast vs. national risk
Map from www.suddenoakdeath.orgKelly, UC-Berkeley
Hazard map: Frank Koch & Bill Smith, 3rd SOD Science
Symposium (2007)
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from: McKelvey, Koch & Smith (2007) SOD Science Symposium III
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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
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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
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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)]}
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Epidemic threshold and network structure
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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
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0.00
0.25
0.50
0.75
1.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30
probability of transmission
prob
abili
ty o
f per
sist
ence local
small-world
random
scale-free
Lower epidemic threshold for scale-free networks
from: Pautasso & Jeger (2008) Ecological Complexity
Epidemic does not develop
Epidemic develops
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Connectance, in-out correlations
and clustering
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Correlation of number of links in and number of links out for wholesalers/retailers
Courtesy of Tom Harwood
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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, Pautasso & Jeger (submitted)
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(a) (b)
(c) (d)
from: Moslonka-Lefebvre et al. (submitted)
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1.0
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
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1.0
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
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-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)
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0.0 0.1 0.2 0.3 0.4 0.50.0
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1.0
0.0 0.1 0.2 0.3 0.4 0.5
local random
small-world scale-free 2
scale-free 0 scale-free 1
thre
shol
d pr
obab
ility
of t
rans
mis
sion
clustering coefficient
(100 links) (200)
(400) (1000)
from: Moslonka-Lefebvre et al. (submitted)
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Starting node and epidemic final size
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0
25
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0 25 50 75 1000
25
50
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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 epidemicfrom: Pautasso, Moslonka-Lefebvre & Jeger (submitted)
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-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)
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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
from: Pautasso et al. (submitted)
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0.0
0.2
0.4
0.6
0.8
1.0
100 200 400 1000links
corr
elat
ion
coef
fici
ent b
etw
een
epid
emic
fin
al s
ize
(0.0
1) a
nd o
ut-
degr
ee o
f st
artin
g no
de localrandomswsf2sf0sf1
A
B B
CED
A A
BC DE
A BC
DDE E
C CA
E
BD
from: Pautasso et al. (submitted)
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links
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
local
random sw sf2 sf0 sf1
corr
elat
ion
betw
een
epid
emic
fi
nal s
ize
(sum
) and
in-d
egre
e of
the
star
ting
node
1002004001000
A
D CB
ABBB
A
A
DB
C
B
CD
A
DC B
D
C
A
B
from: Pautasso et al. (submitted)
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-0.80-0.60-0.40-0.200.000.200.400.600.801.00
100 200 400 1000
links
corr
elat
ion
coef
fici
ent b
etw
een
epid
emic
fin
al s
ize
(0.0
1) a
nd
in-d
egre
e of
sta
rtin
g no
de
localrandomswsf2sf0sf1
A
B CDE E D
BC
A
E E DBC
A
EF CB B
A
E D
from: Pautasso et al. (submitted)
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Main results
1. lower epidemic threshold for scale-free networks
2. in-out correlation more important than clustering
3. out-degree as a predictor of epidemic final size
4. implications for the horticultural trade
Photo: Marin County Fire Department
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ReferencesChiari C, Dinetti M, Licciardello C, Licitra G & Pautasso M (2010) Urbanization and the more-individuals hypothesis. Journal of Animal Ecology 79: 366-371Dehnen-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 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 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: 1300-1309Pecher 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: 272-280Xu 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