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 trade

1. 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
Photo: Marin County Fire Department, CA, USA

2. Disease spread in
a globalized world
number of passengers per day
From: Hufnagel, Brockmann & Geisel (2004) Forecast and control
of epidemics in a globalized world. PNAS 101: 15124-15129

3. Epidemiology is just one of the
many applications of network theory
Network pictures from: NATURAL
Newman (2003)
SIAM Review food webs
cell
metabolism
neural Food web of Little Rock
networks Lake, Wisconsin, US
ant nests sexual
partnerships
DISEASE
SPREAD
family
innovation networks
Internet flows co-authorship HIV
structure railway urban road nets spread
electrical networks networks network
power grids telephone calls
WWW
computing airport Internet E-mail
committees
grids networks software maps patterns
TECHNOLOGICAL SOCIAL
modified from: Jeger, Pautasso, Holdenrieder & Shaw (2007) New Phytologist

4. P. ramorum
Map from www.suddenoakdeath.org confirmations on
Kelly, UC-Berkeley
the US West Coast
vs. national risk
Hazard map: Frank
Koch & Bill Smith,
3rd SOD Science
Symposium (2007)

5. from: McKelvey, Koch & Smith (2007) SOD Science Symposium III

6. Phytophthora ramorum in England & Wales (2003-2006)
511 nurseries/ 168 historic gardens/
garden centres woodlands 122
85
2003- 46
2003-
Jun 2008 Jun
426 2008
Climatic match courtesy of Outbreak maps courtesy of
Richard Baker, CSL, UK David Slawson, PHSI, DEFRA, UK

7. Simple model of infection spread (e.g. P. ramorum) in a network
pt probability of infection transmission
pp probability of infection persistence
node 1 2 3 4 5 6 7 8 … 100
step 1
step 2
step 3
…
step n

8. The four basic types of network structure used
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)]}
local small-
world
random scale-free

9. Epidemic threshold
and network structure

10. 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
local 35 small-world
% nodes with probability of infection > 0.01
1.0 1.0
20
30
0.8 0.8
25
15
0.6 20 0.6
10
15
0.4 0.4
10
5
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
scale-free 70
random
1.4
1.0
50
1.2 60
40 0.8
1.0 50
0.8 30 0.6 40
0.6 30
20 0.4
0.4 20
10 0.2
0.2 10
0.0 0 0.0 0
1 26 51 76 1 26 51 76
from: Pautasso & Jeger (2008) Ecological Complexity

11. Lower epidemic threshold for scale-free networks
1.00
local
probability of persistence
Epidemic develops
0.75 small-world
random
0.50 scale-free
Epidemic
0.25 does not
develop
0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30
probability of transmission
from: Pautasso & Jeger (2008) Ecological Complexity

12. Connectance,
in-out correlations
and clustering

13. Correlation of number of links in and number
of links out for wholesalers/retailers
Courtesy
of Tom
Harwood

14. 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)

15. (a) (b)
(c) (d)
from: Moslonka-Lefebvre et al. (submitted)

16. 1.0 1.0
(100) (200 links)
threshold probability of transmission
0.8 0.8
0.6 0.6
0.4 local random 0.4
small-world scale-free 2 0.2
0.2
scale-free 0 scale-free 1
0.0
0.0
-0.6 -0.4 -0.2 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
1.0 1.0
0.8 (400) 0.8
(1000 links)
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.6 -0.4 -0.2 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
correlation coefficient between in- and out-degree
from: Moslonka-Lefebvre et al. (submitted)

17. 1.0 1.0
threshold probability of transmission
(100 links) (200)
0.8 0.8
0.6 0.6
0.4 local random 0.4
small-world scale-free 2
0.2 0.2
scale-free 0 scale-free 1
0.0 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
1.0 1.0
0.8
(400) 0.8
(1000)
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
clustering coefficient
from: Moslonka-Lefebvre et al. (submitted)

18. Starting node and
epidemic final size

19. 100 100
75
(local) 75
(sw)
(N of nodes with infection status > 0.01) 50 50
25 25
0 0
0 25 50 75 100 0 25 50 75 100
epidemic final size
100 100
(rand)
75 75
(sf2)
50 50
25 25
0 0
0 25 50 75 100 0 25 50 75 100
100 100
75
(sf0) 75 (sf1)
50 50
25 25
0 0
0 25 50 75 100 0 25 50 75 100
starting node of the epidemic
from: Pautasso, Moslonka-Lefebvre & Jeger (submitted)

20. 2.0 3.0
local 2.5 sw
1.5
across all nodes (+0.01 for sf networks) 2.0
sum at equilibrium of infection status
1.0 1.5
1.0
0.5
0.5
0.0 0.0
0 1 2 3 4 5 6 0 2 4 6 8
3.0 1 .0
2.5 rand sf2 (log-log)
2.0
1.5 0 .0
1.0
0.5
0.0 -1 .0
-1 0 1 2 3
0 2 4 6 8 10 12
2.0
2.0
1.5 sf0 (log-log) 1.5 sf1 (log-log)
1.0 1.0
0.5 0.5
0.0 0.0
-0.5 -0.5
-1.0 -1.0
0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0
n of links from starting node n of links from starting node

21. Correlation of epidemic final size with out-degree of
starting node increases with network connectivity
from: Pautasso N replicates = 100; error bars are St. Dev.;
et al. (submitted)
different letters show sign. different means at p < 0.05

22. epidemic final size (0.01) and out- 1.0 C AC B
D
correlation coefficient between
A B
AA C
0.8 DE
degree of starting node
E
C
B
E D
D E
local
A random
0.6 B B sw
D C
E
sf2
0.4
sf0
0.2 sf1
0.0
from: Pautasso 100 200 400 1000
et al. (submitted) links

23. 1.00
A
0.75
final size (sum) and in-degree
correlation between epidemic
0.50
of the starting node
A B
links
A
0.25 A
BBB B C
DC
B
D
C
D 100
0.00
200
-0.25 sw
sf2
sf0
sf1
l
om
ca
D
400
lo
D
nd
C B
ra
A C
-0.50 B 1000
A
-0.75
-1.00
from: Pautasso et al. (submitted)

24. 1.00
0.80
correlation coefficient between
A
epidemic final size (0.01) and
A
in-degree of starting node
0.60 local
0.40 A random
A
0.20 B C
B BC B B sw
C
EED C EE D E F DE
0.00
D
sf2
-0.20 100 200 400 1000 sf0
-0.40 sf1
-0.60
-0.80
from: Pautasso et al. (submitted)
links

25. 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

26. References
Chiari C, Dinetti M, Licciardello C, Licitra G & Pautasso M (2010) Urbanization and the more-individuals hypothesis. Journal of Animal Ecology 79:
366-371
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
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 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
Pecher 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: 272-280
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