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

1. Disease spread in small-size
directed networks
Marco Pautasso, Mathieu
Moslonka-Lefebvre,
& Mike Jeger - Imperial College
London, Silwood Park
Bath University, 2nd July 2009

2. 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

3. Disease spread in
a globalized world
number of passengers per day
Hufnagel et al. (2004) Forecast and control of epidemics in a globalized world. PNAS

4. Understanding human mobility patterns
Matisoo-Smith et al. (1998) Patterns of prehistoric human mobility
in Polynesia indicated by mtDNA from the Pacific rat. PNAS

5. Understanding plant mobility patterns
Vendramin et al. (2008) Genetically depauperate but widespread:
the case of an emblematic Mediterranean pine. Evolution

6. Food webs: an example of small-size networks
Dunne et al. (2002) Food-web structure and network theory:
the role of connectance and size. PNAS

7. 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

8. P. ramorum in Monterey County, California
from: Rizzo et al. (2005) Annual Reviews of Phytopathology, Photo: Susan Frankel

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

10. from: McKelvey et al. (2007) SOD Science Symposium III

11. Phytophthora ramorum in England & Wales (2003-2008)
gardens/ nurseries
woodlands & garden
centres
Climatic match courtesy of Outbreak maps courtesy of
Richard Baker, CSL, UK David Slawson, PHSI, DEFRA, UK

12. 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

13. 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

14. Features of the P. ramorum pathosystem → model
1. spread in the asymmetry in the
ornamental plant trade adjacency matrices
(asymmetric) (directed networks)
2. garden centres/plant
0 < pi < 1
nurseries are not just either
(continuum model)
susceptible or infected
3. nurseries at risk absence of
even after eradication removal/immunization
if still trading susceptible spp (SIS model)

15. 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

16. 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

17. Lower epidemic threshold for scale-free networks
with positive correlation between in- and out-degree
1.00
local
probability of persistence
random
0.75 small-world
scale-free (two-way)
scale-free (uncorrelated)
0.50 scale-free (one way)
0.25
0.00
0.00 0.25 0.50 0.75 1.00
Epidemic probability of transmission
does not
develop Epidemic develops
modified from: Pautasso & Jeger (2008) Ecological Complexity

18. 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

19. 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. (in press) Journal of Theoretical Biology

20. 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

21. 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

22. 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

23. 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

24. Contemporary
ornamental
trade
patterns
From International
Statistics Flower and
Plants 2004, Institut
fuer Gartenbau-
oekonomie der
Universitaet
Hannover,
Germany

25. 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 et al. (2007) New Phytologist

26. Acknowledgements
Jennifer Richard
Parke, Baker, CSL
Univ. of Alan
Oregon Inman, Mike Shaw,
DEFRA University of
Reading
Ottmar
Holdenrieder,
ETHZ, CH
Xiangming Xu,
East Malling Tom
Research Joan Webber, Harwood,
Forest Research, CEP, Imperial
Farnham College

27. References
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
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. Journal of Theoretical Biology 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 and Evolution 11: 157-189
Pautasso 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: 729-755
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
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