Models of disease spread and     establishment in small-size          directed networks                   Mathieu Moslonka...
Disease spread in                                                                 a globalized world                      ...
Epidemiology is just one of the                      many applications of network theoryNetwork pictures from:          NA...
P. ramorumMap from www.suddenoakdeath.org    confirmations on        Kelly, UC-Berkeley                                  t...
from: McKelvey, Koch & Smith (2007) SOD Science Symposium III
Phytophthora ramorum in England & Wales (2003-2006)                    511 nurseries/            168 historic gardens/    ...
Simple model of infection spread (e.g. P. ramorum) in a network                   pt probability of infection transmission...
The four basic types of network structure used SIS Model, 100 Nodes, directed networks, P [i (x, t)] = Σ {p [s] * P [i (y,...
Epidemic thresholdand network structure
Examples of epidemic development in four kinds of                                directed networks of small size (at thres...
Lower epidemic threshold for scale-free networks                             1.00                                         ...
Connectance,in-out correlations  and clustering
Correlation of number of links in and number    of links out for wholesalers/retailersCourtesyof TomHarwood
Lower epidemic threshold for two-way scale-free networks        (unless networks are sparsely connected)                  ...
(a)        (b)                                 (c)         (d)from: Moslonka-Lefebvre et al. (submitted)
1.0                                                              1.0                                                      ...
1.0                                                 1.0threshold probability of transmission                              ...
Starting node andepidemic final size
100                                      100                                             75                               ...
2.0                                                          3.0                                                      loca...
Correlation of epidemic final size with out-degree of      starting node increases with network connectivityfrom: Pautasso...
epidemic final size (0.01) and out-   1.0                                                                       C AC B    ...
1.00                                                                             A                                 0.75fin...
1.00                                   0.80 correlation coefficient between                                               ...
Main results               1. lower epidemic threshold                  for scale-free networks                  2. in-out...
ReferencesChiari C, Dinetti M, Licciardello C, Licitra G & Pautasso M (2010) Urbanization and the more-individuals hypothe...
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Models of disease spread and establishment in small-size directed networks

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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

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Models of disease spread and establishment in small-size directed networks

  1. 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 2009Photo: Marin County Fire Department, CA, USA
  2. 2. Disease spread in a globalized world number of passengers per dayFrom: Hufnagel, Brockmann & Geisel (2004) Forecast and controlof epidemics in a globalized world. PNAS 101: 15124-15129
  3. 3. Epidemiology is just one of the many applications of network theoryNetwork pictures from: NATURALNewman (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 networksInternet flows co-authorship HIVstructure railway urban road nets spread electrical networks networks network power grids telephone calls WWW computing airport Internet E-mail committees grids networks software maps patternsTECHNOLOGICAL SOCIALmodified from: Jeger, Pautasso, Holdenrieder & Shaw (2007) New Phytologist
  4. 4. P. ramorumMap 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. 5. from: McKelvey, Koch & Smith (2007) SOD Science Symposium III
  6. 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 2008Climatic match courtesy of Outbreak maps courtesy ofRichard Baker, CSL, UK David Slawson, PHSI, DEFRA, UK
  7. 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. 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- worldrandom scale-free
  9. 9. Epidemic thresholdand network structure
  10. 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. 11. Lower epidemic threshold for scale-free networks 1.00 localprobability 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. 12. Connectance,in-out correlations and clustering
  13. 13. Correlation of number of links in and number of links out for wholesalers/retailersCourtesyof TomHarwood
  14. 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 < 0.05from: Moslonka-Lefebvre, Pautasso & Jeger (submitted)
  15. 15. (a) (b) (c) (d)from: Moslonka-Lefebvre et al. (submitted)
  16. 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. 17. 1.0 1.0threshold 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. 18. Starting node andepidemic final size
  19. 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. 20. 2.0 3.0 local 2.5 sw 1.5across all nodes (+0.01 for sf networks) 2.0sum 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. 21. Correlation of epidemic final size with out-degree of starting node increases with network connectivityfrom: Pautasso N replicates = 100; error bars are St. Dev.;et al. (submitted) different letters show sign. different means at p < 0.05
  22. 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 0.0from: Pautasso 100 200 400 1000et al. (submitted) links
  23. 23. 1.00 A 0.75final size (sum) and in-degreecorrelation 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 -1.00from: Pautasso et al. (submitted)
  24. 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 -0.80from: Pautasso et al. (submitted) links
  25. 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 tradePhoto: Marin County Fire Department
  26. 26. 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 implicationsfor 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: 3353-3361Jeger 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. NewPhytologist 174: 179-197MacLeod A, Pautasso M, Jeger MJ & Haines-Young R (2010) Evolution of the international regulation of plant pests and challenges for future planthealth. Food Security 2: 49-70Moslonka-Lefebvre M, Pautasso M & Jeger MJ (2009) Disease spread in small-size directed networks: epidemic threshold, correlation betweenlinks 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 inplant 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: 424-432Pautasso M, Xu XM, Jeger MJ, Harwood T, Moslonka-Lefebvre M & Pellis L (2010) Disease spread in small-size directed trade networks: the role ofhierarchical 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 speciesrichness 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 Englandand Wales. Ecography 32: 504-516

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