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

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 Presentation Transcript

  • 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. 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. 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. 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. 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 2008Climatic match courtesy of Outbreak maps courtesy ofRichard 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- worldrandom scale-free
  • 9. Epidemic thresholdand 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 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. Connectance,in-out correlations and clustering
  • 13. Correlation of number of links in and number of links out for wholesalers/retailersCourtesyof TomHarwood
  • 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. (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 1.0threshold 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. Starting node andepidemic 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 1.5across 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. 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. 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. 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. 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. 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. 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