Models of disease spread in small-size directed networks

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Models of disease spread in small-size directed networks, human and plant mobility patterns, epidemic simulations, sudden oak death. Epidemiology is just one of the many applications of network …

Models of disease spread in small-size directed networks, human and plant mobility patterns, epidemic simulations, sudden oak death. Epidemiology is just one of the many applications of network theory. Examples of epidemic development in four kinds of directed networks of small size (at threshold conditions).

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  • 1. Plant disease spread and establishment in small-size directed networks Mathieu Moslonka-Lefebvre, Marco Pautasso & Mike Jeger Imperial College London, Silwood Park IEW 10, Geneva, NY - 10 June 2009Photo: Ottmar Holdenrieder
  • 2. Outline of the talk 1. The relevance of networks for disease epidemiology2. 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 dayHufnagel et al. (2004) Forecast and control of epidemics in a globalized world. PNAS
  • 4. Understanding human mobility patternsMatisoo-Smith et al. (1998) Patterns of prehistoric human mobilityin Polynesia indicated by mtDNA from the Pacific rat. PNAS
  • 5. Understanding plant mobility patternsVendramin et al. (2008) Genetically depauperate but widespread:the case of an emblematic Mediterranean pine. Evolution
  • 6. Plant nurseriesas hubs 100 km2000-2004Brenn et al. (2008) Community structure of Phialocephala fortinii s. lat.in European tree nurseries, and assessment of the potential of theseedlings as dissemination vehicles. Mycological Research
  • 7. 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 et al. (2007) New Phytologist
  • 8. Outline of the talk 1. The relevance of networks for disease epidemiology2. Case study: Phytophthora ramorum 3. Simulations of disease spread in small-size directed networks 4. Conclusions
  • 9. from: McKelvey et al. (2007) SOD Science Symposium III
  • 10. P. ramorumMap 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)
  • 11. 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
  • 12. Outline of the talk 1. The relevance of networks for disease epidemiology2. Case study: Phytophthora ramorum 3. Simulations of disease spread in small-size directed networks 4. Conclusions
  • 13. Fraction of population infected (l) as a function of ρ0 uniform degree distribution scale-free network with P(i) ≈ i-3 ρ0 is coincident with R0 for a uniform degree distribution; for a scale-free network, theory says that R0 = ρ0 + [1 + (CV)2], where CV is the coefficient of variation of the degree distribution May (2006) Network structure and the biology of populations. Trends in Ecology & Evolution
  • 14. 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
  • 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- worldrandom 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 localprobability 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 < 0.05from: Moslonka-Lefebvre et al. (submitted)
  • 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. (submitted)
  • 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 1.5across 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
  • 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 size4. implications for the horticultural trade
  • 24. A very short history of ornamental gardensThebes, ~1500 BCE Florence, 16th century Gardens of Heligan, 17-18th century California, ramorum- California, ramorum-affected affected nursery, 2004 urban setting, 2000
  • 25. Contemporary ornamental trade patterns From InternationalStatistics Flower andPlants 2004, Institut fuer Gartenbau- oekonomie der Universitaet Hannover, Germany
  • 26. AcknowledgementsJennifer RichardParke, Baker, CSLUniv. of AlanOregon Inman, Mike Shaw, DEFRA University of Reading Ottmar Holdenrieder, ETHZ, CHXiangming Xu, East Malling Tom Research Joan Webber, Harwood, Forest Research, CEP, Imperial Farnham College
  • 27. ReferencesDehnen-Schmutz K, Holdenrieder O, Jeger MJ & Pautasso M (2010) Structural change in the international horticulturalindustry: 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 networkand 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 PlantPathology 122: 111-126MacLeod A, Pautasso M, Jeger MJ & Haines-Young R (2010) Evolution of the international regulation of plant pests andchallenges for future plant health. Food Security 2: 49-70Moslonka-Lefebvre M, Pautasso M & Jeger MJ (2009) Disease spread in small-size directed networks: epidemicthreshold, correlation between links to and from nodes, and clustering. Journal of Theoretical Biology 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 and Evolution 11: 157-189Pautasso 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: 729-755Pautasso M, Moslonka-Lefebvre M & Jeger MJ (2010) The number of links to and from the starting node as a predictorof 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-sizedirected trade networks: the role of hierarchical categories. Journal of Applied Ecology 47: 1300-1309Xu 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