Disease spread in small-size directed networks
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Disease spread in small-size directed networks

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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......

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

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  • 1. Disease spread in small-size directed networksMarco Pautasso, MathieuMoslonka-Lefebvre,& Mike Jeger - Imperial CollegeLondon, Silwood ParkBath 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 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. Food webs: an example of small-size networksDunne 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, Californiafrom: Rizzo et al. (2005) Annual Reviews of Phytopathology, Photo: Susan Frankel
  • 9. 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)
  • 10. from: McKelvey et al. (2007) SOD Science Symposium III
  • 11. Phytophthora ramorum in England & Wales (2003-2008) gardens/ nurseries woodlands & garden centresClimatic match courtesy of Outbreak maps courtesy ofRichard 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 theornamental 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/immunizationif 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- 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. (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 1.5across 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 networks2. importance of the in-out correlation 3. out-degree as a predictor of epidemic final size4. implications for biological invasions
  • 24. Contemporary ornamental trade patterns From InternationalStatistics Flower andPlants 2004, Institut fuer Gartenbau- oekonomie der Universitaet Hannover, Germany
  • 25. 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
  • 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: 3353-3361MacLeod 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 & McKinney ML (2007) The botanist effect revisited: plant species richness, county area and humanpopulation size in the US. Conservation Biology 21, 5: 1333-1340Pautasso 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