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Spatial behavior of an epidemic model with migration

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Abstract

It was reported that there are traveling patterns in the spatiotemporal data of epidemics (Cummings et al., Nature 427:344, 2004; Grenfell et al., Nature 414:716, 2001). To well understand the mechanism, we present a spatial epidemic model with migration, which means that the individuals exhibit a correlated motion toward certain direction, and obtain traveling pattern. Our results may be helpful to understand the mechanism of the spatiotemporal epidemics and have potential application of control of the epidemics.

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References

  1. Cummings, D.A., Irizarry, R.A., Huang, N.E., Endy, T.P., Nisalak, A., Ungchusak, K., Burke, D.S.: Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand. Nature 427, 344–347 (2004)

    Article  Google Scholar 

  2. Filipe, J.A.N., Maule, M.M.: Effects of dispersal mechanisms on the spatiotemporal development of epidemics. J. Theor. Biol. 226, 125–141 (2004)

    Article  MathSciNet  Google Scholar 

  3. Ostfeld, R.S., Glass, G.E., Keesing, F.: Spatial epidemiology: an emerging (or re-emerging) discipline. Trends Ecol. Evol. 20, 328–336 (2005)

    Article  Google Scholar 

  4. Hassell, M.P., Comins, H.N., May, R.M.: Species coexistence and self-organizing spatial dynamics. Nature 370, 290–292 (1994)

    Article  Google Scholar 

  5. Kerr, B., Riley, M.A., Feldman, M.W., Bohannan, B.J.M.: Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418, 171–174 (2002)

    Article  Google Scholar 

  6. Eshel, I.: On the neighbor effect and the evolution of altruistic traits. Theor. Popul. Biol. 3, 258–277 (1972)

    Article  MathSciNet  Google Scholar 

  7. Morris, D.W., Diffendorfer, J.E., Lundberg, P.: Dispersal among habitats varying in fitness: source–sink dynamics, balanced dispersal or pulsed migration through ideal habitat selection? Oikos 107, 559–575 (2004)

    Article  Google Scholar 

  8. Bascompte, J., Sole, R.V.: Spatially induced bifurcations in single-species population dynamics. J. Anim. Ecol. 159, 469–480 (1992)

    Google Scholar 

  9. Hanski, I.: Coexistence of competitors in patchy environment. Ecology 64, 493–500 (1983)

    Article  Google Scholar 

  10. Hastings, A.: Spatial heterogeneity and ecological models. Ecology 71, 426–428 (1990)

    Article  Google Scholar 

  11. Mollison, D.: Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39, 283–326 (1977)

    MathSciNet  MATH  Google Scholar 

  12. Courchamp, F., Pontier, D., Langlais, M., Artois, M.: Population dynamics of feline immunodeficiency virus within cat populations. J. Theor. Biol. 175, 553–560 (1995)

    Article  Google Scholar 

  13. Haraguchi, Y., Sasaki, A.: Evolution of parasite virulence and transmission rate in a spatially structured population. J. Theor. Biol. 203, 85–96 (2000)

    Article  Google Scholar 

  14. Hilker, F.M., Langlais, M., Petrovskii, S.V., Malchow, H.: A diffusive SI model with Allee effect and application to FIV. Math. Biosci. 206, 61–80 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Petrovskii, S., Morozov, A., Li, B.L.: Regimes of biological invasion in a predator–prey system with the Allee effect. Bull. Math. Biol. 67, 637–661 (2005)

    Article  MathSciNet  Google Scholar 

  16. Leppnen, T.: Ph.D. Thesis, Helsinki University of Technology, Finland (2004)

  17. Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.-L.: Spatiotemporal complexity of plankton and fish dynamics in simple model ecosystems. SIAM Rev. 44, 311–370 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Petrovskii, S., Li, B.L., Malchow, H.: Transition to spatiotemporal chaos can resolve the paradox of enrichment. Ecol. Complex. 1, 37–47 (2004)

    Article  Google Scholar 

  19. Sherratt, J.A., Lambin, X., Sherratt, T.N.: The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations. Am. Nat. 162, 503–513 (2003)

    Article  Google Scholar 

  20. Bär, M., Or-Guil, M.: Alternative scenarios of spiral breakup in a reaction–diffusion model with excitable and oscillatory dynamics. Phys. Rev. Lett. 82, 1160–1163 (1999)

    Article  Google Scholar 

  21. Xie, F., Xie, D., Weiss, J.N.: Inwardly rotating spiral wave breakup in oscillatory reaction–diffusion media. Phys. Rev. E 74, 026107 (2006)

    Article  Google Scholar 

  22. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, New York (2003)

    MATH  Google Scholar 

  23. Sherratt, J.A., Lambin, X., Thomas, C.J., Sherratt, T.N.: Generation of periodic waves by landscape features in cyclic predator–prey systems. Proc. R. Soc. Lond. B 269, 327–334 (2002)

    Article  Google Scholar 

  24. Bjørnstad, O.N., Bascompte, J.: Synchrony and second order spatial correlation in host–parasitoid systems. J. Anim. Ecol. 70, 924–933 (2001)

    Article  Google Scholar 

  25. Bjørnstad, O.N., Finkenstädt, B.F., Grenfell, B.T.: Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR model. Ecol. Monogr. 72, 169–184 (2002)

    Google Scholar 

  26. Bolker, B.M., Grenfell, B.T.: Chaos and biological complexity in measles dynamics. Proc. R. Soc. Lond. B 251, 75–81 (1993)

    Article  Google Scholar 

  27. Earn, D.J.D., Rohani, P., Bolker, B.M., Grenfell, B.T.: A simple model for complex dynamical transitions in epidemics. Science 287, 667–670 (2000)

    Article  Google Scholar 

  28. Finkenstadt, B.F., Keeling, M.J., Grenfell, B.T.: Patterns of density dependence in measles dynamics. Proc. R. Soc. Lond. B 265, 753–762 (1998)

    Article  Google Scholar 

  29. Rohani, P., Earn, D.J.D., Grenfell, B.T.: Opposite patterns of synchrony in sympatric disease metapopulations. Science 286, 968–971 (1999)

    Article  Google Scholar 

  30. Grenfell, B.T., Bjornstad, O.N., Kappey, J.: Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716–723 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Instrumentation Science and Dynamic Measurement (North University of China), Ministry of Education, Taiyuan, Shanxi, 030051, P.R. China

    Min Cui, Tie-Hua Ma & Xin-E. Li

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  1. Min Cui
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  2. Tie-Hua Ma
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  3. Xin-E. Li
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Correspondence to Min Cui.

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Cui, M., Ma, TH. & Li, XE. Spatial behavior of an epidemic model with migration. Nonlinear Dyn 64, 331–338 (2011). https://doi.org/10.1007/s11071-010-9864-6

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  • DOI: https://doi.org/10.1007/s11071-010-9864-6

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