Bulletin of Mathematical Biology

, Volume 68, Issue 2, pp 401–416 | Cite as

A Model of Spatial Epidemic Spread When Individuals Move Within Overlapping Home Ranges

  • Timothy C. Reluga
  • Jan Medlock
  • Alison P. Galvani
Original Article

Abstract

One of the central goals of mathematical epidemiology is to predict disease transmission patterns in populations. Two models are commonly used to predict spatial spread of a disease. The first is the distributed-contacts model, often described by a contact distribution among stationary individuals. The second is the distributed-infectives model, often described by the diffusion of infected individuals. However, neither approach is ideal when individuals move within home ranges. This paper presents a unified modeling hypothesis, called the restricted-movement model. We use this model to predict spatial spread in settings where infected individuals move within overlapping home ranges. Using mathematical and computational approaches, we show that our restricted-movement model has three limits: the distributed-contacts model, the distributed-infectives model, and a third, less studied advective distributed-infectives limit. We also calculate approximate upper bounds for the rates of an epidemic's spatial spread. Guidelines are suggested for determining which limit is most appropriate for a specific disease.

Keywords

Epidemics Invasions Distributed-contacts Distributed-infectives 

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References

  1. Allen, L., Ernest, R., 2002. The impact of long-range dispersal on the rate of spread in population and epidemic models. In: Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.-A. (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. Springer, New York, pp. 183–197.Google Scholar
  2. Alt, W., 1980. Biased random-walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147–177.CrossRefPubMedMathSciNetMATHGoogle Scholar
  3. Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York.Google Scholar
  4. Arino, J., van den Driessche, P., 2003. A multi-city epidemic model. Math. Pop. Stud. 10, 175–193.CrossRefMathSciNetMATHGoogle Scholar
  5. Bailey, N.T.J., 1975. The Mathematical Theory of Infectious Diseases, 2nd ed. Griffin, London.MATHGoogle Scholar
  6. Bauch, C.T., Galvani, A.P., 2003. Using network models to approximate spatial point-process models. Math. Biosci. 184, 101–114.CrossRefPubMedMathSciNetMATHGoogle Scholar
  7. Bian, L., 2004. A conceptual framework for an individual-based spatially explicit epidemiological model. Environ. Plan. B-Plan. Design 31, 381–395.CrossRefGoogle Scholar
  8. Busenberg, S.N., Travis, C.C., 1983. Epidemic models with spatial spread due to population migration. J. Math. Biol. 16, 181–198.CrossRefPubMedMathSciNetMATHGoogle Scholar
  9. Caraco, T., Glavanakov, S., Chen, G., Flaherty, J.E., Ohsumi, T.K., Szymanski, B.K., 2002. Stage-structured infection transmission and a spatial epidemic: A model for Lyme disease. Am. Natural. 160, 348–359.CrossRefGoogle Scholar
  10. Clark, J.S., Lewis, M., Horvath, L., 2001. Invasion by extremes: Population spread with variation in dispersal and reproduction. Am. Natural. 152, 204–224.CrossRefGoogle Scholar
  11. Daniels, H.E., 1975. The deterministic spread of a simple epidemic. In: Gani, J. (Ed.), Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett on the Occasion of his Sixty-Fifth Birthday. Academic Press, London, pp. 373–386.Google Scholar
  12. Doran, R.J., Laffan, S.W., 2005. Simulating the spatial dynamics of foot and mouth disease outbreaks in feral pigs and livestock in Queensland, Australia, using a susceptible-infected-recovered cellular automata model. Prev. Vet. Med. 70, 133–152.CrossRefPubMedGoogle Scholar
  13. Durrett, R., 1999. Stochastic spatial models. SIAM Rev. 41, 677–718.CrossRefMathSciNetMATHGoogle Scholar
  14. Durrett, R., Levin, S., 1994. The importance of being discrete (and spatial). Theor. Pop. Biol. 46, 363–394.CrossRefMATHGoogle Scholar
  15. Eubank, S., Guclu, H., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Toroczkai, Z., Wang, N., 2004. Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184.CrossRefGoogle Scholar
  16. Evans, M., Hastings, N., Peacock, B., 2000. Statistical Distributions, 2nd ed. Wiley, New York.MATHGoogle Scholar
  17. Fenner, F., Henderson, D.A., Arita, I., Jezek, Z., Ladnyi, I., 1988. Smallpox and Its Eradication. World Health Organization, Geneva.Google Scholar
  18. Fife, P.C., 1979. Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics. Springer-Verlag, New York.Google Scholar
  19. Filipe, J.A.N., Maule, M.M., 2003. Analytical methods for predicting the behaviour of population models with general spatial interactions. Math. Biosci. 183, 15–35.CrossRefPubMedMathSciNetMATHGoogle Scholar
  20. Fuks, H., Lawniczak, A.T., 2001. Individual-based lattice model for spatial spread of epidemics. Discrete Dyn. Nat. Soc. 6, 191–200.CrossRefMATHGoogle Scholar
  21. Goel, N.S., Richter-Dyn, N., 1974. Stochastic Models in Biology. Academic Press, New York.Google Scholar
  22. Hadeler, K.P., 2003. The role of migration and contact distributions in epidemic spread. In: Banks, H.T., Castillo-Chavez, C. (Eds.), Bioterrorism: Mathematical Modeling Applications in Homeland Security. SIAM, Philadelphia, pp. 199–210.Google Scholar
  23. Hillen, T., Othmer, H.G., 2000. The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61, 751–775.CrossRefMathSciNetMATHGoogle Scholar
  24. Keeling, M.J., Gilligan, C.A., 2000. Bubonic plague: A metapopulation model of a zoonosis. Proc. R. Soc. Lond. B 267, 2219–2230.CrossRefGoogle Scholar
  25. Kendall, D.G., 1965. Mathematical models of the spread of infection. In Mathematics and Computer Science in Biology and Medicine. H. M. Stationary Office, London, pp. 213–225.Google Scholar
  26. Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, New York.Google Scholar
  27. Kot, M., Medlock, J., Reluga, T., Walton, D.B., 2004. Stochasticity, invasions, and branching random walks. Theor. Pop. Biol. 66, 175–184.CrossRefGoogle Scholar
  28. Lewis, M.A., 2000. Spread rate for a nonlinear stochastic invasion. J. Math. Biol. 41, 430–454.CrossRefPubMedMathSciNetMATHGoogle Scholar
  29. Lewis, M.A., Pacala, S., 2000. Modeling and analysis of stochastic invasion processes. J. Math. Biol. 41, 387–429.CrossRefPubMedMathSciNetMATHGoogle Scholar
  30. Lloyd, A.L., Jansen, V.A.A., 2004. Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models. Math. Biosci. 188, 1–16.CrossRefMathSciNetMATHGoogle Scholar
  31. McKean, H.P., 1975. Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331.CrossRefMathSciNetMATHGoogle Scholar
  32. Medlock, J., Kot, M., 2003. Spreading disease: Integro-differential equations old and new. Math. Biosci. 184, 201–222.CrossRefPubMedMathSciNetMATHGoogle Scholar
  33. Méndez, V., 1998. Epidemic models with an infected-infectious period. Phys. Rev. E 57, 3622–3624.CrossRefGoogle Scholar
  34. Metz, J.A.J., Mollison, D., van den Bosch, F., 1999. The Dynamics of Invasion Waves. Technical Report IR-99-039, International Institute for Applied Systems Analysis. Laxenburg, Austria.Google Scholar
  35. Metz, J.A.J., van den Bosch, F., 1995. Velocities of epidemic spread. In: Mollison, D. (Ed.), Epidemic Models: Their Structure and Relation to Data. Cambridge University Press, Cambridge, UK, pp. 150–186.Google Scholar
  36. Meyers, L.A., Pourbohloul, B., Newman, M.E.J., Skowronski, D.M., Brunham, R.C., 2005. Network theory and SARS: Predicting outbreak diversity. J. Theor. Biol. 232, 71–81.CrossRefPubMedMathSciNetGoogle Scholar
  37. Mollison, D., 1972. The rate of spatial propagation of simple epidemics. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. 3. University of California Press, Berkeley, CA, pp. 579–614.Google Scholar
  38. Mollison, D., 1977. Spatial contact models for ecological and epidemic spread. J. R. Stat. Soc. B 39, 283–326.MathSciNetMATHGoogle Scholar
  39. Mollison, D., 1991. Dependence of epidemic and population velocities on basic parameters. Math. Biosci. 107, 255–287.CrossRefPubMedMATHGoogle Scholar
  40. Newman, M.E.J., 2002. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128.CrossRefMathSciNetGoogle Scholar
  41. Noble, J.V., 1974. Geographic and temporal development of plagues. Nature 250, 726–729.CrossRefPubMedGoogle Scholar
  42. Othmer, H.G., Dunbar, S.R., Alt, W., 1988. Models of dispersal in biological-systems. J. Math. Biol. 26, 263–298.CrossRefPubMedMathSciNetMATHGoogle Scholar
  43. Othmer, H.G., Hillen, T., 2002. The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250.CrossRefMathSciNetMATHGoogle Scholar
  44. Read, J.M., Keeling, M.J., 2003. Disease evolution on networks: the role of contact structure. Proc. R. Soc. Lond. B 270, 699–708.CrossRefGoogle Scholar
  45. Riley, S., Fraser, C., Donnelly, C.A., Ghani, A.C., Abu-Raddad, L.J., Hedley, A.J., Leung, G.M., Ho, L.M., Lam, T.H., Thach, T.Q., Chau, P., Chan, K.P., Leung, P.Y., Tsang, T., Ho, W., Lee, K.H., Lau, E.M.C., Ferguson, N.M., Anderson, R.M., 2003. Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions. Science 300, 1961–1966.CrossRefPubMedGoogle Scholar
  46. Schinazi, R., 1996. On an interacting particle system modeling an epidemic. J. Math. Biol. 34, 915–925.PubMedMathSciNetMATHGoogle Scholar
  47. Schwetlick, H.R., 2000. Travelling fronts for multidimensional nonlinear transport equations. Ann. Inst. Henri Poincare Phys. Theor. 17, 523–550.MathSciNetMATHGoogle Scholar
  48. Snyder, R.E., 2003. How demographic stochasticity can slow biological invasions. Ecology 84, 1333–1339.CrossRefGoogle Scholar
  49. Thomson, N.A., Ellner, S.P., 2003. Pair-edge approximation for heterogeneous lattice population models. Theor. Pop. Biol. 64, 271–280.CrossRefMATHGoogle Scholar
  50. Uhlenbeck, G.E., Ornstein, L.S., 1930. On the theory of the Brownian motion. Phys. Rev. 36, 823–841.CrossRefGoogle Scholar
  51. van den Bosch, F., Metz, J., Zadoks, J., 1999. Pandemics of focal plant disease, a model. Phytopathology 89, 495–505.CrossRefGoogle Scholar
  52. van den Bosch, F., Verhaar, M., Buiel, A., Hoogkamer,W., Zadoks, J., 1990. Focus expansion in plant disease. IV: Expansion rates in mixtures of resistant and susceptible hosts. Phytopathology 80, 598–602.Google Scholar
  53. van den Bosch, F., Zadoks, J., Metz, J., 1988. Focus expansion in plant disease. I: The constant rate of focus expansion. Phytopathology 78, 54–58.Google Scholar

Copyright information

© Society for Mathematical Biology 2006

Authors and Affiliations

  • Timothy C. Reluga
    • 1
  • Jan Medlock
    • 1
  • Alison P. Galvani
    • 1
  1. 1.Department of Epidemiology and Public HealthYale University School of MedicineNew HavenUS

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