Bulletin of Mathematical Biology

, Volume 63, Issue 3, pp 547–571

Models of infectious diseases in spatially heterogeneous environments



Most models of dynamics of infectious diseases have assumed homogeneous mixing in the host population. However, it is increasingly recognized that heterogeneity can arise through many processes. It is then important to consider the existence of subpopulations of hosts, and that the contact rate within subpopulations is different than that between subpopulations. We study models with hosts distributed in subpopulations as a consequence of spatial partitioning. Two types of models are considered. In the first one there is direct transmission. The second one is a model of dynamics of a mosquito-borne disease, with indirect transmission, and applicable to malaria. The contact between subpopulations is achieved through the visits of hosts. Two types of visit are considered: a first one in which the visit time is independent of the distance travelled, and a second one in which visit time decreases with distance. There are two types of spatial arrangement: one dimensional, and two dimensional. Conditions for the establishment of the disease are obtained. Results indicate that the disease becomes established with greater difficulty when the degree of spatial partition increases, and when visit time decreases. In addition, when visit time decreases with distance, the establishment of the disease is more difficult when the spatial arrangement is one dimensional than when it is two dimensional. The results indicate the importance of knowing the spatial distribution and mobility patterns to understand the dynamics of infectious diseases. The consequences of these results for the design of public health policies are discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderson, R. M. and R. M. May (1992). Infectious Diseases of Humans, Oxford: Oxford University Press.Google Scholar
  2. Andreasen, V. and F. B. Christiansen (1989). Persistence of an infectious disease in a subdivided population. Math. Biosci. 96, 239–253.MathSciNetCrossRefGoogle Scholar
  3. Aron, J. L. and R. M. May (1982). The population dynamics of malaria, in Population Dynamics of Infectious Diseases: Theory and Applications, R. M. Anderson (Ed.), London: Chapman & Hall, pp. 139–179.Google Scholar
  4. Bailey, N. T. J. (1982). The Biomathematics of Malaria, London: Charles Griffin.Google Scholar
  5. Ball, F. (1999). Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156, 41–67.MATHMathSciNetCrossRefGoogle Scholar
  6. Bascompté, J. and R. V. Sole (1998). Modelling Spatiotemporal Dynamics in Ecology, New York: Springer.Google Scholar
  7. Becker, N. G., A. Bahrampour and K. Dietz (1995). Threshold parameters for epidemics in different community settings. Math. Biosci. 129, 189–208.CrossRefGoogle Scholar
  8. Becker, N. G. and K. Dietz (1995). The effect of household distribution on transmission and control of highly infectious diseases. Math. Biosci. 127, 207–219.CrossRefGoogle Scholar
  9. Becker, N. G. and R. Hall (1996). Immunization levels for preventing epidemics in a community of households made up of individuals of various types. Math. Biosci. 132, 205–216.CrossRefGoogle Scholar
  10. Becker, N. G. and D. N. Starczak (1997). Optimal vaccination strategies for a community of households. Math. Biosci. 139, 117–132.CrossRefGoogle Scholar
  11. Begon, M., J. L. Harper and C. R. Townsend (1996). Ecology, Oxford: Blackwell.Google Scholar
  12. Berlin, T. H. and M. Kac (1952). The spherical model of a ferromagnet. Phys. Rev. 86, 821–835.MathSciNetCrossRefGoogle Scholar
  13. Collins, F. H. and S. M. Paskewitz (1995). Malaria: current and future prospects. Ann. Rev. Entomol. 40, 195–219.CrossRefGoogle Scholar
  14. De Jong, M. C. M., O. Diekmann and J. A. P. Heesterbeck (1994). The computation of R 0 for discrete-time epidemic models with dynamic heterogeneity. Math. Biosci. 119, 97–114.CrossRefGoogle Scholar
  15. Diekmann, O., J. A. P. Hesterbeck and J. A. J. Metz (1990). On the definition and the computation of the basic reproduction rate ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382.MathSciNetCrossRefGoogle Scholar
  16. Dietz, K. (1988). Mathematical models for transmission and control of malaria, in Principles and Practice of Malariology, W. Wernsdorfer and Y. McGregor (Eds), Edinburgh: Churchill Livingstone, pp. 1091–1133.Google Scholar
  17. Dye, C. and G. Hasibeder (1986). Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others. Trans. R. Soc. Trop. Med. Hyg. 80, 69–77.CrossRefGoogle Scholar
  18. Gilpin, M. and I. Hanski (Eds) (1991). Metapopulation Dynamics: Empirical and Theoretical Investigations, New York: Academic Press.Google Scholar
  19. Gratz, N. G. (1999). Emerging and resurging vector-borne diseases. Ann. Rev. Entomol. 44, 51–75.CrossRefGoogle Scholar
  20. Grenfell, B. T. and A. P. Dobson (Eds) (1995). Ecology of Infectious Diseases in Natural Populations, Cambridge: Cambridge University Press.Google Scholar
  21. Grenfell, B. T. and J. Harwood (1997). (Meta)populations dynamics of infectious diseases. Trends Ecol. Evol. 12, 395–399.CrossRefGoogle Scholar
  22. Hanski, I. (1999). Metapopulation Ecology, Oxford: Oxford University Press.Google Scholar
  23. Hanski, I. and M. E. Gilpin (1997). Metapopulation Biology. Ecology, Genetics, and Evolution, New York: Academic Press.Google Scholar
  24. Hasibeder, G. and C. Dye (1988). Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environment. Theor. Popul. Biol. 33, 31–53.MathSciNetCrossRefGoogle Scholar
  25. Hess, G. (1996). Disease in metapopulation models: implications for conservation. Ecology 77, 1617–1632.CrossRefGoogle Scholar
  26. Hethcote, H. W. (1978). An immunization model for a heterogeneous population. Theor. Popul. Biol. 14, 338–349.MATHMathSciNetCrossRefGoogle Scholar
  27. Hethcote, H. W. and H. R. Thieme (1985). Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci. 75, 205–277.MathSciNetCrossRefGoogle Scholar
  28. Hethcote, H. W. and J. W. Van Ark (1987). Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs. Math. Biosci. 84, 85–118.MathSciNetCrossRefGoogle Scholar
  29. Lajmanovich, A. and J. A. Yorke (1976). A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 28, 221–236.MathSciNetCrossRefGoogle Scholar
  30. Levins, R., T. Awerbuch, U. Brinkman, I. Eckardt, P. Epstein, N. Makhoul, C. Albuquerque de Posas, C. Puccia, A. Spielman and M. Wilson (1994). The emergence of new diseases. Am. Sci. 82, 52–60.Google Scholar
  31. Longini, I. M., Jr (1988). A mathematical model for predicting the geographical spread of new infectious agents. Math. Biosci. 90, 367–383.MATHMathSciNetCrossRefGoogle Scholar
  32. Longini, I. M., Jr, P. E. M. Fine and S. B. Thacker (1986). Predicting the global spread of new infectious agents. Am. J. Epidemiol. 123, 383–391.Google Scholar
  33. Macdonald, G. (1957). The Epidemiology and Control of Malaria, London: Oxford University Press.Google Scholar
  34. May, R. M. (1974). Stability and Complexity in Model Ecosystems. Monographs in Population Biology 6, Princeton: Princeton University Press.Google Scholar
  35. May, R. M. and R. M. Anderson (1984). Spatial Heterogeneity and the design of immunization programs. Math. Biosci. 72, 83–111.MathSciNetCrossRefGoogle Scholar
  36. Nold, A. (1980). Heterogeneity in disease-tansmission modelling. Math. Biosci. 52, 227–240.MATHMathSciNetCrossRefGoogle Scholar
  37. Post, W. M., D. L. DeAngelis and C. C. Travis (1983). Endemic disease in environments with spatially heterogeneous host populations. Math. Biosci. 63, 289–302.MathSciNetCrossRefGoogle Scholar
  38. Prothero, M. (1991). Resettlement and health: Amazonian and tropical perspective, in A Desordem Ecologico na Amazonia, L. E. Aragon (Ed.), Belem: Editora Universitaria, pp. 161–182.Google Scholar
  39. Ross, R. (1911). The Prevention of Malaria, London: Murray.Google Scholar
  40. Rubio-Palis, Y., R. A. Wirtz and C. F. Curtis (1992). Malaria entomological inoculation rates in western Venezuela. Acta Tropica 52, 167–174.CrossRefGoogle Scholar
  41. Rushton, S. and A. J. Mautner (1955). The deterministic model of a simple epidemic for more than one community. Biometrika 42, 126–132.MathSciNetCrossRefGoogle Scholar
  42. Rvachev, L. A. and I. M. Longini, Jr (1985). A mathematical model for the global spread of influenza. Math. Biosci. 75, 3–22.MathSciNetCrossRefGoogle Scholar
  43. Sandia-Mago, A. (1994). Venezuela: malaria y movilidad humana estacional de las comunidades indígenas del río Riecito del estado Apure. Fermentum 3/4, 102–123.Google Scholar
  44. Sattenspiel, L. and K. Dietz (1995). A structured epidemic model incorporating geographic mobility among regions. Math. Biosci. 128, 71–91.CrossRefGoogle Scholar
  45. Sattenspiel, L. and D. A. Herring (1998). Structured epidemic models and the spread of influenza in the Central Canada Subarctic. Hum. Biol. 70, 91–115.Google Scholar
  46. Sattenspiel, L., A. Mobarry and D. A. Herring (2000). Modeling the influence of settlement structure on the spread of influenza among communities. Am. J. Hum. Biol. 12, 736–748.CrossRefGoogle Scholar
  47. Sattenspiel, L. and C. P. Simon (1988). The spread and persistence of infectious diseases in structured populations. Math. Biosci. 90, 341–366.MathSciNetCrossRefGoogle Scholar
  48. Searle, S. R. (1982). Matrix Algebra Useful for Statistics, New York: Wiley.Google Scholar
  49. Sifontes, R. (1985). VenezuelaLa, in Escuela de Malariología y el Saneamiento Ambiental y la Accíon Sanitaria en las Repúblicas Latinoamericanas, Caracas: Fundación Bicentenario de Simón Bolívar, pp. 519–559.Google Scholar
  50. Thrall, P. H. and J. J. Burdon (1997). Host-pathogen dynamics in a metapopulation context: the ecological and evolutionary consequences of being spatial. J. Ecol. 85, 743–753.Google Scholar
  51. Tilman, D. and P. Kareiva (Eds) (1997). Spatial Ecoloy. Monographs in Population Biology, 30, Princeton: Princeton University Press.Google Scholar
  52. Torres-Sorando, L. J. (1998). Modelos Espacio-temporales y Estudio del Comportamiento Dinámico de la Incidencia de Malaria en Venezuela, PhD thesis, Universidad Central de Venezuela, Caracas.Google Scholar
  53. Torres-Sorando, L. J. and D. J. Rodríguez (1997). Models of spatio-temporal dynamics in malaria. Ecol. Modelling 104, 231–240.CrossRefGoogle Scholar
  54. Travis, C. C. and S. M. Lenhart (1987). Eradication of infectious diseases in heterogeneous populations. Math. Biosci. 83, 191–198.MathSciNetCrossRefGoogle Scholar
  55. Watson, R. K. (1972). On an epidemic in a stratified population. J. Appl. Prob. 9, 659–666.MATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2001

Authors and Affiliations

  1. 1.Instituto de Zoología TropicalUniversidad Central de VenezuelaCaracasVenezuela

Personalised recommendations