Journal of Mathematical Biology

, Volume 25, Issue 4, pp 359–380 | Cite as

Dynamical behavior of epidemiological models with nonlinear incidence rates

  • Wei-min Liu
  • Herbert W. Hethcote
  • Simon A. Levin
Article

Abstract

Epidemiological models with nonlinear incidence rates λIpSqshow a much wider range of dynamical behaviors than do those with bilinear incidence rates λIS. These behaviors are determined mainly by p and λ, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.

Key words

Epidemiological models Nonlinear incidence rates Hopf bifurcation Homoclinic-loop bifurcation 

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References

  1. Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications 2nd edn. London: Griffin 1975Google Scholar
  2. Capasso, V., Serio, G.: A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)Google Scholar
  3. Carr, J.: Applications of centre manifold theory. Berlin Heidelberg New York: Springer 1981Google Scholar
  4. Cunningham, J.: A deterministic model for measles. Z. Naturforsch. 34c, 647–648 (1979)Google Scholar
  5. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, pp. 150–156. Berlin Heidelberg New York Tokyo: Springer 1983Google Scholar
  6. Hale, J. K.: Ordinary differential equations. New York: Wiley-Interscience 1969Google Scholar
  7. Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335–356 (1976)Google Scholar
  8. Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol. 13, 185–198 (1981a)Google Scholar
  9. Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Periodicity and stability in epidemic models: a survey. In: Cooke, K. L. (ed.) Differential equations and applications in ecology, epidemics, and population problems, pp. 65–82. New York London Toronto Sydney San Francisco: Academic Press 1981bGoogle Scholar
  10. Liu, W. M., Levin, S. A.: Influenza and some related mathematical models. In: Levin, S. A., Hallam, T., Gross, L. (eds.) Applied mathematical ecology. Berlin Heidelberg New York: SpringerGoogle Scholar
  11. Liu, W. M., Levin, S. A., Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23, 187–204 (1986)Google Scholar
  12. Liu, W. M.: Dynamics of epidemiological models-recurrent outbreaks in autonomous systems. Ph.D. Thesis, Cornell University (1987)Google Scholar
  13. Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications. Berlin Heidelberg New York: Springer 1976Google Scholar
  14. Saunders, I. W.: A model for myxomatosis. Math. Biosci. 48, 1–15 (1980)Google Scholar
  15. Wang, F. J. S.: Asymptotic behavior of some deterministic epidemic models. SIAM J. Math. Anal. 9, 529–534 (1978)Google Scholar
  16. Wilson, E. B., Worcester, J.: The law of mass action in epidemiology. Proc. Natl. Acad. Sci. USA 31, 24–34 (1945)Google Scholar
  17. Wilson, E. B., Worcester, J.: The law of mass action in epidemiology II. Proc. Natl. Acad. Sci. USA 31, 109–116 (1945)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Wei-min Liu
    • 1
    • 2
  • Herbert W. Hethcote
    • 4
  • Simon A. Levin
    • 1
    • 2
    • 3
  1. 1.Center for Applied MathematicsIthacaUSA
  2. 2.Section of Ecology and Systematics, Corson HallCornell UniversityIthacaUSA
  3. 3.Ecosystems Research CenterIowa CityUSA
  4. 4.Department of MathematicsUniversity of IowaIowa CityUSA

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