Journal of Mathematical Biology

, Volume 31, Issue 5, pp 495–512 | Cite as

A disease transmission model in a nonconstant population

  • W. R. Derrick
  • P. van den Driessche
Article

Abstract

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.

Key words

Epidemiological model Nonlinear incidence function Hopf bifurcation Homoclinic loop Saddle connection 

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References

  1. Busenberg, S., van den Driessche, P.: Analysis of a disease transmission model in a population with varying size. J. Math. Biol. 28, 257–270 (1990)Google Scholar
  2. Busenberg, S., van den Driessche, P. Nonexistence of periodic solutions for a class of epidemiological models. In: Busenberg, S., Martelli, M. (eds.) Biology, Epidemiology, and Ecology. (Lect. Notes Biomath., vol. 92, pp. 70–79) Berlin Heidelberg New York: Springer 1991Google Scholar
  3. Castillo-Chavez, C., Hethcote, H. W., Andreasen, V., Levin, S. A., Liu, W. M.: Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27, 233–258 (1989)Google Scholar
  4. Diekmann, O., Kretzschmar,M.: Patterns in the effects of infectious diseases on population growth. J. Math. Biol. 29, 539–570 (1991)MathSciNetMATHGoogle Scholar
  5. Doedel, E.: AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology (1986)Google Scholar
  6. Ermentrout, B.: PhasePlane: The Dynamical System's Tool, Version 3.0. Pacific Grove, CA: Brooks/Cole 1990Google Scholar
  7. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Berlin Heidelberg New York: Springer 1983Google Scholar
  8. Hahn, W.: Stability of Motion. Berlin Heidelberg New York: Springer 1967Google Scholar
  9. Hethcote, H. W., Levin S. A.: Periodicity in epidemiological models. In: Levin, S. A., Hallam, T. G., Gross, L. J. (eds.) Applied Mathematical Ecology. (Biomath., vol. 18). Berlin Heidelberg New York: Springer 1989Google Scholar
  10. Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., Perry, T.: Modelling and analyzing HIV transmission: the effect of contact patterns. Math. Biosci. 92, 119–199 (1988)Google 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., Hethcote, H. W., Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359–380 (1987)Google Scholar
  13. Mena-Lorca, J., Hethcote, H. W.: Dynamic models of infectious diseases as regulators of population size. J. Math. Biol. 30, 693–716 (1992)Google Scholar
  14. Nold, A.: Heterogeneity in disease-transmission modeling. Math. Biosci. 52, 227–240 (1980)Google Scholar
  15. Revelle, C., Lynn, W. R., Feldmann, F.: Mathematical models for the economic allocation of tuberculosis control activities in developing nations. Am. Rev. Respir. Dis. 96, 893–909 (1967)Google Scholar
  16. Tudor, D.: A deterministic model for herpes infections in human and animal populations. SIAM Rev. 32, 136–139 (1990)Google Scholar
  17. Westphal, H.: Zur Abschätzung der Lösungen nichlinearer parabolischer Differentialgleichungen. Math. Z. 51, 690–695 (1947/49)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • W. R. Derrick
    • 1
  • P. van den Driessche
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of MontanaMissoulaUSA
  2. 2.Department of MathematicsUniversity of VictoriaVictoriaCanada
  3. 3.Mathematisches InstitutUniversität MünchenMünchen 2Germany

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