Advertisement

Journal of Mathematical Biology

, Volume 14, Issue 1, pp 117–127 | Cite as

Prelude to hopf bifurcation in an epidemic model: Analysis of a characteristic equation associated with a nonlinear Volterra integral equation

  • O. Diekmann
  • R. Montijn
Article

Abstract

We discuss a simple deterministic model for the spread, in a closed population, of an infectious disease which confers only temporary immunity. The model leads to a nonlinear Volterra integral equation of convolution type. We are interested in the bifurcation of periodic solutions from a constant solution (the endemic state) as a certain parameter (the population size) is varied. Thus we are led to study a characteristic equation. Our main result gives a fairly detailed description (in terms of Fourier coefficients of the kernel) of the traffic of roots across the imaginary axis. As a corollary we obtain the following: if the period of immunity is longer than the preceding period of incubation and infectivity, then the endemic state is unstable for large population sizes and at least one periodic solution will originate.

Key words

Epidemic model Temporary immunity Nonlinear Volterra integral equation Characteristic equation Hopf bifurcation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Busenberg, S., Cooke, K. L.: The effect of integral conditions in certain equations modelling epidemics and population growth. J. Math. Biol.10, 13–32 (1980)CrossRefGoogle Scholar
  2. 2.
    Cushing, J. M.: Nontrivial periodic solutions of some Volterra integral equations. In: Volterra equations, S.-O. Londen, O. J. Staffans (eds.), Vol. 737, pp. 50–66, Lecture notes in mathematics. Berlin-Heidelberg-New York: Springer 1979Google Scholar
  3. 3.
    Cushing, J. M.: Bifurcation of periodic solutions of nonlinear equations in age-structured population dynamics. To appear in the Proceedings of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Arlington, 1980Google Scholar
  4. 4.
    Cushing, J. M., Simmes, S.D.: Bifurcation of asymptotically periodic solutions of Volterra integral equations. J. Int. Eq.2, 339–361 (1980)Google Scholar
  5. 5.
    Diekmann, O.: Volterra integral equations and semigroups of operators. Math. Centre Report TW 197, 1980Google Scholar
  6. 6.
    Diekmann, O., Van Gils, S. A.: A variation of constants formula for nonlinear Volterra integral equations of convolution type. In: Nonlinear Differential Equations: Invariance, Stability and Bifurcation. P. de Mottoni, L. Salvadori (eds.), pp. 133–143, Academic Press, 1981Google Scholar
  7. 7.
    Diekmann, O., Van Gils, S. A.: Invariant manifolds for Volterra integral equations of convolution type. (Preprint) Math. Centre Report TW 219, 1981Google Scholar
  8. 8.
    Gripenberg, G.: Periodic solutions of an epidemic model. J. Math. Biol.10, 271–280 (1980)CrossRefGoogle Scholar
  9. 9.
    Hale, J. K.: Behavior near constant solutions of functional differential equations. J. Diff. Eq.15, 278–294 (1974)CrossRefGoogle Scholar
  10. 10.
    Hale, J. K.: Nonlinear oscillations in equations with delays. In: F. Hoppensteadt (ed.), Nonlinear oscillations in biology. Providence: AMS 1979Google Scholar
  11. 11.
    Hale, J. K., De Oliveira, J. C. F.: Hopf bifurcation for functional equations. J. Math. Anal. Appl.74, 41–59 (1980)CrossRefGoogle Scholar
  12. 12.
    Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Applied Math.40, 1–9 (1981)CrossRefGoogle Scholar
  13. 13.
    Hethcote, H. W., Stech, H. W., Van den Driessche, P.: Stability analysis for models of diseases without immunity. J. Math. Biol.13, 185–198 (1981)Google Scholar
  14. 14.
    Hethcote, H. W., Tudor, D. W.: Integral equations describing endemic infectious diseases. J. Math. Biol.9, 37–48 (1980)CrossRefGoogle Scholar
  15. 15.
    Hoppensteadt, F.: Mathematical theories of populations: Demographics, genetics, and epidemics. Philadelphia: SIAM 1975Google Scholar
  16. 16.
    Lauwerier, H. A.: Mathematische modellen voor epidemische processen, unpublished manuscript (in Dutch)Google Scholar
  17. 17.
    Lauwerier, H. A.: Mathematical models of epidemics. Math. Centre Tract138, 1981Google Scholar
  18. 18.
    Montijn, R.: Een karakteristieke vergelijking uit de mathematische epidemiologie. Math. Centre Report TN 94, 1980 (in Dutch)Google Scholar
  19. 19.
    Stech, H. W., Williams, M.: Stability in a class of cyclic epidemic models with delay. J. Math. Biol.11, 95–103 (1981)CrossRefGoogle Scholar
  20. 20.
    Turyn, L.: Functional difference equations and an epidemic model. To appear in the Proceedings of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Arlington, 1980Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • O. Diekmann
    • 1
  • R. Montijn
    • 1
  1. 1.Mathematisch CentrumSJ AmsterdamThe Netherlands

Personalised recommendations