Skip to main content
SpringerLink
Log in

Periodic solutions of an epidemic model

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Summary

The existence of periodic solutions of the equation

$$x(t) = k\left( {P - \int_{ - \infty }^t A(t - s)x(s)ds } \right)\int_{ - \infty }^t a(t - s)x(s)ds$$

is established. This equation arises in the study of the spread of a disease which does not induce permanent immunity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atkinson, C.: The existence of steady wave solutions to a nonlocal version of the K.P.P. equation. Studies in Appl. Math. 59, 95–115 (1978)

    Google Scholar 

  2. Bailey, N. T. J.: The mathematical theory of infectious diseases and its applications, 2nd ed. New York: Hafner Press 1975

    Google Scholar 

  3. Busenberg, S., Cooke, K. L.: Periodic solutions of a periodic nonlinear delay differential equation. SIAM J. Appl. Math. 35, 704–721 (1978)

    Google Scholar 

  4. Capasso, V.: Global solutions for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35, 274–284 (1978)

    Google Scholar 

  5. Crandall, M. G., Rabinowitz, P. H.: The Hopf bifurcation theorem. MRC Tech. Summary Report # 1604, University of Wisconsin-Madison 1976

  6. Cushing, J. M.: Bifurcation and periodic solutions of integrodifferential systems and applications to time delay models in population dynamics. SIAM J. Appl. Math. 33, 640–654 (1977)

    Google Scholar 

  7. Cushing, J. M.: Nontrivial periodic solutions of integrodifferential equations. J. Integral Equations 1, 165–181 (1979)

    Google Scholar 

  8. Cushing, J. M.: Nontrivial periodic solutions of some Volterra integral equations. In: Volterra equations (Londen, S.-O., Staffans, O. J., eds.). Lecture Notes in Mathematics # 737. Berlin: Springer-Verlag 1979

    Google Scholar 

  9. Diekmann, O.: Limiting behaviour in an epidemic model. Nonlinear Anal., Theory, Methods, Appl. 1, 459–470 (1977)

    Google Scholar 

  10. Diekmann, O.: Run for your life. A note on the asymptotic speed of propagation of an epidemic. J. Differential Equations 33, 58–73 (1979)

    Google Scholar 

  11. Gripenberg, G.: On some epidemic models. Quart. Appl. Math., to appear

  12. Smith, H.: Periodic solutions of an epidemic model with a threshold. Rocky Mountain J. Math. 9, 131–142 (1979)

    Google Scholar 

  13. Thieme, H. R.: Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J. Reine Angew. Math. 306, 94–121 (1979)

    Google Scholar 

  14. Waltman, P.: Deterministic threshold models in the theory of epidemics. Lecture Notes in Biomathematics, Vol. I. New York: Springer-Verlag 1974

    Google Scholar 

  15. Wierman, J. C.: The front velocity of the simple epidemic. J. Appl. Prob. 16, 409–415 (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Helsinki University of Technology, SF-02150, Espoo 15, Finland

    Gustaf Gripenberg

Authors
  1. Gustaf Gripenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gripenberg, G. Periodic solutions of an epidemic model. J. Math. Biology 10, 271–280 (1980). https://doi.org/10.1007/BF00276986

Download citation

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00276986

Key words

Search

Navigation