Skip to main content
SpringerLink
Log in

Stability in a class of cyclic epidemic models with delay

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

Abstract

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.

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. Grossman, S. I., Miller, R. K.: Perturbation theory for Volterra integrodifferential systems. J. Diff. Equations 8, 457–474 (1971)

    Google Scholar 

  2. Grossman, S. I., Miller, R. K.: Nonlinear Volterra integrodifferential systems with L 1-kernels. J. Diff. Equations 13, 551–556 (1973)

    Google Scholar 

  3. Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335 -356 (1976)

    Google Scholar 

  4. Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. Math., in press (1981)

  5. Hille, E., Phillips, R. S.: Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957

  6. Londen, S. O.: On the variation of the solutions of a nonlinear integral equation. J. Math. Anal. Appl. 52, 430–449 (1975)

    Google Scholar 

  7. Miller, R. K., Nohel, J. A.: A stable manifold theorem for a system of Volterra integrodifferential equations. SIAM J. Math. Anal. 6, 506–522 (1975)

    Google Scholar 

  8. Stech, H. W.: The effect of time lags on the stability of the equilibrium state of a population growth equation. J. Math. Biol. 5, 115–130 (1978)

    Google Scholar 

  9. Stein, E. M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, 1971

  10. Wang, F. J. S.: Perturbation and maximum number of infectives of some SIR epidemic models. SIAM J. Math. Anal. 10, 721–727 (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, 24061, Blacksburg, VA, USA

    Harlan Stech & Michael Williams

Authors
  1. Harlan Stech
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Williams
    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

Stech, H., Williams, M. Stability in a class of cyclic epidemic models with delay. J. Math. Biology 11, 95–103 (1981). https://doi.org/10.1007/BF00275827

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00275827

Key words

Search

Navigation