Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T11:13:27.247Z Has data issue: false hasContentIssue false
  • English
  • Français

On the global stability of a delay epidemic model

Published online by Cambridge University Press:  17 February 2009

Xiaodong Lin
Xiaodong Lin
Affiliation:
Dept. of Applied Maths, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study the asymptotic behavior of an SIRS epidemic model with a time delay in the recovered class and a nonlinear incidence rate. A conjecture of Hethcote et al. [5] on the global stability of the disease-free equilibrium is solved. Moreover, we analyse the model when the contact number takes its threshold value. We show that solutions tend to either the disease-free equilibrium or to a unique positive endemic equilibrium, and there is no periodic solution.

Type
Research Article
Information
The ANZIAM Journal , Volume 34 , Issue 1 , July 1992 , pp. 26 - 34
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Hale, J. K., Theory of functional differential equations, (Springer, Berlin, 1977).Google Scholar
[2] Hethcote, H. W., “Qualitative analyses of communicable disease models,” Math. Biosci. 28 (1976) 335–356.Google Scholar
[3] Hethcote, H. W., Stech, H. W. and Driessche, P. van den, “Nonlinear oscillations in epidemic models”, SIAM J. Appl. Math. 40 (1981) 1–9.CrossRefGoogle Scholar
[4] Hethcote, H. W., Stech, H. W. and Driessche, P. van den, “Periodicity and stability in epidemic models: A survey,” in Differential equations and applications in ecology, epidemics and population problems, (eds. Busenberg, S. W. and Cooke, K. L.), (Academic Press, New York, 1981) 65–82.Google Scholar
[5] Hethcote, H. W., Lewis, M. A. and Driessche, P. van den, “An epidemiological model with a delay and a nonlinear incidence rate,”. Math. Biol. 27 (1989) 49–64.Google Scholar
[6] Liu, W. M., Hethcote, H. W. and Levin, S. A., “Dynamical behavior of epidemiological models with nonlinear incidence rates,” J. Math. Biol. 25 (1987) 359–380.Google Scholar
[7] Liu, W. M., Levin, S. A. and Iwasa, Y., “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” J. Math. Biol. 23 (1986) 187–204.CrossRefGoogle ScholarPubMed