Abstract
An SIS model with periodic maximum infectious force, recruitment rate and the removal rate of the infectives has been investigated in this article. Sufficient conditions for the permanence and extinction of the disease are obtained. Furthermore, the existence and global stability of positive periodic solution are established. Finally, we present a procedure by which one can control the parameters of the model to keep the infectives stay eventually in a desired set.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beretta E., Capsso V., Rinldi F. Global stability results for a generalized Lotka-Volterra system with distributed delays. J. Math. Biol. 26: 661–688 (1988)
Brauer, F. Models for universally fatal diseases, II. In: Differential Equtions Models in Biology, Epidemiology and Ecology, Busenberg, S. and Martelli, M. ed., Lecture Notes in Biomathematics 92, Springer-Verlag, Berlin, Heidelberg, New York, 57–69, 1991
Bruer, F. Models for the spread of universally fatal diseases. J. Math. Biol., 28: 451–462 (1990)
Cooke, K.L. Stability analysis for a vector disease model. Rocky Mount. J. Math., 9: 31–42 (1979)
Cooke, K.L., Yorke, J.A. Some equations modeling growth processes and gonorrhea epidemics. Math. Biosci. 16: 75–101 (1973)
Goplsamy, K. Stability and oscillations in delay differential equations of population dynamics. Kluwer Academic Publishers, Dordrecht, 1992
Greenberg, J.M., Hoppensteadt, F. Asymptotic behavior of solutions to a population equation. SIAM J. Appl. Math. 28: 662–674 (1975)
Herthcote, H.W., Stech, H.W., van Driessche, P. Periodicity and stability in epidemic models: a survey. In: Differential Equations and Applications in Ecology, Epidemics and Population Problems, Busenberg, S.N. and Cooke, K.L. eds., Academic Press, New York, 65–82, 1981
Herthcote, H.W., van den Driessche, P. Two SIS Epidemiologic Models With Delays. J. Math. Biol., 40: 3–26 (2000)
Herthcote, H.W., van den Driessche, P. An SIS epidemic model with variable population size and a delay. J. Math. Biol., 34: 177–194 (1995)
Sawano, K. Some considerations on the fundamental theorems for functional differential equations with infinite delay. Funkcial. Ekvac., 25: 97–104 (1982)
Yuan, S., Ma, Z.E. Global stability and Holf bifurcation of an SIS epidemic model with time delays. Journal of Systems Science and Complexity, 14(3): 327–336 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the National Nature Science Foundation of China (No. 19971066) and the Youth Sciences Foundation of Shanxi Province (No. 20021003).
Rights and permissions
About this article
Cite this article
Yuan, Sl., Ma, Ze. & Jin, Z. Persistence and Periodic Solution on a Nonautonomous SIS Model with Delays. Acta Mathematicae Applicatae Sinica, English Series 19, 167–176 (2003). https://doi.org/10.1007/s10255-003-0093-3
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10255-003-0093-3