×

Global behavior and permanence of SIRS epidemic model with time delay. (English) Zbl 1154.34390

Summary: An autonomous SIRS epidemic model with time delay is studied. The basic reproductive number \(R_{0}\) is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.

MSC:

34K20 Stability theory of functional-differential equations
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
93C23 Control/observation systems governed by functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anderson, R.M.; May, R.M., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)
[2] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Convergence results in SIR epidemic models with varying population sizes, J. nonlinear anal., 28, 1909-1921, (1997) · Zbl 0879.34054
[3] Capasso, V., Mathematical structures of epidemic systems, () · Zbl 1141.92035
[4] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000), Wiley Chichester, New York · Zbl 0997.92505
[5] Hale, J.K., Theory of functional differential equations, (1977), Springer New York
[6] Kermark, M.D.; Mckendrick, A.G., Contributions to the mathematical theory of epidemics, Part I. proc. roy. soc. A, 115, 5, 700-721, (1927) · JFM 53.0517.01
[7] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014
[8] Liu, W.M.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023
[9] Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl. math. lett., 17, 1141-1145, (2004) · Zbl 1071.34082
[10] Ma, W.; Takeuchi, Y.; Hara, T.; Beretta, E., Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J., 54, 581-591, (2002) · Zbl 1014.92033
[11] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical modelling and research of epidemic dynamical systems, (2004), Science Press Beijing
[12] Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
[13] Teng, Z.; Chen, L., Uniform persistence and existence of strictly positive solutions in nonautonomous lotka – volterra competitive systems with delays, Comput. math. appl., 37, 61-71, (1999) · Zbl 0942.34061
[14] Teng, Z.; Chen, L., Permanence and extinction of periodic predator – prey systems in a patchy environment with delay, Nonlinear anal.: RWA, 4, 335-364, (2003) · Zbl 1018.92033
[15] Teng, Z.; Li, Z., Permanence and asymptotic behavior of the N-species nonautonomous lotka – volterra competitive systems, Comput. math. appl., 39, 107-116, (2000) · Zbl 0959.34039
[16] Teng, Z.; Yu, Y., The extinction in nonautonomous prey – predator lotka – volterra systems, Acta math. appl. sinica, 15, 401-408, (1999) · Zbl 1007.92031
[17] Thieme, H.R., Uniform persistence and permanence for nonautonomous semiflows in population biology, Math. biosci., 166, 173-201, (2000) · Zbl 0970.37061
[18] Wang, W., Global behavior of an SEIRS epidemic model with time delays, Appl. math. lett., 15, 423-428, (2002) · Zbl 1015.92033
[19] Wang, W.; Ruan, G., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differential equations, 188, 135-163, (2003) · Zbl 1028.34046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.