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Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity. (English) Zbl 1193.34115
Summary: In this paper, a delayed SIRS epidemic model with saturation incidence and temporary immunity is investigated. The immunity gained by experiencing a disease is temporary, whenever infected the diseased individuals will return to the susceptible class after a fixed period. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.
34D23Global stability of ODE
34K20Stability theory of functional-differential equations
Full Text: DOI
[1] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal. 47, 4107-4115 (2001) · Zbl 1042.34585 · doi:10.1016/S0362-546X(01)00528-4
[2] Beretta, E.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delays, J. math. Biol. 33, 250-260 (1995) · Zbl 0811.92019 · doi:10.1007/BF00169563
[3] Beretta, E.; Takeuchi, Y.: Convergence results in SIR epidemic model with varying population sizes, Nonlinear anal. 28, 1909-1921 (1997) · Zbl 0879.34054 · doi:10.1016/S0362-546X(96)00035-1
[4] Cooke, K. L.: Stability analysis for a vector disease model, Rocky mountain J. Math. 9, 31-42 (1979) · Zbl 0423.92029 · doi:10.1216/RMJ-1979-9-1-31
[5] Gao, S.; Chen, L.; Nieto, J.; Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24, 6037-6045 (2006)
[6] Kyrychkoa, Y. N.; Blyuss, K. B.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. Real world appl. 6, 495-507 (2005) · Zbl 1144.34374 · doi:10.1016/j.nonrwa.2004.10.001
[7] 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 · doi:10.1016/j.aml.2003.11.005
[8] 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 · doi:10.2748/tmj/1113247650
[9] Takeuchi, Y.; Ma, W.: Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dyn. contin. Discrete impuls. Syst. 5, 171-184 (1999) · Zbl 0937.92026
[10] Takeuchi, Y.; Ma, W.; Beretta, E.: Global asymptotic properties of a SIR epidemic model with finite incubation time, Nonlinear anal. 42, 931-947 (2000) · Zbl 0967.34070 · doi:10.1016/S0362-546X(99)00138-8
[11] Wen, L.; Yang, X.: Global stability of a delayed SIRS model with temporary immunity, Chaos solitons fractals 38, 221-226 (2008) · Zbl 1142.34354 · doi:10.1016/j.chaos.2006.11.010
[12] Zhang, T.; Teng, Z.: Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear anal. Real world appl. 9, 1409-1424 (2008) · Zbl 1154.34390 · doi:10.1016/j.nonrwa.2007.03.010
[13] Brauer, F.; Den Driessche, P. Van; Wang, L.: Oscillations in a patchy environment disease model, Math. biosci. 215, 1-10 (2008) · Zbl 1176.34098 · doi:10.1016/j.mbs.2008.05.001
[14] Capasso, V.; Serio, G.: A generalization of the kermack--mckendrick deterministic epidemic model, Math. biosci. 42, 41-61 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8
[15] 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 · doi:10.1007/BF00277162
[16] 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 · doi:10.1007/BF00276956
[17] Hale, J.: Theory of functional differential equations, (1977) · Zbl 0352.34001
[18] Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. anal. 33, 1144-1165 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086