Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate. (English) Zbl 1144.34374

Summary: We derive and study a time-delayed SIR model with a general incidence rate. The time delay represents temporary immunity period, i.e. time from recovery to becoming susceptible again. Both trivial and endemic equilibria are found, and their stability is investigated. Using Lyapunov functional approach, the global stability of an endemic equilibrium is shown. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.


34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology
92D30 Epidemiology


Full Text: DOI


[1] Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear anal., 2, 35-74, (2001) · Zbl 1015.92049
[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
[3] Beretta, E.; Takeuchi, Y., Convergence results in SIR epidemic models with varying population sizes, Nonlinear anal., 28, 1909-1921, (1997) · Zbl 0879.34054
[4] Cooke, K.L., Stability analysis for a vector disease model, Rocky mount. J. math., 9, 253-263, (1979)
[5] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseasesmodel building, analysis and interpretation, (2000), Wiley New York
[6] Kuang, Y., Delay differential equations with applications in population biology, (1993), Academic Press New York
[7] Li, B.; Kuang, Y., Simple food chain in a chemostat with distinct removal rates, J. math. anal. appl., 242, 75-92, (2000) · Zbl 0943.92034
[8] Liu, W.; Hethcote, H.W.; Levin, S.A., Dynamical behaviour of epidemiological models with nonlinear incidence rate, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014
[9] Liu, W.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon behaviour of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023
[10] Moghadas, S.M.; Gumel, A.B., Global stability of a two-stage epidemic model with generalized non-linear incidence rate, Math. comput. simulation, 60, 107-118, (2002) · Zbl 1005.92031
[11] Shampine, L.F.; Thompson, S., Solving DDEs in MATLAB, Appl. numer. math., 37, 441-458, (2001) · Zbl 0983.65079
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.