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On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate. (English) Zbl 1203.92049
Summary: We study an epidemic model for infections with non-permanent acquired immunity (SIRS). The incidence rate is assumed to be a general nonlinear function of the susceptibles and the infectious classes. By using a peculiar Lyapunov function, we obtain necessary and sufficient conditions for the local nonlinear stability of equilibria. Conditions ensuring the global stability are also obtained. Unlike the recent literature on this subject, here no restrictions are required about the monotonicity and concavity of the incidence rate with respect to the infectious class. Among the applications, the noteworthy case of a convex incidence rate is provided.

MSC:
92C60 Medical epidemiology
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
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