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Global properties of infectious disease models with nonlinear incidence. (English) Zbl 1298.92101
Summary: We consider global properties for the classical SIR, SIRS and SEIR models of infectious diseases, including the models with the vertical transmission, assuming that the horizontal transmission is governed by an unspecified function \(f(S,I)\). We construct Lyapunov functions which enable us to find biologically realistic conditions sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. This state can be either endemic, or infection-free, depending on the value of the basic reproduction number.

MSC:
92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
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