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Global dynamics of an SEIR epidemic model with saturating contact rate. (English) Zbl 1021.92040
Summary: J. A. P. Heesterbeek and J. A. P. Metz [J. Math. Biol. 31, 529-539 (1993; Zbl 0770.92021)] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number \(R_0\) is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If \(R_0\leq 1\), the disease-free equilibrium is globally stable and the disease always dies out. If \(R_0>1\), there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium are also analyzed.

MSC:
92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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