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Global stability of a stage-structured epidemic model with a nonlinear incidence. (English) Zbl 1172.92027
Summary: A stage-structured epidemic model with a nonlinear incidence with a factor \(S^p\) is investigated. By using limit theory of differential equations and a theorem of S. Busenberg and P. van den Driessche [J. Math. Biol. 28, No. 3, 257–270 (1990; Zbl 0725.92021)] the global dynamics of the model is rigorously established. We prove that if the basic reproduction number \(\operatorname{Re} _0\) is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if \(\operatorname{Re} _0\) is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of \(p\) on the dynamic behavior of the model.

MSC:
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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