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Conditions for permanence and ergodicity of certain SIR epidemic models. (English) Zbl 1416.34037

Summary: In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
60F17 Functional limit theorems; invariance principles
34F05 Ordinary differential equations and systems with randomness
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