Threshold dynamics and ergodicity of an SIRS epidemic model with semi-Markov switching

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Abstract

This work studies the threshold dynamics and ergodicity of a stochastic SIRS epidemic model with the disease transmission rate driven by a semi-Markov process. The semi-Markov process used in this paper for describing a randomly changing environment is a very large extension of the most common Markov regime-switching process. We define a basic reproduction number for the semi-Markov regime-switching environment and show that its position with respect to 1 determines the extinction or persistence of the disease. In the case of disease persistence, we give mild sufficient conditions for ensuring the existence and absolute continuity of the invariant probability measure. Under the same conditions, we also prove the global attractivity of the Ω-limit set of the system and the convergence in total variation norm of the transition probability to the invariant measure. Compared with the existing results in the Markov regime-switching environment, the results generalized require almost no additional conditions.

MSC

60K15
60H10
37A50
93E15
92D30

Keywords

Epidemic model
Basic reproduction number
Invariant probability measure
Ergodicity
Stochastic stability
Semi-Markov switching

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