# American Institute of Mathematical Sciences

## Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  March 2020 Revised  July 2020 Published  November 2020

Fund Project: The first author is supported by China Postdoctoral Science Foundation, and the Post-doctoral Innovative Research Positions in Hubei Province (Grant No. 1232037), the second author is supported by NSFC (Grant Nos. 11671123, 12071446)

This paper is devoted to a stochastic regime-switching susceptible-infected-susceptible epidemic model with nonlinear incidence rate and Lévy jumps. A threshold $\lambda$ in terms of the invariant measure, different from the usual basic reproduction number, is obtained to completely determine the extinction and prevalence of the disease: if $\lambda>0$, the disease is persistent and there is a stationary distribution; if $\lambda<0$, the disease goes to extinction and the susceptible population converges weakly to a boundary distribution. Moreover, some numerical simulations are performed to illustrate our theoretical results. It is very interesting to notice that random fluctuations (including the white noise and Lévy noise) acting the infected individuals can prevent the outbreak of disease, that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime, and that the prevalence of the disease can also be controlled by reducing the value of transmission rate of disease.

Citation: Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020335
##### References:

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##### References:
Trajectories of solutions to model (2) with parameters (29) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (29)
The joint density distribution of $(S,I)$ of model (2) with parameters (29). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (30) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (30)
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (30)
The joint density distribution of $(S,I)$ of model (2) with parameters (30). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (31) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (31)
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (31)
The joint density distribution of $(S,I)$ of model (2) with parameters (31). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (32) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (32)
The joint density distribution of $(S,I)$ of model (2) with parameters (32). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (33) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (33)
The joint density distribution of $(S,I)$ of model (2) with parameters (33). (a) The case without jumps; (b) The case with jumps
Trajectories of solutions to model (2) with parameters (34) in regimes 1 and 2
Trajectories of susceptible and infected populations of model (2) with parameters (34) in regimes 1 and 2
The weak convergence of $S$ to the stationary solution $\varphi$ of (10) with parameters (34)
The joint density distribution of $(S,I)$ of model (2) with parameters (34). (a) The case without jumps; (b) The case with jumps
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