# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021262
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## Threshold of a stochastic SIQS epidemic model with isolation

 1 School of Natural Sciences Education, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam 2 HUS High School for Gifted Student, Hanoi National University, 182 Luong The Vinh, Thanh Xuan, Hanoi, Vietnam 3 Department of Mathematics, Mechanics and Informatics, , Hanoi National University, , 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

* Corresponding author: Nguyen Thanh Dieu

Received  February 2021 Revised  July 2021 Early access October 2021

The aim of this paper is to give sufficient conditions, very close to the necessary one, to classify the stochastic permanence of SIQS epidemic model with isolation via a threshold value $\widehat R$. Precisely, we show that if $\widehat R<1$ then the stochastic SIQS system goes to the disease free case in sense the density of infected $I_z(t)$ and quarantined $Q_z(t)$ classes extincts to $0$ at exponential rate and the density of susceptible class $S_z(t)$ converges almost surely at exponential rate to the solution of boundary equation. In the case $\widehat R>1$, the model is permanent. We show the existence of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to this invariant measure. Some numerical examples are also provided to illustrate our findings.

Citation: Nguyen Thanh Dieu, Vu Hai Sam, Nguyen Huu Du. Threshold of a stochastic SIQS epidemic model with isolation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021262
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Estimated paths of $\frac{\ln I_z(t)}t$ (in red line), $\frac{\ln Q_z(t)}t$ (in ping line) and $\frac{\ln|S_z(t)- \widetilde S_u^0(t)|}{t}$ (in blue line) in Example 3.1
Trajectories of $(S_z(t), I_z(t), Q_z(t))$ in Example 3.2
Marginal one dimensional densities of $(S_z(t), I_z(t), Q_z(t))$
Marginal two dimensional densities of $(S_z(t), I_z(t), Q_z(t))$
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