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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Nonlinear Dynamical Analysis of a Stochastic SIRS Epidemic System with Vertical Dissemination and Switch from Infectious to Susceptible Individuals

Journal of Applied Nonlinear Dynamics 11(3) (2022) 605--633 | DOI:10.5890/JAND.2022.09.007

Driss Kiouach, Yassine Sabbar

LPAIS Laboratory, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fez, Morocco

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Abstract

This study presents a Susceptible-Infected-Recovered-Susceptible (SIRS) system for the dynamics of epidemics in a non-closed human population. We consider the vertical transmission of the epidemic and incorporate varying periods of the individuals' immunity. The stochastic version of our new epidemic model is obtained by simultaneously introducing the stochastic transmission and the proportional perturbation. For the permanence case, the ergodicity of the perturbed system is proved by employing a new approach different from the Lyapunov method. Under the same condition of the existence of the unique ergodic stationary distribution, the persistence in the mean of the epidemic is shown. When the basic reproduction number $\mathcal{R}_0>1$, the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model is provided. For the extinction case, sufficient conditions for the disappearance of the epidemic are obtained. When $\mathcal{R}_0\leq1$, the analysis of the asymptotic behavior around the disease-free equilibrium of the deterministic model is also investigated. In order to make the readers understand our study better, we present some numerical simulations to illustrate our theoretical results.

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