Journal of Applied Nonlinear Dynamics
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) 605633  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 SusceptibleInfectedRecoveredSusceptible (SIRS) system for the dynamics of epidemics in a nonclosed 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 diseasefree 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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