Abstract
In this paper, we study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination. We show the existence and positivity of the solution of the SIRS stochastic differential equation. We defined a number $\mathcal{R}$ and we prove the disease free equilibrium is almost sure exponentially stable if $\mathcal{R}<1$. We studying the behavior around the endemic equilibrium E*. Numerical simulations presented our theoretical results.
Highlights
Mathematical epidemiology play an important role in the study and control of the infectious diseases, the objective of this study is to implement measures to combat and terminate theA
We study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination
In recent years many authors have been developed the mathematics model for the transmission dynamics of infectious diseases, amid these models there are the classical deterministic SIR epidemic model [1, 2], the population are divide into three classes or S represents the number of the individuals susceptible, I represents the number of infective individuals, R represents the number of recovered individuals with temporary immunity acquired from a disease
Summary
Mathematical epidemiology play an important role in the study and control of the infectious diseases, the objective of this study is to implement measures to combat and terminate the. Greenhalgh et al [3] have used The technique of parameter perturbation by the white noise They proved Almost sure exponential stability of the disease free equilibrium and stability in probability. Adnani et al.[4] and Lahrouz et al.[13] have utilized The technique of perturbations stochastic by withe noise around the endemic equilibrium state. They have proved the asymptotically mean square stable of stochastic linearized system. We consider the stochastic version of SIRS model (2)with a general incidence rate find it by perturbing the parameter β by the white noise : m) pdI γ.
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