Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model

A Lahrouz,D Kiouach,L Omari

doi:10.15388/na.16.1.14115

Abstract

We present in this paper an SIRS epidemic model with saturated incidence rate and disease-inflicted mortality. The Global stability of the endemic equilibrium state is proved by constructing a Lyapunov function. For the stochastic version, the global existence and positivity of the solution is showed, and the global stability in probability and pth moment of the system is proved under suitable conditions on the intensity of the white noise perturbation.

Highlights

One of the basic and important research subjects in mathematical epidemiology is the global stability of the equilibrium states of the epidemic models
The section will deal with the global behavior of the system (1) by constructing a Lyapunov function and we demonstrate that the endemic equilibrium state is globally asymptotically stable under the simple condition that the reproduction number is greater than one
This paper presented a mathematical study describing the dynamical behavior of an SIRS epidemic model with saturated incidence rate and disease-inflicted mortality

Summary

Introduction

One of the basic and important research subjects in mathematical epidemiology is the global stability of the equilibrium states of the epidemic models. Even a system is known to be stable, one often still needs explicit Lyapunov function to estimate, for example, the rate of convergence to an equilibrium state, or to study the stability of the stochastic version of the determinist models [14, 15]. By combining the quadratic form (R − R∗) and the function (I − I∗ ln I), O’Regan et al [22] have recently constructed a Lyapunov function to prove the global stability of the equilibria of an SIRS model with standard bilinear incidence rate. The section will deal with the global behavior of the system (1) by constructing a Lyapunov function and we demonstrate that the endemic equilibrium state is globally asymptotically stable under the simple condition that the reproduction number is greater than one.

The global stability of the endemic point

Perturbed SIRS model

Moment exponential stability

Almost sure exponential stability

Numerical simulations

Conclusion