Dynamics of a stochastic SIS epidemic model with nonlinear incidence rates

Ning Gao,Yi Song,Xinzeng Wang,Jianxin Liu

doi:10.1186/s13662-019-1980-0

Abstract

In this paper, considering the impact of stochastic environment noise on infection rate, a stochastic SIS epidemic model with nonlinear incidence rate is proposed and analyzed. Firstly, for the corresponding deterministic system, the threshold which determines the extinction or permanence of the disease is obtained by analyzing the stability of the equilibria. Then, for the stochastic system, the global dynamics is investigated by using the theory of stochastic differential equations; especially the threshold dynamics is explored when the stochastic environment noise is small. The results show that the condition for the epidemic disease to go to extinction in the stochastic system is weaker than that of the deterministic system, which implies that stochastic noise has a significant impact on the spread of infectious diseases and the larger stochastic noise is conducive to controlling the epidemic diseases. To illustrate this phenomenon, we give some computer simulations with different intensities of the stochastic noise.

Highlights

Infectious diseases are the public enemy of mankind and have brought great catastrophe to mankind
6 Conclusion The aim of this paper is to make contributions to understand the dynamics of SIS epidemic models with nonlinear incidence rate
We establish a stochastic system by introducing the white noise disturbance into the deterministic system

Summary

Introduction

Infectious diseases are the public enemy of mankind and have brought great catastrophe to mankind. Mathematical models have been used to study the spread and evolution of infectious diseases in the human population. Lemma 2.2 For any initial value (S0, I0) ∈ R2+, there exists a unique solution (S(t), I(t)) to system (3) on t ≥ 0, and the solution will remain in R2+ with probability one, namely, (S(t), I(t)) ∈ R2+ for all t ≥ 0 almost surely. Proof Firstly, we know that, for any initial value (S0, I0) ∈ R2+, because the coefficients of system (3) are locally Lipschitz continuous, there exists a unique local solution on [0, τ ) where τ is the explosion time To prove this solution is global, we need to show τ = ∞ almost surely.

Dynamics of the deterministic system

Extinction Let us introduce

Numerical simulation

Conclusion