A stochastic SIR epidemic model with density dependent birth rate

Ling Zhu,Hongxiao Hu

doi:10.1186/s13662-015-0669-2

Abstract

In this paper, we introduce stochasticity into a model of SIR with density dependent birth rate. We show that the model possesses non-negative solutions as desired in any population dynamics. We also carry out the globally asymptotical stability of the equilibrium through the stochastic Lyapunov functional method if R 0 ≤ 1 $R_{0}\le1$ . Furthermore, when R 0 > 1 $R_{0}>1$ , we give the asymptotic behavior of the stochastic system around the endemic equilibrium of the deterministic model and show that the solution will oscillate around the endemic equilibrium. We consider that the disease will prevail when the white noise is small and the death rate due to disease is limited.

Highlights

From the pioneering work of Kermack and Mckendrick on SIR [ ], many models for the transmission of infectious have descended
When R0 > 1, we give the asymptotic behavior of the stochastic system around the endemic equilibrium of the deterministic model and show that the solution will oscillate around the endemic equilibrium
Tornatore et al [ ] proposed a stochastic SIR model with or without distributed time delay, they gave a sufficient condition for the asymptotic stability of the disease-free equilibrium

Summary

Introduction

From the pioneering work of Kermack and Mckendrick on SIR [ ], many models for the transmission of infectious have descended (see [ – ]). Tornatore et al [ ] proposed a stochastic SIR model with or without distributed time delay, they gave a sufficient condition for the asymptotic stability of the disease-free equilibrium. Lin and Jiang [ ] considered a stochastic SIR model with perturbed disease transmission coefficient.

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