Persistence and stability of the disease-free equilibrium in a stochastic epidemic model with imperfect vaccine

Dianli Zhao,Sanling Yuan

doi:10.1186/s13662-016-1010-4

Dianli Zhao, Sanling Yuan

Open Access

https://doi.org/10.1186/s13662-016-1010-4

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Abstract

This paper concerns the dynamics of a stochastic SIVR epidemic model with imperfect vaccine where, differently from the epidemic model with perfect vaccine, the vaccinated is perturbed by the noise. This difference is the main difficulty to be conquered to give the threshold $R_{0}^{S}$ . Firstly, $R_{0}^{S}>1$ is proved to be sufficient for persistence in mean of the system. Then, three conditions for the disease to die out are given, which improve the previously known results on extinction of the disease. In case that the disease goes extinct, we show that the disease-free equilibrium is almost surely stable by using the nonnegative semimartingale convergence theorem.

Highlights

1 Introduction Up to now, there are a lot of literatures on studying the disease showing that the deterministic and stochastic differential equations are often effective tools in describing the spread of a disease in real world; for more information, we refer to [ – ] and the references therein
The related known results show us that finding the threshold value for the stochastic epidemic systems is extremely difficult and exciting; see, for example, [ – ]
Inspired by works [, ], Tornatore et al [ ] formulated and studied a stochastic SIVR epidemic model by assuming that the average number of contacts per infective per unit time β is perturbed by environmental noise with β → β + σ B (t), where B(t) is a standard Brownian motion on complete probability space (, F, (Ft)t≥,P) with intensity σ >

Summary

Introduction

There are a lot of literatures on studying the disease showing that the deterministic and stochastic differential equations are often effective tools in describing the spread of a disease in real world; for more information, we refer to [ – ] and the references therein. Inspired by works [ , ], Tornatore et al [ ] formulated and studied a stochastic SIVR epidemic model by assuming that the average number of contacts per infective per unit time β is perturbed by environmental noise with β → β + σ B (t), where B(t) is a standard Brownian motion on complete probability space ( , F , (Ft)t≥ ,P) with intensity σ >. In Section , the obtained results are extended to study the threshold value of a stochastic SIVS epidemic model. Leads to the conclusion that RS can be considered as the threshold whose value above one or below one completely determines the persistence and extinction of the disease in case that the noise is small. Theorem . gives the weakened conditions for extinction of the infective class than those in [ , ]

Extended results: threshold of the stochastic SIVS epidemic model

Methods