Threshold properties of a stochastic epidemic model with a variable vaccination rate

Abstract

This paper aims to improve the analysis of a stochastic SIVR epidemic model with an imperfect vaccination process, taking into consideration the fact that a fraction of vaccinated individuals becomes susceptible to infection. The uniqueness of the positive solution is shown. Further, we obtain the threshold of the stochastic SIVR model which determines whether the epidemic will persist or die out. In the extinction case, we prove that the solution converges almost surely toward the disease-free equilibrium of the deterministic SIVR model. Some numerical illustrations are given to confirm our theoretical results.