Stationary distribution and extinction of an HCV transmission model with protection awareness and environmental fluctuations

We propose a stochastic HCV model with protection awareness and exposed-acute-chronic phases in this study. First of all, we show that the stochastic HCV model admits a unique global positive solution for any given positive initial values. Then, we verify that HCV model has a unique stationary distribution under the sufficient criterion R₀^s>1, which indicates that HCV transmission undergoes the persistence in the long term. Furthermore, we derive the sufficient conditions for HCV extinction under the condition R₀^e<1. As a consequence, we derive the relationships among the stochastic persistence index R₀^s, the stochastic extinction index R₀^e and the threshold (the basic reproduction number R₀) of the model without fluctuations. The condition R₀>R₀^s>1 reveals that the existence of the white noises triggers the less stochastic persistence index. While, the condition R₀<R₀^e<1 reveals that, when the intensities of the white noises are enhanced, the value R₀^e triggers the stochastic extinction.