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 $R_0^s>1$, which indicates that HCV transmission undergoes the persistence in the long term. Furthermore, we derive the sufficient conditions for HCV extinction under the condition $R_0^e<1$. As a consequence, we derive the relationships among the stochastic persistence index $R_0^s$, the stochastic extinction index $R_0^e$ and the threshold (the basic reproduction number $R_0$) of the model without fluctuations. The condition $R_0>R_0^s>1$ reveals that the existence of the white noises triggers the less stochastic persistence index. While, the condition $R_0<R_0^e<1$ reveals that, when the intensities of the white noises are enhanced, the value $R_0^e$ triggers the stochastic extinction.