Threshold dynamics of a HCV model with virus to cell transmission in both liver with CTL immune response and the extrahepatic tissue.

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number $R_{01}$ and immune reproduction number $R_{02}$ are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when $R_{01}<1$ , the virus will be cleared out eventually and the CTL immune response will also disappear; when $R_{02}<1<R_{01}$ , the virus persists within the host, but the CTL immune response disappears eventually; when $R_{02}>1$ , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.