Stability and Hopf bifurcation on an immunity delayed HBV/HCV model with intra- and extra-hepatic coinfection and saturation incidence

hepatitis B or C virus (HBV or HCV) epidemic model with intra- and extra-hepatic coinfection, immune delay and saturation incidence, as well as antiviral therapy is proposed in this paper. The existence of equilibria (infection-free, immune-free and immune-activated), the basic reproduction numbers, i.e., R_0 , R_1 , are given respectively, by which the criteria on (local and global) stability of above equilibria are established. Furthermore, if the immune delay τ >τ _0 , both the existence of subcritical (supercritical) Hopf bifurcation on the immune-activated equilibrium E^* , and the stability of bifurcating periodic solutions are obtained. Finally, the theoretical results are demonstrated by numerical simulations. We derive that the immune delay and intra- and extra-hepatic coinfection have significant influence on the transmission of HBV/HCV, could cause more complicated dynamics at E^* from stability to unstablity untill bifurcation, which greatly increases the difficulty of disease control. While effective antiviral therapy could evidently decrease the spread of HBV/HCV.