Global stability of a delay HCV dynamics model with cellular proliferation

In this work, we propose and investigate a delay cell population model of hepatitis C virus (HCV) infection with cellular proliferation, absorption effect and a nonlinear incidence function. First of all, after having shown the existence of the local solutions of our model, we show the existence of the global solutions and positivity. Moreover, we determine the uninfected equilibrium point and the basic reproduction rate R0, which is a threshold number in mathematical epidemiology. After showing the existence and uniqueness of the infected equilibrium point, we proceed to the study of the local and global stability of this equilibrium. We show that if  R0 < 1, the uninfected equilibrium point is globally asymptotically stable, which means that the disease will disappear and if  R0 > 1, we have a unique infected equilibrium that is globally asymptotically stable under some conditions. Finally, we perform some numerical simulations to illustrate the obtained theoretical results.