Global stability analysis of a delay cell-population model of hepatitis B infection with humoral immune response

In this work, we propose and investigate a delay cell population model of hepatitis B virus (HBV) infection. We suppose spatial diffusion of free HBV particles, and use a Beddington-DeAngelis incidence function to describe viral infection. The model takes into account the exposed hepatocytes and the usually neglected humoral immune response. Moreover, a time delay is introduced to account for the transformation processes necessary for actual HBV production. We naturally find two threshold parameters, namely the basic reproduction number R-0 and the humoral immune response reproduction number R-1 which completely determine the global stability of the spatially homogeneous equilibria of the model obtained. By constructing appropriate Lyapunov functionals and using LaSalle's invariance principle we show that, if R-0 <= 1, the disease-free equilibrium is globally asymptotically stable. Furthermore, we prove that the endemic equilibrium without humoral immune response and the endemic equilibrium with humoral immune response are globally asymptotically stable if R-1 <= 1 < R-0 and R-1 > 1, respectively. Finally, in one dimensional space, we perform some numerical simulations to illustrate the theoretical results obtained.