Mathematical analysis of a fractional differential model of HBV infection with antibody immune response

Abstract

Fractional differential mathematical model describing the dynamics of hepatitis B viral infection with DNA-containing capsids, the liver hepatocytes and the humoral immune response is presented and investigated in this paper. The humoral immunity is represented by antibodies, the principal role of those antibodies is to attack the free viruses. In order to describe the time needed for the interaction between biological liver cells and viral particles and also the time needed for the activation of the humoral immune response, a memory term represented by a fractional derivative will be added in each equation of our suggested model. The positivity and boundedness of all solutions with non negative initial condition will be proved which is consistent biologically. Moreover, the disease-free equilibrium, the infection steady state without humoral immunity and the infection steady state with humoral immunity are given. By constructing some suitable Lyapunov functionals, the global stability of all equilibria are proven depending on the basic reproduction number and on the antibody immune response reproduction number. Finally, different numerical simulations using the multistage generalized differential method are established in order to illustrate our theoretical findings. It was revealed from both the theoretical and the numerical results that the order of the fractional derivative has no effect on the three equilibria stability. However, for increased values of the fractional derivative order, which describes the long memory behavior, each solution converge more rapidly to its stationary state.