Effect of humoral immunity on HIV-1 dynamics with virus-to-target and infected-to-target infections

We consider an HIV-1 dynamics model by incorporating (i) two routes of infection via, respectively, binding of a virus to a receptor on the surface of a target cell to start genetic reactions (virus-to-target infection), and the direct transmission from infected cells to uninfected cells through the concept of virological synapse in vivo (infected-to-target infection); (ii) two types of distributed-time delays to describe the time between the virus or infected cell contacts an uninfected CD4+ T cell and the emission of new active viruses; (iii) humoral immune response, where the HIV-1 particles are attacked by the antibodies that are produced from the B lymphocytes. The existence and stability of all steady states are completely established by two bifurcation parameters, R0(the basic reproduction number) and R1(the viral reproduction number at the chronic-infection steady state without humoral immune response). By constructing Lyapunov functionals and using LaSalle’s invariance principle, we have proven that, if R0≤1, then the infection-free steady state is globally asymptotically stable, if R1≤11, then the chronic-infection steady state with humoral immune response is globally asymptotically stable. We have performed numerical simulations to confirm our theoretical results