Stability and Hopf bifurcation of a cytokine-enhanced HIV infection model with antibody immune response delay

In this paper, the dynamical behaviors of a new cytokine-enhanced HIV infection model with intracellular delay [Formula: see text], virus replication delay [Formula: see text] and immune response delay [Formula: see text] are investigated. The positivity and boundedness of all solutions for the model with non-negative initial values have been proved. Moreover, two important biological parameters, called the virus reproductive number [Formula: see text] and the antibody immune reproductive number [Formula: see text] are established. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, the global dynamics of the equilibria is completely determined by [Formula: see text] and [Formula: see text]. On the one hand, the results show that intracellular delay [Formula: see text], viral replication delay [Formula: see text] and immune response delay [Formula: see text] have no effect on the stability of [Formula: see text], [Formula: see text], and if [Formula: see text], [Formula: see text], [Formula: see text], the endemic equilibrium with the presence of antibody response [Formula: see text] is globally asymptotically stable. On the other hand, when [Formula: see text], [Formula: see text], [Formula: see text], numerical analysis confirms the theorems and suggests that time delay play a positive role in virus infection, with the increase of [Formula: see text], the dynamic behavior of the equilibrium [Formula: see text] will change as follows: locally asymptotically stable [Formula: see text] unstable; Hopf bifurcation appears.