Global properties and bifurcation analysis of an HIV-1 infection model with two target cells

In this paper, the authors consider a four-compartmental HIV epidemiological model, which describes the interaction between HIV virus and two target cells, CD4 T cells and macrophages in vivo. It is proved that the bilinear incidence can cause the backward bifurcation, where a locally asymptotically stable disease-free equilibrium co-exists with a locally asymptotically stable endemic equilibrium when the basic reproduction number $\left(R_0\right)$ is less than unity. It is shown that a sequence of Hopf bifurcations occur at the endemic equilibrium by choosing one parameter of the model as the bifurcation parameter. Meanwhile, the global asymptotic stabilities of the equilibria are established by constructing suitable Lyapunov functions under some conditions. Furthermore, the authors develop an extended model by incorporating with the intracellular delays and derive global asymptotic stability of the delayed model by constructing Lyapunov functions. Some numerical simulations for justifying the theoretical analysis results are also given.