Stability and backward bifurcation for an HIV model with macrophages and CD4+T cells with latent reservoirs

Human immunodeficiency virus (HIV) primarily attacks CD4+ T cells but can also attack macrophage and dendritic cells leading to AIDS disease. Antiretroviral therapies are unavoidable to treat patients with HIV. However, the existing therapies failed in eliminating viral particles in infected patients. The establishment of a latency state in these patients could play a major role in this failure. To better understand the impact of different drug therapies on HIV dynamics, we propose an in-host model describing the dynamics of HIV and its interaction with both CD4+T and macrophage cells. This model incorporates the CD4+T latent reservoir as well. Results on the positivity, boundedness, and stability of solutions are shown. Furthermore, the basic reproduction number R0 is obtained using the next generation-matrix method. It is shown that the model is locally asymptotically stable at disease-free equilibrium (DFE) when the basic reproduction number R0<1. Moreover, the model exhibits either forward or backward bifurcation when R0=1. The global asymptotic stability of equilibria is investigated using suitable Lyapunov functions. To study the impact of our model parameters on the model dynamics, a sensitivity analysis is performed using the partial rank correlation coefficient method and the Latin hypercube sampling. Finally, numerical simulations are carried out to assess several drug therapy strategies to reduce HIV infection and improve health outcomes for HIV infected patients.