Dynamics analysis of a HIV Infection model with the effect release of inflammatory cytokines

HIV disease is a type of chronic disease that over time causes a decrease in the body’s immune system cells, especially T cells. One of the main questions about this disease is that T cells depletion can take many years but slow decline are not well understood. In some studies they showed that infected T cells die in the lymph nodes by a type of cell death called pyroptosis cell death. In this type of cell death, inflammatory cytokines can be released, thus attracting more T cells. In this paper, a model for HIV infection with the effect release of inflammatory cytokines on interaction between the T cells and virus is considered. This model includes the dynamic behavior of the uninfected and infected T cell, and the HIV viral load. In this model, the proliferation of T cells is a logistic growth and the infection function of T cells is Beddington–DeAngelis function. We prove that if the basic reproduction number is less than one then the infection free equilibrium point of the model is globally asymptotically stable and if more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. In the end, we showed the existence of the condition by creating an orbitally asymptotically stable periodic solution numerical results.