Stability of delayed general HIV-1 with cell-to-cell and inflammatory cytokines infection model with immune system response

Abstract

In this paper, we propose a dynamical model for the immune system’s reaction to HIV-1 that includes three temporal delays distributed in the presence of inflammatory cytokines. The model contains six components: uninfected CD4+T cells, infected CD4+T cells, inflammatory cytokines, HIV-1 virus, CTLs, and antibodies. The model contains the general functions for the incidence rates of the healthy CD4+T cells with viruses, infected cells, and inflammatory cytokines. Moreover, the production/proliferation and removal/death rates of the virus and cells are represented by general functions, too. The model was developed to better understand HIV-1 infection by describing two immune responses (antibody and cytotoxic T lymphocyte (CTLs)), three forms of distributed time delays, and inflammatory cytokines in the context of an infection through cell-to-cell (CTC) transmission. First, the system’s basic properties were investigated; then the equilibria points of the system were determined. The reproduction numbers for the model have been calculated and investigated, too. The model has five reproduction numbers, basic one (ℜ₀), CTLs immune response (ℜ₁), antibody immune response (ℜ₂), CTLs immune competitive (ℜ₃), and antibody immune competitive (ℜ₄). Using the Lyapunov approach and LaSalle’s invariance principle, we have demonstrated the global asymptotic stability of all equilibria under a set of restrictions on the general functions and the threshold parameters. The numerical simulations has been performed to demonstrate the theoretical results. By adding a time delay for the model can reduce the quantities of free HIV-1 particles and infected cells while also dramatically raising the concentration of healthy CD4+T cells. However, the addition of CTC transmission raises the concentrations of infected cells and free HIV-1 particles while decreasing the concentration of healthy CD4+T cells.