Stability, Hopf Bifurcation and Numerical Simulation of an HIV Model with Two Modes of Transmission and with Cellular and Humoral Immunity

Abstract

In this paper, we incorporate immune systems containing Cytotoxic T lymphocyte and humoral immunity into a general human immunodeficiency viruses infection model, which also considers logistic growth for target cells and both modes of spread, cell-to-cell and cell-free represents, by linear functions. We derive five threshold parameters which are used to study the existence of equilibria. By considering the characteristic equations, the local stability of disease-free and immune-free equilibria is investigated. Lyapunov functions and LaSalle’s invariance are constructed to prove the global stability of all steady states. Global dynamics of the human immunodeficiency viruses model can be accurately expressed by threshold parameters; also the existence of Hopf bifurcation is discussed and the results are expressed in the form of a proposition. Furthermore, numerical simulations confirm the corresponding theoretical results.