Mathematical analysis of an HTLV-I infection model with the mitosis of CD4+ T cells and delayed CTL immune response

Abstract

In this paper, we consider an improved Human T-lymphotropic virus type I (HTLV-I) infection model with the mitosis of CD4+ T cells and delayed cytotoxic T-lymphocyte (CTL) immune response by analyzing the distributions of roots of the corresponding characteristic equations, the local stability of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium when the CTL immune delay is zero is established. And we discuss the existence of Hopf bifurcation at the immunity-activated equilibrium. We define the immune-inactivated reproduction ratio R₀ and the immune-activated reproduction ratio R₁. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that if R₀ < 1, the infection-free equilibrium is globally asymptotically stable; if R₁ < 1 < R₀, the immunity-inactivated equilibrium is globally asymptotically stable; if R₁ > 1, the immunity-activated equilibrium is globally asymptotically stable when the CTL immune delay is zero. Besides, uniform persistence is obtained when R₁ > 1. Numerical simulations are carried out to illustrate the theoretical results.