Asymptotic Behavior Of HTLV-Ⅰ Model With Nonlinear Immune Response And Infection Delay

HTLV-ⅠVirus maily infectes CD4⁺T cell througn the transmission way ofcell-to-cell contact. The HTLV-I-specific CD8⁺cytotoxic T lymphocytes （CTL） playsignificantly important roles of killing the infected nuclear and reducing the numberof proviral loads, but the high levels of CTL and their cytotoxicity are the chiefcontributor to the development of HAM/TSP.Horacio Gómez-Acevedo et al proposed an ODE model for HTLV-Ⅰintection ofCD4⁺T cells that incorporated CTL response. First of all, in order to describe thetime between from a uninfected cell contacts infected cells to that cell is infected byHTLV-Ⅰvirus and this period of time between from monitoring to infected cells toproducing an CTL immune response, based on the Horacio Gómez-Acevedo et al,this paper established a mathematical model with the time delayτ1of infection andthe time delayτ₂of CTL immunity. Secondly, In this paper, there are also twothreshold parametersR₀andR₁, the basic reproduction number for viralpersistence and CTL response, respectively. Further more, the existence and stabilityof the equilibrium points are discussed. Our mathematical analysis shows that, theinfection-free equilibriumP0is globally asymptotically stable, and the eliminationof disease whenR₁≤1; The asymptomatic carriers equilibrium is globallyasymptotically stable, and the infection becomes long-term and without lasting CTLresponse WhenR₁<1<R₀. During the study the dynamic behavior of HAM/TSPequilibrium, we divided into two kinds of cases, HAM/TSP equilibrium is globallyattractive; The parameters conditions to ensure the stability of the HAM/TSPequilibrium for the time delayτ₂are given whenτ₁>0，τ₂=0. We also discuss theinfluences of the time delayτ1of infection and the time delayτ₂of CTLimmunity to the stability of model （3-2） at the HAM/TSP equilibrium. Finally, theconclusions above are proved by using numerical simulation.