A mathematical model of HTLV-I infection with two time delays

In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxicT lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection,where one is the intracellular infection delay and the other is the immune delay to account for aseries of immunological events leading to the CTL response. We show that the global dynamicsof the model system are determined by two threshold values $R_0$, the correspondingreproductive number of a viral infection, and $R_1$, the corresponding reproductive numberof a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globallyasymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-freeequilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with nopersistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-Iinfection becomes chronic with a persistent CTL response. Moreover, we show that the immunedelay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numericalsimulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize theHAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, thestability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"effects from the intracellular delay with those ``destabilizing" influences from immune delay.