HIV infection and CD4+ T cell dynamics

We study a mathematical model for the interaction of HIV infectionand CD4$^+$ T cells. Local and global analysis is carried out. Let$N$ be the number of HIV virus produced per actively infected Tcell. After identifying a critical number $N_{crit}$, we show thatif $N\le N_{crit},$ then the uninfected steady state $P_{0}$ is theonly equilibrium in the feasible region, and $P_{0}$ is globallyasymptotically stable. Therefore, no HIV infection persists. If$N>N_{crit},$ then the infected steady state $P^$* emerges asthe unique equilibrium in the interior of the feasible region,$P_{0}$ becomes unstable and the system is uniformly persistent.Therefore, HIV infection persists. In this case, $P^$* can beeither stable or unstable. We show that $P^$* is stable only for $r$(the proliferation rate of T cells) small or large and unstable forsome intermediate values of $r.$ In the latter case, numericalsimulations indicate a stable periodic solution exists.