# American Institute of Mathematical Sciences

November  2006, 6(6): 1417-1430. doi: 10.3934/dcdsb.2006.6.1417

## HIV infection and CD4+ T cell dynamics

 1 Department of Mathematics & Statistics, Kennesaw State University, Kennesaw, GA 30144, United States 2 Department of Mathematics and Statistics, Kennesaw State University, 1000 Chastain Road, Kennesaw, GA 30144, United States

Received  September 2004 Revised  February 2006 Published  August 2006

We study a mathematical model for the interaction of HIV infection and CD4$^+$ T cells. Local and global analysis is carried out. Let $N$ be the number of HIV virus produced per actively infected T cell. After identifying a critical number $N_{crit}$, we show that if $N\le N_{crit},$ then the uninfected steady state $P_{0}$ is the only equilibrium in the feasible region, and $P_{0}$ is globally asymptotically stable. Therefore, no HIV infection persists. If $N>N_{crit},$ then the infected steady state $P^$* emerges as the 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 be either stable or unstable. We show that $P^$* is stable only for $r$ (the proliferation rate of T cells) small or large and unstable for some intermediate values of $r.$ In the latter case, numerical simulations indicate a stable periodic solution exists.
Citation: Liancheng Wang, Sean Ellermeyer. HIV infection and CD4+ T cell dynamics. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1417-1430. doi: 10.3934/dcdsb.2006.6.1417
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