Heterogeneous Viral Environment in a HIV Spatial Model

We consider the basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a 2D heterogenous environment. It consists of two ODEs for the non-infected and infected $CD_4^+$ $T$ lymphocytes, $T$ and $I$, and a parabolic PDE for the virus $V$. We define a new parameter $\lambda_0$ as an eigenvalue of some Sturm-Liouville problem, which takes the heterogenous reproductive ratio into account. For $\lambda_0<0$ the trivial non-infected solution is the only equilibrium. When $\lambda_0>0$, the former becomes unstable whereas there is only one positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a universal attractor. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment. The latter justifies the classical approach via ODE systems.