Heterogeneous Viral Environment in a HIV Spatial Model

TL;DR

This paper models HIV dynamics in a 2D heterogeneous environment, introducing a new eigenvalue parameter to analyze stability and equilibrium, and justifies classical ODE models through homogenization in alternating source structures.

Contribution

It introduces a novel eigenvalue parameter to account for heterogeneity and proves stability results, connecting PDE models with classical ODE approaches via homogenization.

Findings

Existence of a universal attractor for the HIV model.

Stability of the non-infected and infected equilibria depending on the eigenvalue.

Homogenization justifies classical ODE models in structured environments.

Abstract

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.