Heterogeneous viral environment in a HIV spatial model

Abstract

We consider a basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a two-dimensional heterogenous environment. It consists of two ODEs for the uninfected and infected CD4+ T lymphocytes, T and I, and a parabolic PDE for the free virus particles V. We introduce a new parameter λ0 which is the largest eigenvalue of some Sturm-Liouville problem and takes the heterogenous reproductive ratio into account. For λ0 < 0 the uninfected steady state is the only equilibrium. When λ0 > 0, it becomes unstable and there is a unique positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a positively invariant region. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment which justifies the classical approach via ODE systems.