BRAUNER, Claude-Michel; JOLLY, Danaelle; LORENZI, Luca; THIÉBAUT, Rodolphe

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BRAUNER, Claude-Michel
Institut de Mathématiques de Bordeaux [IMB]

JOLLY, Danaelle
Institut de Mathématiques de Bordeaux [IMB]

LORENZI, Luca
Dipartimento di Matematica

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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.< Leer menos

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Mathematical models

HIV-host interaction

Diffusion

Equilibrium solutions

Stability

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