Mathematical analysis of a HIV model with quadratic logistic growth term

Xinyue Fan (1) , Claude-Michel Brauner (1, 2) , Linda Wittkop (3)

1 School of Mathematical Sciences
2 IMB - Institut de Mathématiques de Bordeaux
3 Epidémiologie et Biostatistique [Bordeaux]

We consider a model of disease dynamics in the modeling of Human Immunodeficiency Virus (HIV). The system consists of three ODEs for the concentrations of the target T cells, the infected cells and the virus particles. There are two main parameters, $N$, the total number of virions produced by one infected cell, and $r$, the logistic parameter which controls the growth rate. The equilibria corresponding to the infected state are asymptotically stable in a region $(\mathcal I)$, but unstable in a region $(\mathcal P)$. In the unstable region, the levels of the various cell types and virus particles oscillate, rather than converging to steady values. Hopf bifurcations occurring at the interfaces are fully investigated via several techniques including asymptotic analysis. The Hopf points are connected through a ''snake" of periodic orbits. Numerical results are presented.

Mots clés: HIV modeling; stability; Hopf bifurcation; orbits; snakes

Domaines: Systèmes dynamiques [math.DS]; Sciences du Vivant [q-bio]

Xinyue Fan, Claude-Michel Brauner, Linda Wittkop. Mathematical analysis of a HIV model with quadratic logistic growth term. Discrete and Continuous Dynamical Systems - Series B, 2012, 17 (7), pp.27. ⟨10.3934/dcdsb.2012.17.2359⟩. ⟨hal-00537467v2⟩