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Abstract
We consider a Human Immunodeficiency Virus (HIV) model with a logistic growth term and continue the analysis of the previous article [6]. We now take the viral diffusion in a two-dimensional environment. The model consists of two ODEs for the concentrations of the target T cells, the infected cells, and a parabolic PDE for the virus particles. We study the stability of the uninfected and infected equilibria, the occurrence of Hopf bifurcation and the stability of the periodic solutions.
Highlights
Over the past thirty years, there has been much research in the mathematical modeling of Human Immunodeficiency Virus (HIV), the virus which causes AIDS (Acquired Immune Deficiency Syndrome)
The major target of HIV infection is a class of lymphocytes, or white blood cells, known as CD4+ T cells
When the CD4+ T-cell count, which is normally around 1000 mm−3, reaches 200 mm−3 or below in an HIV-infected patient, that person is classified as having AIDS
Summary
Over the past thirty years, there has been much research in the mathematical modeling of Human Immunodeficiency Virus (HIV), the virus which causes AIDS (Acquired Immune Deficiency Syndrome). The paper is organized as follows: in Section 2, we prove that System (1.1)-(1.3) admits, for any value of the parameters r and N, the uninfected steady state Xu = (Tu, 0, 0) and that, in a region of the space of parameters, there exists another steady-state solution, the so-called infected steady state Xi = (Ti, Ii, Vi), where Ti, Ii and Vi are positive. In [6], we have exhibited an unbounded subdomain P in I in which the positive infected equilibrium becomes unstable whereas it is asymptotically stable in the rest of I In this unstable region, the levels of the various cell types and virus particles oscillate, rather than converging to steady values. The levels of the various cell types and virus particles oscillate, rather than converging to steady values This subdomain P may be biologically interpreted as a perturbation of the infection by a specific or unspecific immune response against HIV. We denote by Id the identity operator, and by (·)+ the positive part of the number in brackets
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