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International Journal of Differential Equations
Volume 2018, Article ID 3294268, 9 pages
https://doi.org/10.1155/2018/3294268
Research Article

Spatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activation

1Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955 Sidi Othman, Casablanca, Morocco
2Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, 20340 Casablanca, Morocco

Correspondence should be addressed to Mehdi Maziane; moc.liamg@enaizamidhem

Received 20 August 2017; Accepted 31 January 2018; Published 1 March 2018

Academic Editor: Peiguang Wang

Copyright © 2018 Mehdi Maziane et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction number and the CTL immune response reproduction number . The stability of the last equilibrium depends on and as well as time delay in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when passes through a certain critical value.