Mehdi Maziane

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 R ₀ and the CTL immune response reproduction number R ₁ . The stability of the last equilibrium depends on R ₀ and R ₁ as well as time delay τ 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 τ passes through a certain critical value.

In this work, we propose an HIV infection model with cure of infected cells in eclipse stage and delay in the activation of immune response. The stability of the equilibria and the existence of the Hopf bifurcation are investigated. Moreover, numerical simulations are carried out to illustrate our theoretical results.

In this paper, we propose a viral infection model governed by three stochastic differential equations. The global existence and positivity of solutions is investigated. Further, we give sufficient conditions for the stability in probability of the endemic equilibrium by using the direct Lyapunov method. Moreover, an application and numerical simulations are given to illustrate our theoretical results.

In this paper, we propose two HIV infection models with specific nonlinear incidence rate by including a class of infected cells in the eclipse phase. The first model is described by ordinary differential equations (ODEs) and generalizes a set of previously existing models and their results. The second model extends our ODE model by taking into account the diffusion of virus. Furthermore, the global stability of both models is investigated by constructing suitable Lyapunov functionals. Finally, we check our theoretical results with numerical simulations.

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.