Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate

We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results

Three-Dimensional Cellular Automaton for Modeling the Hepatitis B Virus Infection

Hepatitis B is considered as the most common hepatic in the world and may lead to cirrhosis and liver cancer. It is caused by the hepatitis B virus, which attacks and can damage the liver. In this paper we investigate a new mathematical model to study the dynamic process of HBV infection on the liver. This model is based on a three dimensional cellular automaton, which is composed of four state variables. The model takes into account the heterogeneous feature and the spatial localization of the population studied. Furthemore, since the virus doesn’t remain only on the liver surface but penetrates into the organ, our model describes better the behavior of interactions between cells and hepatitis B virus in the liver than the previous works found in the literature, which have used only two cellular automata in their models

A delayed SIR epidemic model with a general incidence rate

A delayed SIR epidemic model with a generalized incidence rate is studied. The time delay represents the incubation period. The threshold parameter, $R_0(\tau)$ is obtained which determines whether the disease is extinct or not. Throughout the paper, we mainly use the technique of Lyapunov functional to establish the global stability of both the disease-free and endemic equilibrium

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

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 R0 and the CTL immune response reproduction number R1. The stability of the last equilibrium depends on R0 and R1 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

TIC Dans L’enseignement Des Mathematiques Au Lycee Marocain

In this work, we investigate the use of information and communication technologies (ICT) in the teaching of mathematics at Moroccan high school. We first start with the analysis of school books from the three years of Baccalaureate. Furthermore, we analyze the pedagogical orientations concerning the use of these technologies. The results obtained from this study show that the use of ICT in mathematics education at Moroccan high school is still too limited

Presentation of Malaria Epidemics Using Multiple Optimal Controls

An existing model is extended to assess the impact of some antimalaria control measures, by reformulating the model as an optimal control problem. This paper investigates the fundamental role of three type of controls, personal protection, treatment, and mosquito reduction strategies in controlling the malaria. We work in the nonlinear optimal control framework. The existence and the uniqueness results of the solution are discussed. A characterization of the optimal control via adjoint variables is established. The optimality system is solved numerically by a competitive Gauss-Seidel-like implicit difference method. Finally, numerical simulations of the optimal control problem, using a set of reasonable parameter values, are carried out to investigate the effectiveness of the proposed control measures

On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering