Bedr'Eddine Ainseba

A model for ovine brucellosis incorporating direct and indirect transmission.

In this work, we construct and analyse an ovine brucellosis mathematical model. In this model, the population is divided into susceptible and infected subclasses. Susceptible individuals can contract the disease in two ways: (i) direct mode – caused by contact with infected individuals; (ii) indirect mode – related to the presence of virulent organisms in the environment. We derive a net reproductive number and analyse the global asymptotic behaviour of the model. We also perform some numerical simulations, and investigate the effect of a slaughtering policy.

Two-step genetic algorithm to solve the inverse problem in electrocardiography for cardiac sources

This study is concerned with the numerical analysis of the inverse problem in electrocardiography. We propose a new method (two-step algorithm) based on the application of genetic algorithm with the Tikhonov method to calculate the electrical sources on the heart. In the literature, the solution reconstructed by the classical Tikhonov method is based on the minimum energy norm using a closed geometry. This solution could be very different from the original source on the heart. Compared with the classical inverse problem method, the two-step algorithm improves the accuracy of the solution with respect to the original distribution. Finally, to validate our results, we create a distribution over the heart by using a model of electrical activity (Monodomain model), and then using our method, we make the reconstruction of the potential distribution.

Control Strategies for TB Epidemics

A model for tuberculosis (TB) that includes immigration of susceptible and infected individuals is presented and analyzed. Infected individuals are structured by time since infection to include a long and variable latency period, and individuals with active TB have an increased mortality rate. Two control problems are formulated and analyzed, minimizing the impact of infection by controlling infected immigrants and/or screening for detection and treatment of infected individuals before they develop active TB.

Computing the electrical activity of the heart with a dynamic inverse monodomain operator

The inverse problem in electrocardiography is to reconstruct the voltage in the surface of the heart, using a high density electrocardiogram. This problem is usually solved using regularization techniques, which tend to give the minimum energy response in a static scheme. In our work, we propose to calculate a dynamic inverse solution using the Monodomain as a model of electrical heart activity, thus constraining the family of solutions to one that satisfies the model.

Parameters identification for a model of T cell homeostasis.

In this study, we consider a model of T cell homeostasis based on the Smith-Martin model. This nonlinear model is structured by age and CD44 expression. First, we establish the mathematical well-posedness of the model system. Next, we prove the theoretical identifiability regarding the up-regulation of CD44, the proliferation time phase and the rate of entry into division, by using the experimental data. Finally, we compare two versions of the Smith-Martin model and we identify which model fits the experimental data best.

An Age-Structured Model for T Cell Homeostasis in Vivo

In this study, we consider a model of T cell proliferation in vivo which is structured by age and CD44 expression. This model is rewritten as an age-structured model system without the CD44 structure, and we investigate its asymptotic behavior. We find that there exists one or three stationary solutions when cells undergo at least five divisions and only one stationary solution when cells undergo at most three divisions. The limiting case with four divisions is numerically handled. By applying the Lyapunov method, we prove that the stationary solution is globally asymptotically stable in some regions of parameter space.