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An epidemic model is formulated by a reaction–diffusion system where the spatial pattern formation is driven by cross-diffusion. The reaction terms describe the local dynamics of susceptible and infected species, whereas the diffusion terms account for the spatial distribution dynamics. For both self-diffusion and cross-diffusion, nonlinear constitutive assumptions are suggested. To simulate the pattern formation two finite volume formulations are proposed, which employ a conservative and a non-conservative discretization, respectively. An efficient simulation is obtained by a fully adaptive multiresolution strategy. Numerical examples illustrate the impact of the cross-diffusion on the pattern formation.

This project presents the theoretical background in electrophysiology that is used as a basis in the development of numerical methods for the simulation of heart’s electrical activity. We discuss the mathematical models currently used in cardiac electrophysiology research to simulate the propagation of an action potential wave, and then focus on the description of the ionic currents. This study allows us to anticipate the numerical model sensitivity with regard to the discretization, with the aim of recovering the physiological behaviors of the tissue (i.e. conduction velocity, electrical restitution, etc). Thereafter, we introduce the pseudo-ECG signal reconstruction method from the action potential map. This leads us to examine the approximation of such a signal using the finite element basis coming from the discretization of the monodomain equations. We discuss then a particular cardiac rhythm disorder: action potential alternans. In a second step, we perform an extended simulation study in order to validate the previous numerical methods. We carefully check the convergence of the conduction velocity with respect to the mesh size, and we bring out the advantage of phenomenological approach in terms of electrical restitutions and computational cost. We reproduce typical rhythm disorders and show that the pseudo-ECG signal is able to detect them. Finally, we discuss the influence of anisotropy in cardiac wave propagation and alternans development, characterizing the effect of fibers direction and pacing site in the resulting spatio-temporal distribution.

Ce projet présente les bases physiques nécessaires à la modélisation de l’activité électrique dans le coeur et les oreillettes ainsi que plusieurs modèles mathématiques développés à cet effet. On donne les spécificités de chacun de ceux-ci en terme de rendu du potentiel de la membrane cellulaire. On étudie ensuite plus en détails le modèle de Courtemanche-Ramirez-Nattel pour le potentiel dans les oreillettes dont on valide le code implémenté en LifeV. La seconde partie du travail présente un problème de contrôle optimal : on dérive le modèle théorique pour les équations monodomaines avec le modèle de Rogers-McCulloch, puis on applique ces résultats à la défibrillation d’un domaine excité.