Impulsive differential equations are often used in mathematical modelling to simplify complicated hybrid models. We propose an inverse framework inspired by impulsive differential equations, called impulse extension equations, which can be used as a tool to determine when these impulsive models are accurate. The linear theory is the primary focus, for which theorems analoguous to ordinary and impulsive differential equations are derived. Results explicitly connecting the stability of impulsive differential equations to related impulse extension equations are proven in what we call time scale consistency theorems. Opportunities for future research in this direction are discussed. Following the work of Smith? and Hove-Musekwa on malaria vector control by impulsive insecticide spraying, we propose a novel autonomous vector control scheme based on human disease incidence. Existence and stability of periodic orbits is established. We compare the implementation cost of the incidence-based control to a fixed-time spraying schedule. Hybrid control strategies are discussed.

L'infection d'un organisme par un agent pathogène déclenche l'activation des lymphocytes T-CD8 et l'initiation de la réponse immunitaire. Il s'ensuit un programme complexe de prolifération et de différenciation des lymphocytes T-CD8, contrôlé par l'évolution de leur contenu moléculaire. Dans ce manuscrit, nous présentons deux modèles mathématiques de la réponse T-CD8. Le premier se présente comme une équation différentielle à impulsions grâce à laquelle nous étudions l'effet du partage inégal des protéines lors des divisions cellulaires sur la régulation de l'hétérogénéité moléculaire. Le second est un modèle à base d'agents couplant la description d'une population discrète de lymphocytes T-CD8 à celle du contenu moléculaire de ces derniers. Ce modèle s'avère capable de reproduire les différentes phases caractéristiques de la réponse T-CD8 aux échelle cellulaire et moléculaire. Ces deux travaux supportent l'hypothèse que la dynamique cellulaire observée in vivo est le reflet de l'hétérogénéité moléculaire qui structure la population de lymphocytes T-CD8 / Infection of an organism by a pathogen triggers the activation of the CD8 T-cells and the initiation of the immune response. The result is a complex program of proliferation and differentiation of the CD8 T-cells, controlled by the evolution of their molecular content. In this manuscript, we present two mathematical models of the CD8 T-cell response. The first one is presented as an impulsive differential equation by which we study the effect of unequal molecular partitioning at cell division on the regulation of molecular heterogeneity. The second one is an agent-based-model that couples the description of a discrete population of CD8 T-cells and that of their molecular content. This model can reproduce the different typical phases of the CD8 T-cell response at both the cellular and the molecular scales. These two studies support the hypothesis that the cell dynamics observed in vivo is a consequence of the molecular heterogeneity structuring the CD8 T-cell population