Structured models and bifurcations in distributed delay differential equations

Abstract

This work deals with a recent within-host malaria infection model with multistrain for the parasites and a spatial modeling of anopheles mosquito dynamics population. In this work, we also consider models of infectious disease into the host population structured by age. Namely, Hepatitis B virus (HBV) model and Susceptible-Infected-Lost of sight (SIL) model for the spread of a directly transmitted infectious disease taking into account demographic process and vertical transmission of the disease. The work is organized into five majors chapters: 1. General introduction, 2. Biology of Malaria and Hepatitis B Virus, 3. Hepatitis B virus and within-host malaria models, 4. Within-host malaria infection and Anopheles mosquito dynamics, 5. Population Models Structured by Age (Hepatitis B and SIL models). The true chapters of this work are Chapters 4 and 5. Fundamentals tools of this work are Hille-Yosida operator, Strongly continuous semi-group, Integrated semigroup, Invariant manifold, Bifurcation, Lyapunov stability, linearized stability and Numerical analysis. For each model, we derived the existence of a unique maximal bounded dissipative semiflow. We also performed the asymptotic behavior of the models with respect to a specific threshold parameter. Thematic results are provided for Within-host malaria infection, Anopheles mosquito dynamics, HBV and HIV dynamics in age-structured population. For example, for within-host multi-strain malaria infection dynamics, our study allowed for the observation of competitive suppression, the reduction of parasites numbers due to the presence of another parasite, and competitive release, the improved performance of a parasite after the removal of a competitor. These studies demonstrated that the presence of two parasites led to the reduction in density of at least one parasite.