Étude d’équations à retard appliquées à la régulation de la production de plaquettes sanguines
The object of this thesis is the study, using mathematical models, of the regulation mechanism maintaining an optimal quantity of blood platelets. The first chapter presents the biological and mathematical context of the thesis. In a second chapter, we introduce a model for platelet production assuming a regulation by the platelet quantity of both the differentiation rate of stem cells to the platelet cell line and the amount of platelets produced by each megakaryocyte. We show that the dynamic of this model corresponds to a delay differential equation x'(t) = −γx(t)+f(x(t))g(x(t−τ)), and we obtain for this equation new sufficient conditions for stability and for the oscillation of solutions. In a third chapter, we analyze a second model for platelet production in which the regulation is continuous through the maturation speed of megakaryocyte progenitors. The stability analysis requires to adapt a pre-existing framework to problems where the bifurcation parameter is not the delay, and allows to show that increasing the death rate of megakaryocyte progenitors leads to the onset of periodic solutions, in agreement with clinical observation of amegakaryocytic cyclical thrombocytopenia. The last chapter covers a differential equation with two delays that appears among others in a model of platelet production which considers that platelet death can both age-independent and age-dependent.
oscillations platelet cyclical thrombocytopenia stability analysis oscillations delay transcendental equation Hopf bifurcation