Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

Oscillations et convergence asymptotique pour une équation différentielle à retard issue de la modélisation de la production des plaquettes

Loïs Boullu (1, 2) , Mostafa Adimy (1) , Fabien Crauste (1) , Laurent Pujo-Menjouet (1)

1 DRACULA - Multi-scale modelling of cell dynamics : application to hematopoiesis
2 Département de Mathématiques et de Statistiques [UdeM- Montréal]

We analyze the existence of oscillating solutions and the asymptotic convergence for a nonlinear delay differential equation arising from the modeling of platelet production. We consider four different cell compartments corresponding to different cell maturity levels: stem cells, megakaryocytic progenitors, megakaryocytes, and platelets compartments, and the quantity of circulating thrombopoietin (TPO), a platelet regulation cytokine. Our initial model consists in a nonlinear age-structured partial differential equation system, where each equation describes the dynamics of a single compartment. This system is reduced to a single nonlinear delay differential equation describing the dynamics of the platelet population, in which the delay accounts for a differentiation time. After introducing the model, we prove the existence of a unique steady state for the delay differential equation. Then we determine necessary and sufficient conditions for the existence of oscillating solutions. Next we set up conditions to get local asymptotic stability and asymptotic convergence of this steady state. Finally we present a short analysis of the influence of the conditions at t < 0 on the proof for asymptotic convergence.

Mots clés: Megakaryopoiesis; platelet; oscillations; stability; delay

Domaines: Systèmes dynamiques [math.DS]

Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete and Continuous Dynamical Systems - Series B, In press. ⟨hal-01835435v1⟩