Asymptotic Behavior of a Discrete Maturity StructuredSystem of Hematopoietic Stem Cell Dynamicswith Several Delays

Abstract

We propose and analyze a mathematical model of hematopoietic stem cell dynamics. This model takesinto account a finite number of stages in blood production, characterized by cell maturity levels,which enhance the difference, in the hematopoiesis process, between dividing cells thatdifferentiate (by going to the next stage) and dividing cells that keep the same maturity level (bystaying in the same stage). It is described by a system of n nonlinear differential equationswith n delays. We study some fundamental properties of the solutions, such as boundedness andpositivity, and we investigate the existence of steady states. We determine some conditions for thelocal asymptotic stability of the trivial steady state, and obtain a sufficient condition for itsglobal asymptotic stability by using a Lyapunov functional. Then we prove the instability of axialsteady states. We study the asymptotic behavior of the unique positive steady state and obtain theexistence of a stability area depending on all the time delays. We give a numerical illustration ofthis result for a system of four equations.