Asymptotic Behavior of a Discrete Maturity Structured System of Hematopoietic Stem Cell Dynamics with Several Delays

Abstract

We propose and analyze a mathematical model of hematopoietic stem cell dy- namics. This model takes into account a flnite number of stages in blood production, char- acterized by cell maturity levels, which enhance the difierence, in the hematopoiesis process, between dividing cells that difierentiate (by going to the next stage) and dividing cells that keep the same maturity level (by staying in the same stage). It is described by a system of n nonlinear difierential equations with n delays. We study some fundamental properties of the solutions, such as boundedness and positivity, and we investigate the existence of steady states. We determine some conditions for the local asymptotic stability of the trivial steady state, and obtain a su-cient condition for its global asymptotic stability by using a Lyapunov functional. Then we prove the instability of axial steady states. We study the asymptotic behavior of the unique positive steady state and obtain the existence of a stabil- ity area depending on all the time delays. We give a numerical illustration of this result for a system of four equations.