Research Article

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

M. Adimy^a1a2, F. Crauste^a3 and A. El Abdllaoui^a1

^a1 Laboratoire de Mathématiques Appliquées CNRS UMR 5142, Université de Pau et des Pays de l'Adour, 64000 Pau, France

^a2 ANUBIS Team, INRIA Futurs

^a3 Université de Lyon, Université Lyon 1, CNRS UMR 5208 Institut Camille Jordan F - 69200 Villeurbanne, France

Abstract

We propose and analyze a mathematical model of hematopoietic stem cell dynamics. This model takes into 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 that differentiate (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 differential 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 sufficient 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 stability area depending on all the time delays. We give a numerical illustration of this result for a system of four equations.

(Online publication May 15 2008)

Key Words:

• Hematopoiesis modelling;
• system of delay equations;
• global and local asymptotic stability;
• Lyapunov functional

Mathematics Subject Classification:

• 34K20;
• 92C37;
• 34C23;
• 34D20;
• 34K99

Correspondence:

^c1 crauste@math.univ-lyon1.fr