Ordinary Differential Equations/Dynamical Systems
On the stability of hematopoietic model with feedback control
[Stabilité dʼun modèle hématopoïétique avec contrôle rétroactif]
Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 173-176.

Nous proposons et étudions un modèle mathématique de production et régulation des cellules sanguines dans la moelle osseuse (hématopoïèse). Ce modèle décrit la dynamique des cellules souches hématopoïétiques primitives (PHSC), les trois lignées de cellules souches progéniteurs quʼelles génèrent ainsi que les cellules matures correspondantes (globules rouges, globules blancs et plaquettes). Le modèle mathématique obtenu est un système non linéaire dʼéquations différentielles avec plusieurs retards représentant les durées de cycles cellulaires de chaque type de cellules. Nous étudions la stabilité locale du point dʼéquilibre trivial par lʼétude de lʼéquation caractéristique, puis nous prouvons sa stabilité globale par la méthode de Lyapunov. Ce résultat illustre lʼextinction de la population des cellules dans certains cas pathologiques.

We propose and analyze a mathematical model of the production and regulation of blood cell population in the bone marrow (hematopoiesis). This model includes the primitive hematopoietic stem cells (PHSC), the three lineages of their progenitors and the corresponding mature blood cells (red blood cells, white cells and platelets). The resulting mathematical model is a nonlinear system of differential equations with several delays corresponding to the cell cycle durations for each type of cells. We investigate the local asymptotic stability of the trivial steady state by analyzing the roots of the characteristic equation. We also prove by a Lyapunov function the global asymptotic stability of this steady state. This situation illustrates the extinction of the cell population in some pathological cases.

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DOI : 10.1016/j.crma.2012.01.014
Marquet, Catherine 1 ; Adimy, Mostafa 2

1 Université de Pau, laboratoire de mathématiques appliquées, CNRS UMR 5142, avenue de lʼuniversité, 64000 Pau, France
2 INRIA Rhône-Alpes, université Lyon 1, institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France
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Marquet, Catherine; Adimy, Mostafa. On the stability of hematopoietic model with feedback control. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 173-176. doi : 10.1016/j.crma.2012.01.014. http://www.numdam.org/articles/10.1016/j.crma.2012.01.014/

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