[Stabilité dʼun modèle hématopoïétique avec contrôle rétroactif]
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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@article{CRMATH_2012__350_3-4_173_0,
author = {Marquet, Catherine and Adimy, Mostafa},
title = {On the stability of hematopoietic model with feedback control},
journal = {Comptes Rendus. Math\'ematique},
pages = {173--176},
publisher = {Elsevier},
volume = {350},
number = {3-4},
year = {2012},
doi = {10.1016/j.crma.2012.01.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2012.01.014/}
}
TY - JOUR AU - Marquet, Catherine AU - Adimy, Mostafa TI - On the stability of hematopoietic model with feedback control JO - Comptes Rendus. Mathématique PY - 2012 SP - 173 EP - 176 VL - 350 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.01.014/ DO - 10.1016/j.crma.2012.01.014 LA - en ID - CRMATH_2012__350_3-4_173_0 ER -
%0 Journal Article %A Marquet, Catherine %A Adimy, Mostafa %T On the stability of hematopoietic model with feedback control %J Comptes Rendus. Mathématique %D 2012 %P 173-176 %V 350 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.01.014/ %R 10.1016/j.crma.2012.01.014 %G en %F CRMATH_2012__350_3-4_173_0
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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