Modelling and Asymptotic Stability of a Growth Factor Dependent Stem Cells
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Modelling and Asymptotic Stability of a Growth Factor-Dependent Stem Cells Dynamics Model with Distributed Delay? Mostafa Adimy and Fabien Crauste Year 2006 Laboratoire de Mathematiques Appliquees, UMR 5142, Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France. ANUBIS project, INRIA–Futurs Email: (M. Adimy), (F. Crauste) Abstract Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathe- matical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modelling the dynamics of the stem cell population coupled with a delay dif- ferential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the character- istic equation with delay-dependent coefficients obtained while linearizing our system.
Voir
- cell mortality
- red blood
- cells dynamics
- age structured
- dynamics has
- stem cells
- age variable
- growth factors
- stem cell
Modelling and Asymptotic Stability of a Growth Factor-Dependent Stem Cells Dynamics Model with Distributed Delay∗
MostafaAdimyand FabienCrauste
Year 2006
LaboratoiredeMathematiquesAppliquees,UMR5142, UniversitedePauetdesPaysdel’Adour, Avenuedel’universite,64000Pau,France. ANUBIS project, INRIA–Futurs Email: mostafa.adimy@univ-pau.fr (M. Adimy), fabien.crauste@univ-pau.fr (F. Crauste)
Abstract Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells dierentiate and divide, so as to produce blood cells. Growth factors act at dierent levels in the dierentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathe-matical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modelling the dynamics of the stem cell population coupled with a delay dif-ferential equation describing the evolution of the growth factor concentration. We rst reduce our system of three dierential equations to a system of two nonlinear dierential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the character-istic equation with delay-dependent coecients obtained while linearizing our system. We obtain necessary and sucient conditions for the global stability of the steady state describing the cell population’s dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation.
Keywords:Age structured model, dierential equations, distributed delay, asymptotic sta-bility, Lyapunov function, Hopf bifurcation, blood cells model, stem cells, growth factors.
1 Introduction
Hematopoietic stem cells are undierentiated cells, located in the bone marrow, with unique capacities of dierentiation (the ability to produce cells committed to one blood cell lin-eage: white cells, red blood cells or platelets) and self-renewal (the ability to produce a ∗To appear in Discrete and Continuous Dynamical Systems Series B
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