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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Age-structured and delay differential-difference model of hematopoietic stem cell dynamics

Pages: 2765 - 2791, Volume 20, Issue 9, November 2015      doi:10.3934/dcdsb.2015.20.2765

 
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Mostafa Adimy - Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France (email)
Abdennasser Chekroun - Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France (email)
Tarik-Mohamed Touaoula - Department of Mathematics, University Aboubekr Belkaid, Tlemcen, Algeria (email)

Abstract: In this paper, we investigate a mathematical model of hematopoietic stem cell dynamics. We take two cell populations into account, quiescent and proliferating one, and we note the difference between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential-difference system, and we investigate the existence and stability of the steady states. We give sufficient conditions for boundedness and unboundedness properties for the solutions of this system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state, the most biologically meaningful one, and the existence of a Hopf bifurcation allow the determination of a stability area, which is related to a delay-dependent characteristic equation. Numerical simulations illustrate our results on the asymptotic behavior of the steady states and show very rich dynamics of this model. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders.

Keywords:  Age-structured partial differential equations, delay differential-difference system, stability switch, Hopf bifurcation, Lyapunov function, cell dynamic, hematopoietic stem cells.
Mathematics Subject Classification:  34D20, 34D23, 34K06, 92C37.

Received: September 2014;      Revised: June 2015;      Available Online: September 2015.

 References