Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T15:14:51.830Z Has data issue: false hasContentIssue false
  • English
  • Français

Delay Model of Hematopoietic Stem Cell Dynamics:Asymptotic Stability and Stability Switch

Published online by Cambridge University Press:  26 March 2009

F. Crauste
F. Crauste*
Affiliation:
Université de Lyon, Université Lyon1, CNRS UMR 5208 Institut Camille Jordan - F - 69200 Villeurbanne Cedex, France
*
crauste@math.univ-lyon1.fr
Get access

Abstract

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis.

Keywords

nonlinear delay differential systemsecond degree exponential polynomialLyapunov functionstability switchhematopoietic stem cell dynamics
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 28 - 47
Copyright
© EDP Sciences, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adimy, M., Crauste, F.. Global stability of a partial differential equation with distributed delay due to cellular replication. Nonlinear Analysis, 54 (2003), 1469–1491. CrossRef
Adimy, M., Crauste, F.. Modelling and asymptotic stability of a growth factor-dependent stem cells dynamics model with distributed delay. Discret. Cont. Dyn. Sys. Ser. B, 8 (2007), No. 1, 19–38.
Adimy, M., Crauste, F., Pujo-Menjouet, L.. On the stability of a maturity structured model of cellular proliferation. Discret. Cont. Dyn. Sys. Ser. A, 12 (2005), No. 3, 501–522.
Adimy, M., Crauste, F., Ruan, S.. Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics. Nonlinear Analysis: Real World Applications, 6 (2005), No. 4, 651–670. CrossRef
Adimy, M., Crauste, F., Ruan, S.. Periodic Oscillations in Leukopoiesis Models with Two Delays. J. Theo. Biol., 242 (2006), 288–299. CrossRef
Bélair, J., Mackey, M.C., Mahaffy, J.M.. Age-structured and two-delay models for erythropoiesis. Math. Biosci., 128 (1995), 317–346. CrossRef
Beretta, E., Kuang, Y.. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal., 33 (2002), No. 5, 1144–1165. CrossRef
Bernard, S., Bélair, J., Mackey, M.C.. Oscillations in cyclical neutropenia: New evidence based on mathematical modeling. J. Theor. Biol., 223 (2003), 283–298. CrossRef
Burns, F.J., Tannock, I.F.. On the existence of a G0 phase in the cell cycle. Cell Tissue Kinet., 19 (1970), 321–334.
Colijn, C., Foley, C., Mackey, M.C.. G-CSF treatment of canine cyclical neutropenia: A comprehensive mathematical model. Exper. Hematol., 35 (2007), No. 6, 898–907. CrossRef
Crauste, F.. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Math. Bio. Eng., 3 (2006), No. 2, 325–346. CrossRef
Fortin, P., Mackey, M.C.. Periodic chronic myelogenous leukemia: Spectral analysis of blood cell counts and etiological implications. Brit. J. Haematol., 104 (1999), 336–345. CrossRef
J. Hale, S.M. Verduyn Lunel. Introduction to functional differential equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993.
Haurie, C., Dale, D.C., Mackey, M.C.. Cyclical neutropenia and other hematological disorders: A review of mechanisms and mathematical models. Blood, 92 (1998), No. 8, 2629–2640.
L.G. Lajtha. On DNA labeling in the study of the dynamics of bone marrow cell populations, in: Stohlman, Jr., F. (Ed), The Kinetics of Cellular Proliferation, Grune and Stratton, New York (1959) 173–182.
Mackey, M.C.. Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis. Blood, 51 (1978), 941–956.
Mackey, M.C., Rudnicki, R.. Global stability in a delayed partial differential equation describing cellular replication. J. Math. Biol., 33 (1994), 89–109. CrossRef
Mackey, M.C., Rudnicki, R.. A new criterion for the global stability of simultaneous cell replication and maturation processes. J. Math. Biol., 38 (1999), 195–219. CrossRef
Mahaffy, J.M., Bélair, J., Mackey, M.C.. Hematopoietic model with moving boundary condition and state dependent delay. J. Theor. Biol., 190 (1998), 135–146. CrossRef
Pujo-Menjouet, L., Bernard, S., Mackey, M.C.. Long period oscillations in a G0 model of hematopoietic stem cells. SIAM J. Appl. Dyn. Systems, 4 (2005), No. 2, 312–332. CrossRef
Pujo-Menjouet, L., Mackey, M.C.. Contribution to the study of periodic chronic myelogenous leukemia. C. R. Biologies, 327 (2004), 235–244. CrossRef
Shampine, L.F., Thompson, S.. Solving DDEs in Matlab . Appl. Numer. Math., 37 (2001), 441–458. CrossRef