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

Mathematical Modelling of Natural Phenomena (2009)

- Volume: 4, Issue: 2, page 28-47
- ISSN: 0973-5348

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topCrauste, F.. "Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch." Mathematical Modelling of Natural Phenomena 4.2 (2009): 28-47. <http://eudml.org/doc/222258>.

@article{Crauste2009,

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.
},

author = {Crauste, F.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {nonlinear delay differential system; second degree exponential
polynomial; Lyapunov function; stability switch; hematopoietic stem cell
dynamics; nonlinear delay differential systems; second degree exponential polynomial; Lyapunov functional; hematopoietic stem cell dynamics},

language = {eng},

month = {3},

number = {2},

pages = {28-47},

publisher = {EDP Sciences},

title = {Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch},

url = {http://eudml.org/doc/222258},

volume = {4},

year = {2009},

}

TY - JOUR

AU - Crauste, F.

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

JO - Mathematical Modelling of Natural Phenomena

DA - 2009/3//

PB - EDP Sciences

VL - 4

IS - 2

SP - 28

EP - 47

AB -
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.

LA - eng

KW - nonlinear delay differential system; second degree exponential
polynomial; Lyapunov function; stability switch; hematopoietic stem cell
dynamics; nonlinear delay differential systems; second degree exponential polynomial; Lyapunov functional; hematopoietic stem cell dynamics

UR - http://eudml.org/doc/222258

ER -

## References

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

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