Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T12:19:41.540Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Dynamics of an Impulsive Model of Hematopoiesis

Published online by Cambridge University Press:  26 March 2009

C. Kou ,
M. Adimy  and
A. Ducrot
C. Kou*
Affiliation:
Department of Applied Mathematics, Donghua University, Shanghai 201620, P. R. China
M. Adimy
Affiliation:
Laboratoire de Mathématiques Appliquées, UMR CNRS 5142 & INRIA, ANUBIS, Université de Pau, 64000 Pau, France
A. Ducrot
Affiliation:
Institut Mathématiques de Bordeaux, UMR CNRS 5251 & INRIA, ANUBIS Université Victor Segalen Bordeaux 2, 33076 Bordeaux, France
*
kouchunhai@dhu.edu.cn
Get access

Abstract

We propose and analyze a nonlinear mathematical model of hematopoiesis,describing the dynamics of stem cell population subject to impulsiveperturbations. This is a system of two age-structured partial differentialequations with impulses. By integrating these equations over theage, we obtain a system of two nonlinear impulsive differential equations withseveral discrete delays. This system describes the evolution of the totalhematopoietic stem cell populations with impulses. We first examine theasymptotic behavior of the model in the absence of impulsions.Secondly, we add the impulsive perturbations and we investigate the qualitativebehavior of the model including the global asymptotic stability of the trivialsolution and the existence of periodic solution in the case of periodicimpulsive perturbations. Finally, numerical simulations are carriedout to illustrate the behavior of the model. This study maybe helpful tounderstand the reactions observed in the hematopoietic system after differentforms of stress as direct destruction by some drugs or irradiation.

Keywords

model of hematopoiesis; impulsion; delay; asymptotic stability;Lyapunov functional; periodic solution
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 2: Delay equations in biology , 2009 , pp. 68 - 91
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 celluar replication. Nonlinear Analysis, 54 (2003), No. 8, 1469-1491. CrossRef
Adimy, M., Crauste, F.. Existence, positivity and stability for a nonlinear model of celluar proliferation. Nonlinear Analysis: Real World Applications, 6 (2005), No. 2, 337-366. CrossRef
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.
M. Adimy, F. Crauste, S. Ruan. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM J. Appl. Math., 65 (2005), No. 4, 1328-1352 .
Adimy, M., Pujo-Menjouet. Asymptotic behaviour of a singular transport equation modeling cell division. Discret. Cont. Dyn. Sys. B, 3 (2003), No. 3, 439-456.
Bernard, S., Belair, J., Mackey, M.C.. Oscillations in cyclical neutropenia: New evidence based on mathematical modeling. J. Theor. Biol., 223 (2003), No. 3, 283-298. CrossRef
Bernard, S., Belair, J., Mackey, M.C.. Bifurcations in a white-blood-cell production model. C. R. Biologies, 327 (2004), No. 3, 201-210. CrossRef
Burns, F.J., Tannock, I.F.. On the existence of a G 0 phase in the cell cycle. Cell. Tissue Kinet., 19 (1970), No. 4, 321-334.
C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis, I. Periodic chronic myelogenous leukemia. J. Theor. Biol., 237 (2005), No. 2, 117-132.
C. Colijn, M.C. Mackey. A mathematical model of hematopoiesis, II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), No. 2, 133-146.
Ferrell, J.J.. Tripping the switch fantastic: How protein kinase cascade convert graded into switch-like outputs. TIBS, 21 (1996), No. 12, 460-466.
Gopalsamy, K., Zhang, B.G.. On delay differential equation with impulses. J. Math. Anal. Appl., 139 (1989), No. 11, 110-122. CrossRef
I. Gyori, G. Ladas. Oscillation theory of delay differential equations with applications. Clarendon, Oxford, 1991.
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 periodic hematological diseases: A review of mechanisms and mathematical models. Blood, 92 (1998), No. 8, 2629-2640.
Y. Kuang. Delay differential equations with application in population dynamics. Academic Press. Boston, MA, 1993.
V. Lakshmikantham, D.D. Bainov, P.S. Simeonov. Theory of impulsive differential equations. World Scientific. Singapore, 1989.
Loeffler, M., Wichmann, H.E.. A comprehensive mathematical model of stem cell proliferation which reproduces most of the published experimental results. Cell Tissue Kinet., 13 (1980), No. 5, 543-561.
Mackey, M.C.. Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis. Blood, 51 (1978), No. 5, 941-956.
M.C. Mackey. Dynamic hematological disorders of stem cell origin. In Biophysical and Biochemical Information Transfer in Recognition, J.G. Vassileva-Popova and E.V. Jensen, eds., Plenum Publishing, New York, 1979, 373-409.
M.C. Mackey. Mathematical models of hematopoietic cell replication and control. in The Art of Mathematical Modelling: Case Studies in Ecology, Physiology and Biofluids, Prentice-Hall, Upper Saddle River, NJ, 1997, 149-178.
Mackey, M.C., Pujo-Menjouet, L., Wu, J.. Period oscillations of blood cell populations in periodic myelogenous leukemia. SIAM J. Math. Anal., 38 (2006), No. 1, 166-187. CrossRef
Mackey, M.C., Rey, A.. Propagation of population pulses and fronts in a cell replication problem: non-locality and dependence on the initial function. Physica D, 86 (1995), No. 3, 373-395.
Mackey, M.C., Rudnicki, R.. Global stability in a delayed partial differential equation describing cellular replication. J. Math. Bio., 33 (1994), No. 1, 89-109. CrossRef
Pujo-Menjouet, L., Mackey, M.C.. Contribution to the study of periodic chronic myelogenous leukemia. C. R. Biologies, 327 (2004), No. 3, 235-244. CrossRef
Pujo-Menjouet, L., Bernard, S., Mackey, M.C.. Long Period Oscillations in a G 0 Model of Hematopoietic Stem Cells. SIAM J. Appl. Dynam. Systems, 4 (2005), No. 2, 312-332. CrossRef
Rubino, S.I., Lebowitz, J.L.. A mathematical model of neutrophil production and control in normal man. J. Math. Bio., 1 (1975), No. 3, 187-225. CrossRef
Sachs, L.. The molecular control of hemopoiesis and leukemia. C. R. Acad. Sci. Paris, 316 (1993), No. 9, 882-891.
Saker, S.H., Alzabut, J.O.. Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model. Nonlinear Analysis: real world applications, 8 (2007), No. 4, 1029-1039. CrossRef
G.F. Webb. Theory of Nonlinear Age-dependent Population Dynamics. Monogr. Textbooks Pure Appl. Math., 89, Dekker, New York, 1985.
Yan, J., Zhao, A.. Oscillation and stability of linear impulsive delay differential equations. J. Math. Anal. Appl., 227 (1998), No. 1, 187-194. CrossRef
J. Yan, A. Zhao, J.J. Nieto. Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra Systems. Mathematical and Computer Modelling, 40 (2004), No 5-6, 509-518.