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Existence and Stability of Limit Cycles in a Two-delays Modelof Hematopoiesis Including Asymmetric Division

Published online by Cambridge University Press:  07 February 2014

A. Halanay ,
D. Cândea  and
I. R. Rădulescu
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Abstract

A two dimensional two-delays differential system modeling the dynamics of stem-like cellsand white-blood cells in Chronic Myelogenous Leukemia is considered. All three types ofstem cell division (asymmetric division, symmetric renewal and symmetric differentiation)are present in the model. Stability of equilibria is investigated and emergence ofperiodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventuallyshown. The stability of these limit cycles is studied using the first Lyapunovcoefficient.

Keywords

leukemiaasymmetric divisionstabilityHopf bifurcationlimit cycle
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 1: Issue dedicated to Michael Mackey , 2014 , pp. 58 - 78
Copyright
© EDP Sciences, 2014

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References

Adimy, M.. Integrated semigroups and delay differential equations. J. of Math. Anal. and Appl., 177 (1993), no. 1, 125–134. CrossRefGoogle Scholar
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. CrossRefGoogle Scholar
Adimy, M., Crauste, F., Ruan, S.. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM Journal on Applied Mathematics, 65 (2005), no. 4, 1328–1352. CrossRefGoogle Scholar
Adimy, M., Crauste, F., Ruan, S.. Periodic oscillations in leukopoiesis models with two delays. J. Theor. Biol., 242 (2006), 288-299. CrossRefGoogle ScholarPubMed
Adimy, M., Crauste, F., Halanay, A., Neamtu, M., Opris, D.. Stability of limit cycles in a pluripotent stem cell dynamics model. Chaos, Solitons and Fractals, 27 (2006), 4, 1091-1107. CrossRefGoogle Scholar
Adimy, M., Crauste, F.. Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling. Math. Model. Nat. Phenom., Vol. 7 (2012), No. 6, 1-22.CrossRefGoogle Scholar
Belair, J., Mackey, M.C.. A model for the regulation of mammalian platelet production. Annals of the New York Academy of Sciences, 504 (1987), no. 1, 280–282. CrossRefGoogle Scholar
Beretta, E., Kuang, Y.. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal., 33 (2002), no. , 1144-1165. CrossRefGoogle Scholar
Bernard, S., Belair, J., Mackey, M.C.. Oscillations in cyclical neutropenia: new evidence based on mathematical modelling. J. Theor. Biology, 223 (2003), 283-298. CrossRefGoogle Scholar
Chafee, N.. A bifurcation problem for a functional differential equation of finitely retarded type. J. Math. Anal. and Appl., 35 (1991), 312-348. CrossRefGoogle Scholar
Colijn, C., Fowler, A.C., Mackey, M.C.. High frequency spikes in long period blood cell oscillations. Journal of mathematical biology, 53 (2006), no. 4, 499-519. CrossRefGoogle ScholarPubMed
Colijn, C., Mackey, M.C.. A mathematical model of hematopoiesis I-Periodic chronic myelogenous leukemia. J. Theor. Biology, 237 (2005), 117-132. CrossRefGoogle ScholarPubMed
Cooke, K., Grossman, Z.. Discrete Delay, Distributed Delay and Stability Switches. J. Math. Anal. Appl., 86 (1982), 592-627. CrossRefGoogle Scholar
Cooke, K., van den Driessche, P.. On zeros of some transcendental equations. Funkcialaj Ekvacioj, 29 (1986), 77-90. Google Scholar
Deininger, M.W.N., Goldman, J.M., Melo, J.V.. The molecular biology of chronic myeloid leukemia. Blood, 96 (2000), no. 10, 3343–3356. Google ScholarPubMed
Drobnjak, I., Fowler, A.C.. A Model of Oscillatory Blood Cell Counts in Chronic Myelogenous Leukaemia. Bulletin of mathematical biology, 73 (2011), no. 12, 2983–3007. CrossRefGoogle ScholarPubMed
L.E. El’sgol’ts, S.B. Norkin. Introduction to the theory of differential equations with deviating arguments. (in Russian). Nauka, Moscow, 1971.
Engelborghs, K., Luzyanina, T., Roose, D.. Numerical bifurcation analysis of delay differential equations using dde-biftool. ACM Transactions on Mathematical Software (TOMS), 28 (2002), no. 1, 1–21. CrossRefGoogle Scholar
Friedman, A.. Cancer as Multifaceted Disease. Math. Model. Nat. Phenom., Vol. 7 (2012), no. 1, 3–28.CrossRefGoogle Scholar
Goldman, J.M., Melo, J.V.. Chronic myeloid leukemia—advances in biology and new approaches to treatment. New England Journal of Medicine, 349 (2003), no. 15, 1451–1464. CrossRefGoogle Scholar
J. Hale. Introduction to Functional Differential Equations. Springer, New York, 1977.
J. Hale, S.M. Verduyn Lunel. Theory of Functional Differential Equations, Springer, New York, 1993.
B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. Theory and Applications of Hopf Bifurcation. London Mathem. Soc. Lecture Note Series 41, Cambridge University Press, 1981.
Haurie, C., Dale, D.C., Rudnicki, R., Mackey, M.C.. Modeling complex neutrophil dynamics in the grey collie. Journal of theoretical biology, 204 (2000), no. 4, 505–519. CrossRefGoogle ScholarPubMed
Komarova, N.L., Wodarz, D.. Drug resistance in cancer: principles of emergence and prevention. Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), no. 27, 9714–9719. CrossRefGoogle Scholar
M.C. Mackey. Mathematical models of hematopoietic cell replication and control. Case Studies in Mathematical Modeling–Ecology, Physiology and Cell Biology. New Jersey, Prentice-Hall, 151-182, 1997.
Mahaffy, J.M., Belair, J., Mackey, M.C.. Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis. Journal of theoretical biology, 190 (1998), no. 2, 135–146. CrossRefGoogle ScholarPubMed
Marciniak-Czochra, A., Stiehl, T., Wagner, W.. Modeling of replicative senescence in hematopoietic development. Aging, 1 (2009), no. 8, 723–732. CrossRefGoogle ScholarPubMed
Michor, F., Hughes, T.P., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C.L., Nowak, M.A.. Dynamics of chronic myeloid leukaemia. Nature, 435 (2005), 7046, 1267-1270. CrossRefGoogle Scholar
Michor, F., Iwasa, Y., Nowak, M.A.. Dynamics of cancer progression. Nature Reviews Cancer 4 (2004), no. 3, 197–205. CrossRefGoogle ScholarPubMed
Ozbay, H., Bonnet, C., Benjelloun, H. and Clairambault, J.. Stability Analysis of Cell Dynamics in Leukemia. Math. Model. Nat. Phenom., Vol. 7 (2012), no. 1, 203–234.CrossRefGoogle Scholar
Pujo-Menjouet, L., Bernard, S., Mackey, M. C.. Long period oscillations in a G0 model of hematopoietic stem cells. SIAM J. Appl. Dynam. Sys., 4 (2005), no. 2, 312–332. CrossRefGoogle Scholar
Pujo-Menjouet, L., Mackey, M.C.. Contribution to the study of periodic chronic myelogenous leukemia. C. R. Biologies, 327 (2004), 235-244. CrossRefGoogle ScholarPubMed
Roeder, I., Horn, M., Glauche, I., Hochhaus, A., Mueller, M.C., Loeffler, M.. Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nature medicine, 12 (2006), no. 10, 1181–1184. CrossRefGoogle ScholarPubMed
Sawyers, C.L.. Chronic myeloid leukemia. New England Journal of Medicine, 340 (1999), no. 17, 1330–1340. CrossRefGoogle ScholarPubMed
Shampine, L., Thompson, S.. Solving ddes in matlab. Applied Numerical Mathematics, 37 (2001), no. 4, 441–458. CrossRefGoogle Scholar
Stiehl, T., Marciniak-Czochra, A.. Mathematical Modeling of Leukemogenesis and Cancer Stem Cell Dynamics. Math. Model. Nat. Phenom., Vol. 7 (2012), no. 1, 166–202.CrossRefGoogle Scholar
Tomasetti, C., Levy, D.. Role of symmetric and asymmetric division of stem cells in developing drug resistance. PNAS, 107 (2010), 39, 16766-16771.CrossRefGoogle ScholarPubMed