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Periodic Solutions in a Mathematical Model for the Treatment of Chronic Myelogenous Leukemia

Published online by Cambridge University Press:  25 January 2012

A. Halanay
A. Halanay*
Affiliation:
Department of Mathematics I, Politehnica University of Bucharest, 060042 Bucharest, Romania
*
⋆ E-mail: halanay@mathem.pub.ro
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Abstract

Existence and stability of periodic solutions are studied for a system of delay differential equations with two delays, with periodic coefficients. It models the evolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenous leukemia under a periodic treatment that acts only on mature cells. Existence of a guiding function leads to the proof of the existence of a strictly positive periodic solution by a theorem of Krasnoselskii. The stability of this solution is analysed.

Keywords

periodic solutionguiding functionindex of an isolated solutionstabilitychronic myelogenous leukemia
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 235 - 244
Copyright
© EDP Sciences, 2012

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References

Abbott, L.H., Michor, F.. Mathematical models of targeted cancer therapy. British Journal of Cancer 95 (2006), 1136–1141. CrossRefGoogle ScholarPubMed
Adimy, M., Crauste, F., Halanay, A., Neamţu, M., Opriş, D.. Stability of Limit Cycles in a Pluripotent Stem Cell Dynamics Model. Chaos, Solitons&Fractals, 27(4) (2006), 1091–1107. CrossRefGoogle Scholar
Adimy, M., Crauste, F., Ruan, S.. A mathematical study of the hematopoiesis process with application to chronic myelogenous leukemia. SIAM J. Appl. Math. 65(4) (2005), 1328–1352. CrossRefGoogle Scholar
Adimy, M., Crauste, F., Ruan, S.. Periodic oscillations in leukopoiesis models with two delays. Journal of Theoretical Biology 242 (2006), 288-299. CrossRefGoogle ScholarPubMed
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
Clarkson, B., Strife, A., Wisniewski, D., Lambek, U., Liu, C.. Chronic myelogenous leukemia as a paradigm of early cancer and possible curative strategies. Leukemia 17 (2003), 1211–1262. 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
Aristide Halanay, Differential Equations : stability, oscilations, time lags. Academic Press, 1966.
A. Halanay. Periodic Solutions in Mathematical Models for Hematological Diseases under Treatment. IEEE Proceedings of the 8-th IFAC Workshop on Time-Delay Systems, Sept. 1-3, Sinaia, Romania, 2009.
Halanay, A.. Stability analysis for a mathematical model of chemotherapy action in hematological diseases. Bull. Sci. Soc. Roumaine Sci. Math. 53 (101) (2010), no. 1, 3-10. Google Scholar
A. Halanay. Treatment induced periodic solutions in some mathematical models of tumoral cell dynamics. Mathematical Reports !2(62) (2010), no. 4, in press.
J. Hale. Theory of Functional Differential Equations. Springer, New York, 1977.
V. Kolmanovskii, A. Myshkis. Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 1992.
M.A. Krasnoselskii. Shift operator on orbits of differential equations. Nauka, Moskow, 1966 (in Russian).
Mackey, M.C., A unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood 51 (1978), 941–956. Google Scholar
Mackey, M.C., Ou, C., Pujo-Menjouet, L., Wu, J.. Periodic oscillations of blood cell population in chronic myelogenous leukemia. SIAM J. Math. Anal. 38 (2006), 166–187. CrossRefGoogle Scholar
Michor, F., Hughes, T., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C., Novak, M.. Dynamics of chronic myeloid leukemia. Nature 435 (2005), 1267–1270. CrossRefGoogle Scholar
Moore, H., Li, N.K.. A mathematical model for chronic myelogenous leukemia (CML) and T-cell interaction. J. Theor. Biol. 227 (2004), 513–523. CrossRefGoogle Scholar
Pujo-Menjouet, L., Mackey, C.. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biol, 327 (2004), 235–244. CrossRefGoogle ScholarPubMed
Sawyers, C.. Chronic Myeloid Leukemia. N. Engl. J. Med. 340 (2000), 1330–1340. CrossRefGoogle ScholarPubMed