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Mathematical Analysis of a Delayed Hematopoietic Stem Cell Model with Wazewska–Lasota Functional Production Type

  • Radouane YafiaEmail author
  • M. A. Aziz Alaoui
  • Abdessamad Tridane
  • Ali Moussaoui
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 14)

Abstract

In this chapter, we consider a more general model describing the dynamics of a hematopoietic stem cell (HSC) model with Wazewska–Lasota functional production type describing the cycle of proliferating and quiescent phases. The model is governed by a system of two ordinary differential equations with discrete delay. Its dynamics are studied in terms of local stability and Hopf bifurcation. We prove the existence of the possible steady state and their stability with respect to the time delay and the apoptosis rate of proliferating cells. We show that a sequence of Hopf bifurcations occurs at the positive steady state as the delay crosses some critical values. We illustrate our results with some numerical simulations.

Keywords

Hematopoietic Stem Cell Hopf Bifurcation Daughter Cell Delay Differential Equation Autoimmune Hemolytic Anemia 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are very grateful to the editors and to Professor M. C. Mackey for their valuable discussions.

References

  1. 1.
    Adimy, M., Crauste, F.: Existence, positivity and stability for a nonlinear model of cellular proliferation. Nonlinear Anal. Real World Appl. 6, 337–366 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Adimy, M., Crauste, F., Pujo-Menjouet, L.: On the stability of a nonlinear maturity structured model of cellular proliferation. Discret. Cont. Dyn. Syst. 12, 501–522 (2005)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Adimy, M., Crauste, F., Hbid, M.Y.L., Qesmi, R.: Stability and Hopf bifurcation for a cell population model with state-dependent delay. SIAM J. Appl. Math. 70(5), 1611–1633 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alaoui, H.T., Yafia, R.: Stability and Hopf bifurcation in approachable hematopoietic stem cells model. Math. Biosci. 206, 176–184 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Alaoui, H.T., Yafia, R., Aziz Alaoui, M.A.: Dynamics and Hopf bifurcation analysis in a delayed haematopoietic stem cells model. Arab J. Math. Math. Sci. 1(1), 35–49 (2007)Google Scholar
  6. 6.
    Bélair, J., Mahaffy, J.M., Mackey, M.C.: Age structured and two delay models for erythropoiesis. Math. Biosci. 128, 317–346 (1995)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bullough, W.S.: Mitotic control in adult mammalian tissues. Biol. Rev. 50, 99–127 (1975)CrossRefGoogle Scholar
  8. 8.
    Burns, F., Tannock, I.: On the existence of a G 0 phase in the cell cycle. Cell Tissue Kinet. 3, 321–334 (1970)Google Scholar
  9. 9.
    Chikkappa, G., Burlington, H., Borner, G., Chanana, A.D., Cronkite, E.P., Ohl, S., Pavelec, M., Robertso, J.S.: Periodic oscillation of blood leukocytes, platelets, and reticulocytes in a patient with chronic myelocytic leukemia. Blood 47, 1023–1030 (1976)Google Scholar
  10. 10.
    Colijn, C., Mackey, M.C.: A mathematical model of hematopoiesis: Periodic chronic myelogenous leukemia, part I, J. Theor. Biol. 237, 117–132 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Crauste, F., Pujo-Menjouet, L., Genieys, S., Molina, C., Gandrillon, O.: Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis. J. Theor. Biol. 250, 322–338 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Diekmann, O., Van Giles, S., Verduyn Lunel, S., Walter, H.: Delay Equations. Springer, New York (1995)zbMATHCrossRefGoogle Scholar
  13. 13.
    Ferrell, J.J.: Tripping the switch fantastic: how protein kinase cascade convert graded into switch-like outputs. Trends Biochem. Sci. 21, 460–466 (1996)CrossRefGoogle Scholar
  14. 14.
    Fortin, P., Mackey, M.C.: Periodic chronic myelogenous leukemia: spectral analysis of blood cell counts and etiological implications. Br. J. Haematol. 104, 336–345 (1999)CrossRefGoogle Scholar
  15. 15.
    Guerry, D., Dale, D., Omine, D.C., Perry, S., Wol, S.M.: Periodic hematopoiesis in human cyclic neutropenia. J. Clin. Inves. 52, 3220–3230 (1973)CrossRefGoogle Scholar
  16. 16.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  17. 17.
    Haurie, C., Dale, D.C., Mackey, M.C.: Occurrence of periodic oscillations in the differential blood counts of congenital, idiopathic and cyclical neutropenic patients before and during treatment with G-CSF. Exp. Hematol. 27, 401–409 (1999)CrossRefGoogle Scholar
  18. 18.
    Haurie, C., Dale, D.C., Rudnicki, R., Mackey, M.C.: Mathematical modeling of complex neutrophil dynamics in the grey collie. J. Theor. Biol. 204, 505–519 (2000)CrossRefGoogle Scholar
  19. 19.
    Mackey, M.C.: Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis. Blood 51(5), 941–956 (1978)Google Scholar
  20. 20.
    Mackey, M.C.: Cell kinetic status of haematopoietic stem cells. Cell Prolif. 34, 71–83 (2001)CrossRefGoogle Scholar
  21. 21.
    Mackey, M.C., Dormer, P.: Continuous maturation of proliferating erythroid precursors. Cell Tissue Kinet. 15, 381–392 (1982)Google Scholar
  22. 22.
    Mahaffy, J.M., Bélair, J., Mackey, M.C.: Hematopoietic model with moving boundary condition and state dependent delay. J. Theor. Biol. 190, 135–146 (1998)CrossRefGoogle Scholar
  23. 23.
    Orr, J.S., Kirk, J., Gray, K.G., Anderson, J.R.: A study of the interdependence of red cell and bone marrow stem cell populations. Br. J. Haematol. 15, 23–24 (1968)CrossRefGoogle Scholar
  24. 24.
    Othmer, H.G., Adler, F.R., Lewis, M.A., Dalton, J.C.: The Art of Mathematical Modeling: Case Studies in Ecology, Physiology and Biofluids. Prentice Hall, New York (1997)Google Scholar
  25. 25.
    Pitchford, S.C., Furze, R.C., Jones, C.P., Wengner, A.M., Rankin, S.M.: Differential mobilization of subsets of progenitor cells from the bone marrow. Cell Stem Cell 4, 62–72 (2009)CrossRefGoogle Scholar
  26. 26.
    Santillan, M., Mahaffy, J.M., Bélair, J., Mackey, M.C.: Regulation of platelet production: the normal response to perturbation and cyclical platelet disease. J. Theor. Biol. 206, 585–603 (2000)CrossRefGoogle Scholar
  27. 27.
    Smith, J.A., Martin, L.: Do cells cycle? Proc. Natl. Acad. Sci. USA 70, 1263–1267 (1973)CrossRefGoogle Scholar
  28. 28.
    Swinburune, J., Mackey, M.C.: Cyclical thrombocytopenia: characterization by spectral analysis and a review. J. Theor. Med. 2, 81–91 (2000)CrossRefGoogle Scholar
  29. 29.
    Wazewska-Czyzewska, M., Lasota, A.: Mathematical problems of the dynamics of the red blood cell system. Mathematyka Stosowana 6, 23–40 (1976)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Yanabu, M., Nomura, S., Fukuroi, T., Kawakatsu, T., Kido, H., Yamaguchi, K., Suzuki, M., Kokawa, T., Yasunaga, K.: Periodic production of antiplatelet autoantibody directed against GP IIIa in cyclic thrombocytopenia. Acta Haematol. 89, 155–159 (1993)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Radouane Yafia
    • 1
    Email author
  • M. A. Aziz Alaoui
    • 2
    • 3
    • 4
  • Abdessamad Tridane
    • 5
  • Ali Moussaoui
    • 6
  1. 1.Polydisciplinary Faculty of OuarzazateIbn Zohr UniversityOuarzazateMorocco
  2. 2.Normandie UniversityLe HavreFrance
  3. 3.ULH, LMAHLe HavreFrance
  4. 4.FR CNRS 3335Le HavreFrance
  5. 5.Department of Mathematical SciencesUnited Arab Emirates UniversityAl AinUnited Arab Emirates
  6. 6.Department of MathematicsUniversity of TlemcenTlemcenAlgeria

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