Skip to main content
SpringerLink
Log in
SpringerLink
Log in

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

  • Chapter
  • First Online:
Mathematical Modeling and Applications in Nonlinear Dynamics

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 14))

  • 1347 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alaoui, H.T., Yafia, R.: Stability and Hopf bifurcation in approachable hematopoietic stem cells model. Math. Biosci. 206, 176–184 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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. Bélair, J., Mahaffy, J.M., Mackey, M.C.: Age structured and two delay models for erythropoiesis. Math. Biosci. 128, 317–346 (1995)

    Article  MATH  Google Scholar 

  7. Bullough, W.S.: Mitotic control in adult mammalian tissues. Biol. Rev. 50, 99–127 (1975)

    Article  Google Scholar 

  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. 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. Colijn, C., Mackey, M.C.: A mathematical model of hematopoiesis: Periodic chronic myelogenous leukemia, part I, J. Theor. Biol. 237, 117–132 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Diekmann, O., Van Giles, S., Verduyn Lunel, S., Walter, H.: Delay Equations. Springer, New York (1995)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  16. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. Mackey, M.C.: Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis. Blood 51(5), 941–956 (1978)

    Google Scholar 

  20. Mackey, M.C.: Cell kinetic status of haematopoietic stem cells. Cell Prolif. 34, 71–83 (2001)

    Article  Google Scholar 

  21. Mackey, M.C., Dormer, P.: Continuous maturation of proliferating erythroid precursors. Cell Tissue Kinet. 15, 381–392 (1982)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  27. Smith, J.A., Martin, L.: Do cells cycle? Proc. Natl. Acad. Sci. USA 70, 1263–1267 (1973)

    Article  Google Scholar 

  28. Swinburune, J., Mackey, M.C.: Cyclical thrombocytopenia: characterization by spectral analysis and a review. J. Theor. Med. 2, 81–91 (2000)

    Article  Google Scholar 

  29. Wazewska-Czyzewska, M., Lasota, A.: Mathematical problems of the dynamics of the red blood cell system. Mathematyka Stosowana 6, 23–40 (1976)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Polydisciplinary Faculty of Ouarzazate, Ibn Zohr University, 638, Ouarzazate, Morocco

    Radouane Yafia

  2. Normandie University, Le Havre, France

    M. A. Aziz Alaoui

  3. ULH, LMAH, 76600, Le Havre, France

    M. A. Aziz Alaoui

  4. FR CNRS 3335, 25 rue Philippe Lebon, 76600, Le Havre, France

    M. A. Aziz Alaoui

  5. Department of Mathematical Sciences, United Arab Emirates University, 15551, Al Ain, United Arab Emirates

    Abdessamad Tridane

  6. Department of Mathematics, University of Tlemcen, 13000, Tlemcen, Algeria

    Ali Moussaoui

Authors
  1. Radouane Yafia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Aziz Alaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdessamad Tridane
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ali Moussaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radouane Yafia .

Editor information

Editors and Affiliations

  1. Dept of Mech.Eng & Industrial Engg., Southern Illinois Univ, Edwardsville, Edwardsville, Illinois, USA

    Albert C.J. Luo

  2. TOBB Univ. of Economics and Technology, Department of Mathematics, Ankara, Turkey

    Hüseyin Merdan

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Yafia, R., Alaoui, M.A.A., Tridane, A., Moussaoui, A. (2016). Mathematical Analysis of a Delayed Hematopoietic Stem Cell Model with Wazewska–Lasota Functional Production Type. In: Luo, A., Merdan, H. (eds) Mathematical Modeling and Applications in Nonlinear Dynamics. Nonlinear Systems and Complexity, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-26630-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26630-5_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26628-2

  • Online ISBN: 978-3-319-26630-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation