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Journal of Mathematical Biology

, Volume 48, Issue 5, pp 545–562 | Cite as

Optimal HIV treatment by maximising immune response

  • Rebecca V. CulshawEmail author
  • Shigui Ruan
  • Raymond J. Spiteri
Article

Abstract.

We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy CD4+ T cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.

Keywords

HIV Immune response Optimal control Cytotoxic lymphocytes Numerical methods 

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References

  1. 1.
    Arnaout, R.A., Nowak, M.A., Wodarz, D.: HIV-1 dynamics revisited: biphasic decay by cytotoxic T lymphocyte killing? Proc. Roy. Soc. Lond. B 265, 1347–1354 (2000)CrossRefGoogle Scholar
  2. 2.
    Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ, 1988Google Scholar
  3. 3.
    Ascher, U.M., Spiteri, R.J.: Collocation software for boundary value differential-algebraic equations. SIAM J. Sci. Comput. 15, 938–952 (1994)Google Scholar
  4. 4.
    Carr, A., Emery, S., Kelleher, A., Law, M., Cooper, D.A.: CD8+ lymphocyte responses to antiviral therapy of HIV infection. J. AIDS Hum. Retrovir. 13, 320–326 (1996)Google Scholar
  5. 5.
    Cocchi, F., et al.: Identification of RANTES, MIP-1 alpha, MIP-1 beta as the major HIV-suppressive factors produced by CD8+ T-cells. Sci. 270, 1811–1815 (1995)Google Scholar
  6. 6.
    Culshaw, R.V.: Immune Response Models of HIV Infection and Treatment. Ph.D. thesis, Dalhousie University, 2002Google Scholar
  7. 7.
    Ermentrout, B.: Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. SIAM, Philadelphia, 2002Google Scholar
  8. 8.
    Fister, K.R., Lenhart, S., McNally, J.S.: Optimizing chemotherapy in an HIV model. Elect. J. Diff. Eqs. 32, 1–12 (1998)Google Scholar
  9. 9.
    Fleming, W., Rishel, R.: Deterministic and Stochastic Optimal Control. Springer-Verlag, New York, 1975Google Scholar
  10. 10.
    Gray, C.M., Lawrence, J., Schapiro, J.M., Altman, J.D., Winters, M.A., Crompton, M., Loi, M., Kundu, S.K., Davis, M.M., Merigan, T.C.: Frequency of class I HLA-restricted anti-HIV CD8+ T cells in individuals receiving highly-active antiretroviral therapy. J. Immunol. 162, 1780–1788 (1999)Google Scholar
  11. 11.
    Haseltine, W.A., Wong-Staal, F.: The molecular biology of the AIDS virus. Scientific American, Oct. 1988, pp. 52–62Google Scholar
  12. 12.
    Keller, H.B.: Numerical Solution of Two-Point Boundary Value Problems. Regional Conference Series in Applied Mathematics, No. 24, SIAM, Philadelphia, 1976Google Scholar
  13. 13.
    Kirschner, D.E., Perelson, A.S.: A model for the immune system response to HIV: AZT treatment studies. In: Arino, O., Axelrod, D., Kimmel, M., Langlais, M. (eds), Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics, Wuerz Pub. Ltd., Winnipeg, Canada, 1995, pp. 295–310Google Scholar
  14. 14.
    Kirschner, D.E., Lenhart, S., Serbin, S.: Optimal control of the chemotherapy of HIV. J. Math. Biol. 35, 775–792 (1997)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kirschner, D.E., Webb, G.: A model for treatment strategy in the chemotherapy of AIDS. Bull. Math. Biol. 58, 167–190 (1996)CrossRefGoogle Scholar
  16. 16.
    Kirschner, D.E., Webb, G.: Immunotherapy of HIV-1 Infection. J. Biol. Systems 6, 71–83 (1998)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kirschner, D.E., Webb, G., Cloyd, M.: Model of HIV-1 disease progression based on virus-induced lymph node homing and homing-induced apoptosis of CD4+ lymphocytes. J. AIDS 24, 352–362 (2000)Google Scholar
  18. 18.
    Klach, A.M.: Lippincott’s Nursing Drug Guide 2001. Lippincott, Williams and Wilkins, 2001Google Scholar
  19. 19.
    Murray, J.D.: Mathematical Biology, Springer-Verlag, Berlin-Heidelberg, 1989Google Scholar
  20. 20.
    Musey, L., et al.: Cytotoxic T cell responses, viral load and disease progression in early HIV-type 1 infection. N. Engl. J. Med. 337, 1267–1274 (1997)CrossRefGoogle Scholar
  21. 21.
    Nowak, M.A., May, R.M.: Mathematical biology of HIV infections: Antigenic variation and diversity threshold. Math. Biosci. 106, 1–21 (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ogg, G.S., et al.: Decay kinetics of human immunodeficiency virus-specific effector cytotoxic T lymphocytes after combination antiretroviral therapy. J. Virol. 73, 797–800 (1999)Google Scholar
  23. 23.
    Perelson, A.S., Kirschner, D.E., DeBoer, R.: Dynamics of HIV infection of CD4+ T cells. Math. Biosci. 114, 81–125 (1993)CrossRefzbMATHGoogle Scholar
  24. 24.
    Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44 (1999)zbMATHGoogle Scholar
  25. 25.
    Spouge, J.L., Shrager, R.I., Dimitrov, D.S.: HIV-1 infection kinetics in tissue cultures. Math. Biosci. 138, 1–22 (1996)CrossRefzbMATHGoogle Scholar
  26. 26.
    Walker, C.M., Moody, D.T., Stites, D.P., Levy, J.A.: CD8+ lymphocytes can control HIV infection in vitro by suppressing virus replication. Sci. 234, 1563–1566 (1986)Google Scholar
  27. 27.
    Weber, J.N., Weiss, R.A.: HIV infection: The cellular picture. Scientific American, Oct. 1988, pp. 101–109Google Scholar
  28. 28.
    Wein, L.M., Zenios, S.A., Nowak, M.A.: Dynamic multidrug therapies for HIV: A control theoretic approach. J. Theor. Biol. 185, 15–29 (1997)CrossRefGoogle Scholar
  29. 29.
    Wodarz, D., Klenerman, P., Nowak, M.A.: Dynamics of cytotoxic T-lymphocyte exhaustion. Proc. Roy. Soc. Lond. B 265, 191–203 (1998)CrossRefGoogle Scholar
  30. 30.
    Wodarz, D., Nowak, M.: Specific therapies could lead to long-term immunological control of HIV. Proc. Natl. Acad. Sci. 96, 464–469 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rebecca V. Culshaw
    • 1
    Email author
  • Shigui Ruan
    • 2
  • Raymond J. Spiteri
    • 3
  1. 1.Department of MathematicsClarke CollegeDubuqueUSA
  2. 2.Department of Mathematics and Statistics / Department of MathematicsSchool of Biomedical Engineering, Dalhousie University / University of MiamiNova Scotia/Coral GablesCanada/USA
  3. 3.Department of Mathematics and Statistics, Faculty of Computer ScienceDalhousie UniversityHalifaxCanada

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