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A parameter sensitivity methodology in the context of HIV delay equation models

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Abstract.

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

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References

  1. Adelman, H.M., Haftka, R.T.: Sensitivity analysis of discrete structural systems. A.I.A.A. J. 24, 823–832 (1986)

    Google Scholar 

  2. Akaike, H.: A new look at the statistical model identification. IEEE Transactions on Automatic Control 19, 716–723 (1974)

    Google Scholar 

  3. Banks, H.T.: Identification of nonlinear delay systems using spline methods. In: V. Lakshmikantham, (ed.), Nonlinear Phenomena in Mathematical Sciences, Academic Press, Inc., New York, NY, 1982, pp. 47–55

  4. Banks, H.T., Bortz, D.M., Holte, S.E.: Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics. Mathematical Biosciences 183, 63–91 (2003)

    CAS  PubMed  Google Scholar 

  5. Banks, H.T., Kappel, F.: Spline approximations for functional differential equations. J. Differential Equations 34, 496–522 (1979)

    Google Scholar 

  6. Blower, S.M., Dowlatabadi, H.: Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. International Statistics Review 62, 229–243 (1994)

    Google Scholar 

  7. Bode, H.W.: Network Analysis and Feedback Amplifier Design. Van Nostrand New York, NY, 1945

  8. Borowiak, D.S.: Model Discrimination for Nonlinear Regression Models. Vol. 101, of Stastics: textbooks and monographs. Marcel Dekker, Inc. New York, NY, 1989

  9. Bortz, D.M.: Modeling, Analysis, and Estimation of an in vitro HIV Infection Using Functional Differential Equations. Ph.D. dissertation North Carolina State University, Raleigh, NC, 2002

  10. Bortz, D.M., Nelson, P.W.: Sensitivity analysis of nonlinear lumped parameter models of HIV infection dynamics. Bulletin of Mathematical Biology 66 (2004), pp. 1009–1026.

    Google Scholar 

  11. Bozdogan, H.: Akaike’s information criterion and recent developments in information compelexity. J. Math. Psychology 44, 62–91 (2000)

    MathSciNet  Google Scholar 

  12. Bozdogan, H., Haughton, D.M.A.: Informational complexity criteria for regression models. Comput. Stat. Data Anal. 28, 51–76 (1998)

    MathSciNet  Google Scholar 

  13. Callaway, D.S., Perelson, A.S.: HIV-1 Infection and low steady state viral loads. Bulletin of Math. Biol. 64, 29–64 (2002)

    Google Scholar 

  14. Christie, S.H.: The Bakerian Lecture: Experimental determination of the laws of magneto-electric induction in different masses of the same metal and of its intensity in different metals. Philosophical Transactions of the Royal Society of London 123, 95–142 (1833)

    Google Scholar 

  15. Christini, D.J., Bennett, F.M., Lutchen, K.R., Ahmed, H.M., Hausdorff, J.M., Oriol, N.: Application of linear and nonlinear time-series modeling to heart-rate dynamics analysis. IEEE Transactions on Biomedical Engineering 42(4), 411–415 (1995)

    Google Scholar 

  16. Cruz, J.B.: System Sensitivity Analysis. Dowden Hutchinson & Ross, Inc., Stroudsburg, PA, 1973

  17. Eslami, M.: Theory of Sensitivity in Dynamic Systems: An Introduction. Springer-Verlag, Berlin, 1994

  18. Frank, P.M.: Introduction to System Sensitivity Theory. Academic Press, Inc., New York, NY, 1978

  19. Grossman, Z., Feinberg, M., Kuznetsov, V., Dimitrov, D., Paul, W.: HIV infection: how effective is drug combination treatment?. Immunology Today 19, 528–532 (1998)

    CAS  PubMed  Google Scholar 

  20. Grossman, Z., Polis, M., Feinberg, M.B., Grossman, Z., Levi, I., Jankelevich, S., Yarchoan, R., Boon, J., de Wolf, F., Lange, J.M.A., Goudsmit, J., Dimitrov, D.S., Paul, W.E.: Ongoing HIV dissemination during HAART Nature Medicine 5, 1099–1104 (1999)

  21. Herz, A.V.M., Bonhoeffer, S., Anderson, R.M., May, R.M., Nowak, M.A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proceedings of the National Academy of Sciences, USA 93, 7247–7251 1996

  22. Ho, D.D., Neumann, A.U., Perelson, A.S., Chen, W., Leonard, J.M., Markowitz, M.: Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection Nature 373, 123–126 (1995)

  23. R. L. Iman and J. C. Helton An investigation of uncertainty and sensitivity analysis techniques for computer models. Risk Analysis 8, 71–90 (1988)

  24. Kamina, A., Makuch, R.W., Zhao, H.: Stochastic modeling of early HIV-1 population dynamics. Mathematical Biosciences 170, 187–198 (2001)

    CAS  PubMed  MathSciNet  Google Scholar 

  25. D. Kirschner and S. Lenhart and S. Serbin Optimal control of chemotherapy of HIV. J. Math. Biol. 35, 775–792 (1997)

  26. Kleiber, M., Antúnez, H., Hien, T.D., Kowalczyk, P.: Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations. John Wiley & Sons New York, NY, 1997

  27. Kramer, I.: Modeling the dynamical impact of HIV on the immune system: Viral clearance, infection, and AIDS Mathematical and Computer Modelling 29, 95–112 (1999)

  28. Kubiak, S., Lehr, H., Levy, R., Moeller, T., Parker, A., Swim, E.: Modeling control of HIV infection through structured treatment interruptions with recommendations for experimental protocol. In: Proceedings of the 2001 Industrial Mathematics Modeling Workshop for Graduate Students, no. CRSC-TR01-27 in Center for Research in Scientific Computation Techical Report, North. Carolina. State University, Raleigh, NC, Nov. 2001

  29. Lang, S.: Analysis II, 1969 Addison-Welsey Publishing Company. Inc., Reading, MA, 1969

  30. Lloyd, A.L.: The dependence of viral parameter estimates on the asumed viral load life cycle: limitations of studies of viral load data, Proceedings of the Royal Society of London Series B 268, 847–854 (2001)

  31. Mittler, J.E., Markowitz, M., Ho, D.D., Perelson, A.S.: Improved estimates for HIV-1 clearance rate and intracellular delay. AIDS 13, 1415–1417 (1999)

    CAS  PubMed  Google Scholar 

  32. Mittler, J.E., Sulzer, B., Neumann, A.U., Perelson, A.S.: Influence of delayed viral production on viral dynamics in HIV-1 infected patients. Mathematical Biosciences 152, 143–163 (1998)

    CAS  PubMed  Google Scholar 

  33. Murray, J.M., Kaufmann, G., Kelleher, A.D., Cooper, D.A.: A model of primary HIV-1 infection. Mathematical Biosciences 154, 57–85 (1998)

    CAS  PubMed  Google Scholar 

  34. Nelson, P.W., Mittler, J.E., Perelson, A.S.: Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV viral dynamic parameters. J. Acquired Immune Deficiency Syndromes 26, 405–412 (2001)

    CAS  Google Scholar 

  35. Nelson, P.W., Murray, J.D., Perelson, A.S.: A model of HIV-1 pathogenesis that includes an intracellular delay. Mathematical Biosciences 163, 201–215 (2000)

    CAS  PubMed  Google Scholar 

  36. Nelson, P.W., Perelson, A.S.: Mathematical analysis of delay differential equation models of HIV-1 infection. Mathematical Biosciences 179, 73–94 (2002)

    PubMed  Google Scholar 

  37. Nowak, M.A., Bonhoeffer, S., Shaw, G.M., May, R.M.: Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations. J. Theoretical Biology 184, 203–217 (1997)

    CAS  Google Scholar 

  38. Nowak, M.A., May, R.M.: Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Inc., New York, NY, 2000

  39. Pease, C.M., Mattson, D.J.: Demography of the yellowstone grizzly bears. Ecology 80 957–975 (1999)

    Google Scholar 

  40. Perelson, A.S.: Modeling viral and immune system dynamics. Nature Reviews Immunology 2, 28–36 (2002)

    CAS  PubMed  Google Scholar 

  41. Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Review 41, 3–44 (1999)

    MathSciNet  Google Scholar 

  42. Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D.: HIV-1 dynamics in vivo: virion clearance rate infected cell life-span and viral generation time Science. 271, 1582–1586 (1996)

  43. Phillips, A.N.: Reduction of HIV concentration during acute infection: Independence from a specific immune response Science. 271, 497–499 (1996)

  44. Ramratnam, B., Bonhoeffer, S., Binley, J., Hurley, A., Zhang, L., Mittler, J.E., Markowitz, M., Moore, J.P., Perelson, A.S., Ho, D.D.: Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis. The Lancet 354, 1782–1785 (1999)

    CAS  Google Scholar 

  45. Rogel, M.E., Wu, L.I., Emerman, M.: The human immunodeficiency virus type 1 vpr gene prevents cell proliferation during chronic infection. J. Virology 69, 882–888 (1995)

    CAS  PubMed  Google Scholar 

  46. Saltelli, A., Chan, K., Scott, E.M. eds.: Sensitivity Analysis, Wiley Series in Probability and Statistics. John Wiley & Sons New York, NY, 2000

  47. Smith, B.P., Brier, M.E.: Statistical approach to neural network model building for gentamicin peak predictions. J. Pharmaceutical Sciences 85, 65–69 (1996)

    CAS  Google Scholar 

  48. Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S.: Modeling plasma virus concentration during primary HIV infection. J. Theoretical Biology 203, 285–301 (2000)

    CAS  Google Scholar 

  49. Stanley, L.G.: Computational Methods for Sensitivity Analysis with Applications for Elliptic Boundary Value Problems. Ph.D. dissertation Virginia Polytechnic Institute and State University Blacksburg, VA, 1999

  50. Stilianakis, N.I., Dietz, K., Schenzle, D.: Analysis of a model for the pathogenesis of AIDS Mathematical Biosciences. 145, 27–46 (1997)

  51. Tan, W., Wu, H.: Stochastic modeling of the dynamics of CD4+ T-cell infection by HIV and some monte carlo studies. Mathematical Biosciences 147, 173–205 (1998)

    CAS  PubMed  MathSciNet  Google Scholar 

  52. Tuckwell, H.C., Le Corfec, E.: A Stochastic model for early HIV-1 population dynamics. J. Theoretical Biology 195, 451–463 (1998)

    CAS  Google Scholar 

  53. Verotta, D., Schaedeli, F.: Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials. Mathematical Biosciences 176, 163–183 (2002)

    PubMed  MathSciNet  Google Scholar 

  54. Wei, X., Ghosh, S.K., Taylor, M.E., Johnson, V.A., Emini, E.A., Deutsch, P., Lifson, J.D., Bonhoeffer, S., Nowak, M.A., Hahn, B.H., Saag, M.S., Shaw, G.M.: Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373, 117–122 (1995)

    Article  CAS  PubMed  Google Scholar 

  55. Wein, L.M., D’Amato, R.M., Perelson, A.S.: Mathematical analysis of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads. J. Theoretical Biology 192, 81–98 (1998)

    CAS  Google Scholar 

  56. Wein, L.M., Zeinos, S.A., Nowak, M.A.: Dynamic multidrug therapies for HIV: A control theoretic approach. J. Theoretical Biology 185, 15–29 (1997)

    CAS  Google Scholar 

  57. Wick, D., Self, S.G.: Early HIV Infection in vivo: Branching-process model for studying timing of immune responses and drug therapy. Mathematical Biosciences 165, 115–134 (2000)

    CAS  PubMed  Google Scholar 

  58. Wierzbicki, A.: Models and Sensitivity of Control Systems, no. 5 in Studies in Automation and Control. Elsevier Science Publishing Company. Inc., New York, NY, 1984

  59. Wodarz, D., Jansen, V.A.A.: The role of T cell help for anti-viral CTL responses. J. Theoretical Biology 211, 419–432 (2001)

    CAS  Google Scholar 

  60. Wodarz, D., Lloyd, A.L., Jansen, V.A.A., Nowak, M.A.: Dynamics of macrophage and t cell infection by HIV. J. Theoretical Biology 196, 101–113 (1999)

    CAS  Google Scholar 

  61. Wu, H., Ding, A.A., de Gruttola, V.: Estimation of HIV dynamic parameters. Statistics in Medicine 17, 2463–2485 (1998)

    CAS  PubMed  Google Scholar 

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Banks, H., Bortz, D. A parameter sensitivity methodology in the context of HIV delay equation models. J. Math. Biol. 50, 607–625 (2005). https://doi.org/10.1007/s00285-004-0299-x

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  • DOI: https://doi.org/10.1007/s00285-004-0299-x

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