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Proceedings of Sixth International Conference on Soft Computing for Problem Solving pp 376–391Cite as

Effects of Delay and Drug on HIV Infection

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Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 547))

Abstract

This article discusses delayed model of HIV infection with combination therapy consisting of RTI and PI drug. The delay included in this article two kinds of delays viz. immune response delay and intracellular delay. A well known growth law so called logistic growth is assumed for uninfected and healthy T cell. Local properties of the infection free equilibrium point is discussed in terms of \(R_0\), the basic reproduction number. The existence of Hopf bifurcation with respect to delayed parameter is verified using geometric switching conditions numerically because of delay dependent parameters in the model. Extensive numerical simulations have been carried out on the model to ascertain the effects of drug on viral dynamic and disease progression.

S.K. Sahani—Author is grateful to South Asian University for providing financial assistance to present this paper in the conference.

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References

  1. Allen, L.J., Brauer, F., Van den Driessche, P., Wu, J.: Mathematical Epidemiology. Springer, Heidelberg (2008)

    Google Scholar 

  2. Andersen, R.M., May, R.M.: Epidemiological parameters of HIV transmission. Nature 333(6173), 514–519 (1988)

    Article  Google Scholar 

  3. Anderson, R.M.: Mathematical and statistical studies of the epidemiology of HIV. AIDS 3(6), 333–346 (1989)

    Article  Google Scholar 

  4. Bachar, M., Dorfmayr, A.: HIV treatment models with time delay. C.R. Biol. 327(11), 983–994 (2004)

    Article  Google Scholar 

  5. Bailey, J.J., Fletcher, J.E., Chuck, E.T., Shrager, R.I.: A kinetic model of CD4+ lymphocytes with the human immunodeficiency virus (HIV). BioSystems 26(3), 177–183 (1992)

    Article  Google Scholar 

  6. Bairagi, N., Adak, D.: Global analysis of HIV-1 dynamics with hill type infection rate and intracellular delay. Appl. Math. Model. 38(21), 5047–5066 (2014)

    Article  MathSciNet  Google Scholar 

  7. Banks, H., Bortz, D.: A parameter sensitivity methodology in the context of HIV delay equation models. J. Math. Biol. 50(6), 607–625 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banks, H., Bortz, D., Holte, S.: Incorporation of variability into the modeling of viral delays in HIV infection dynamics. Math. Biosci. 183(1), 63–91 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beretta, E., Kuang, Y.: Modeling and analysis of a marine bacteriophage infection with latency period. Nonlinear Anal. Real World Appl. 2(1), 35–74 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay-dependent parameters. SIAM J. Math. Anal. 33(31), 144–1165 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Bonhoeffer, S., May, R.M., Shaw, G.M., Nowak, M.A.: Virus dynamics and drug therapy. Proc. Nat. Acad. Sci. 94(13), 6971–6976 (1997)

    Article  Google Scholar 

  12. Brauer, F., Castillo-Chavez, C., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, vol. 1. Springer, New York (2001)

    Book  MATH  Google Scholar 

  13. Chiyaka, C., Garira, W., Dube, S.: Modelling immune response and drug therapy in human malaria infection. Comput. Math. Methods Med. 9(2), 143–163 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Culshaw, R.V., Ruan, S.: A delay-differential equation model of HIV infection of CD4\(^+\) T-cells. Math. Biosci. 165(1), 27–39 (2000)

    Article  MATH  Google Scholar 

  15. Culshaw, R.V., Ruan, S., Webb, G.: A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. J. Math. Biol. 46(5), 425–444 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deans, J.A., Cohen, S.: Immunology of malaria. Annu. Rev. Microbiol. 37(1), 25–50 (1983)

    Article  Google Scholar 

  17. Dixit, N.M., Markowitz, M., Ho, D.D., Perelson, A.S.: Estimates of intracellular delay and average drug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy. Antivir. Ther. 9, 237–246 (2004)

    Google Scholar 

  18. Granich, R.M., Gilks, C.F., Dye, C., De Cock, K.M., Williams, B.G.: Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: a mathematical model. Lancet 373(9657), 48–57 (2009)

    Article  Google Scholar 

  19. Grossman, Z., Polis, M., Feinberg, M.B., Grossman, Z., Levi, I., Jankelevich, S., Yarchoan, R., Boon, J., de Wolf, F., Lange, J.M., et al.: Ongoing HIV dissemination during haart. Nat. Med. 5(10), 1099–1104 (1999)

    Article  Google Scholar 

  20. Hale, J.K.: Functional Differential Equations. Springer, New York (1971)

    Book  MATH  Google Scholar 

  21. Haynes, B.F., Gilbert, P.B., McElrath, M.J., Zolla-Pazner, S., Tomaras, G.D., Alam, S.M., Evans, D.T., Montefiori, D.C., Karnasuta, C., Sutthent, R., et al.: Immune-correlates analysis of an HIV-1 vaccine efficacy trial. N. Engl. J. Med. 366(14), 1275–1286 (2012)

    Article  Google Scholar 

  22. Herz, A., Bonhoeffer, S., Anderson, R.M., May, R.M., Nowak, M.A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Nat. Acad. Sci. 93(14), 7247–7251 (1996)

    Article  Google Scholar 

  23. Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ho, D.D.: Toward HIV eradication or remission: the tasks ahead. Science 280(5371), 1866–1867 (1998)

    Article  Google Scholar 

  25. Ho, D.D., Neumann, A.U., Perelson, A.S., Chen, W., Leonard, J.M., Markowitz, M., et al.: Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373(6510), 123–126 (1995)

    Article  Google Scholar 

  26. Holder, B.P., Beauchemin, C.A.: Exploring the effect of biological delays in kinetic models of influenza within a host or cell culture. BMC Public Health 11(Suppl 1), S10 (2011)

    Article  Google Scholar 

  27. Hraba, T., Doležal, J., čelikovský, S.: Model-based analysis of CD4+ lymphocyte dynamics in HIV infected individuals. Immunobiology 181(1), 108–118 (1990)

    Article  Google Scholar 

  28. Jiang, X., Zhou, X., Shi, X., Song, X.: Analysis of stability and hopf bifurcation for a delay-differential equation model of HIV infection of CD4+ T-cells. Chaos, Solitons Fractals 38(2), 447–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 115, 700–721 (1927). The Royal Society

    Article  MATH  Google Scholar 

  30. Kirschner, D.: Using mathematics to understand HIV immune dynamics. AMS Not. 43(2), 191–202 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Kirschner, D.E., Webb, G.F.: A mathematical model of combined drug therapy of HIV infection. Comput. Math. Methods Med. 1(1), 25–34 (1997)

    MATH  Google Scholar 

  32. Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  33. Law, M.G., Prestage, G., Grulich, A., Van de Ven, P., Kippax, S.: Modelling the effect of combination antiretroviral treatments on HIV incidence. AIDS 15(10), 1287–1294 (2001)

    Article  Google Scholar 

  34. Li, D., Ma, W.: Asymptotic properties of a HIV-1 infection model with time delay. J. Math. Anal. Appl. 335(1), 683–691 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, M.Y., Shu, H.: Impact of intracellular delays and target-cell dynamics on in vivo viral infections. SIAM J. Appl. Math. 70(7), 2434–2448 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, S., Wang, L.: Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Math. Biosci. Eng. 7(3), 675–685 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Merrill, S.J.: Modeling the interaction of HIV with cells of the immune system. In: Castillo-Chavez, C. (ed.) Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomathematics, vol. 83, pp. 371–385. Springer, Heidelberg (1989)

    Chapter  Google Scholar 

  38. Mittler, J.E., Markowitz, M., Ho, D.D., Perelson, A.S.: Improved estimates for HIV-1 clearance rate and intracellular delay. AIDS 13(11), 1415 (1999)

    Article  Google Scholar 

  39. Mittler, J.E., Sulzer, B., Neumann, A.U., Perelson, A.S.: Influence of delayed viral production on viral dynamics in HIV-1 infected patients. Math. Biosci. 152(2), 143–163 (1998)

    Article  MATH  Google Scholar 

  40. Murray, J.M., Emery, S., Kelleher, A.D., Law, M., Chen, J., Hazuda, D.J., Nguyen, B.Y.T., Teppler, H., Cooper, D.A.: Antiretroviral therapy with the integrase inhibitor raltegravir alters decay kinetics of HIV, significantly reducing the second phase. AIDS 21(17), 2315–2321 (2007)

    Article  Google Scholar 

  41. 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. JAIDS J. Acquir. Immune Defic. Syndr. 26(5), 405–412 (2001)

    Article  Google Scholar 

  42. Nelson, P.W., Murray, J.D., Perelson, A.S.: A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163(2), 201–215 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Nowak, M., May, R.M.: Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  45. Nowak, M.A., Bangham, C.R.: Population dynamics of immune responses to persistent viruses. Science 272(5258), 74–79 (1996)

    Article  Google Scholar 

  46. Nowak, M.A., May, R.M.: Virus Dynamics (2000)

    Google Scholar 

  47. Ouifki, R., Witten, G.: Stability analysis of a model for HIV infection with RTI and three intracellular delays. BioSystems 95(1), 1–6 (2009)

    Article  Google Scholar 

  48. Pawelek, K.A., Liu, S., Pahlevani, F., Rong, L.: A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math. Biosci. 235(1), 98–109 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. Perelson, A.S.: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2(1), 28–36 (2002)

    Article  MathSciNet  Google Scholar 

  50. Perelson, A.S., Kirschner, D.E., De Boer, R.: Dynamics of HIV infection of CD4\(^+\) T cells. Math. Biosci. 114(1), 81–125 (1993)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. 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(5255), 1582–1586 (1996)

    Article  Google Scholar 

  53. Pitchaimani, M., Monica, C.: Global stability analysis of HIV-1 infection model with three time delays. J. Appl. Math. Comput. 48, 1–27 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Pitchaimani, M., Monica, C., Divya, M.: Stability analysis for HIV infection delay model with protease inhibitor. Biosystems 114(2), 118–124 (2013)

    Article  Google Scholar 

  55. Sahani, S.K.: Effects of intracellular delay and immune response delay in HIV model. Neural Parallel Sci. Comput. 23, 357–366 (2015)

    MathSciNet  Google Scholar 

  56. Sahani, S.K.: A delayed model for HIV infection incorporating intracellular delay. Int. J. Appl. Comput. Math., 1–20 (2016). DOI:10.1007/s40819-016-0190-7

  57. Smith, H.L., De Leenheer, P.: Virus dynamics: a global analysis. SIAM J. Appl. Math. 63(4), 1313–1327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  58. Sun, Z., Xu, W., Yang, X., Fang, T.: Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays. Chaos, Solitons Fractals 31(1), 39–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wang, L., Li, M.Y.: Mathematical analysis of the global dynamics of a model for HIV infection of CD4\(^+\) T cells. Math. Biosci. 200(1), 44–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  60. Wein, L.M., Zenios, S.A., Nowak, M.A.: Dynamic multidrug therapies for HIV: a control theoretic approach. J. Theoret. Biol. 185(1), 15–29 (1997)

    Article  Google Scholar 

  61. Wodarz, D., Lloyd, A.L.: Immune responses and the emergence of drug-resistant virus strains in vivo. Proc. R. Soc. Lond.-B 271(1544), 1101–1110 (2004)

    Article  Google Scholar 

  62. Wodarz, D., Nowak, M.A.: Mathematical models of HIV pathogenesis and treatment. BioEssays 24(12), 1178–1187 (2002)

    Article  Google Scholar 

  63. Xiang, H., Feng, L.X., Huo, H.F.: Stability of the virus dynamics model with beddington-deangelis functional response and delays. Appl. Math. Model. 37(7), 5414–5423 (2013)

    Article  MathSciNet  Google Scholar 

  64. Zhu, H., Zou, X.: Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete Contin. Dyn. Syst. Ser. B 12(2), 511–524 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India

    Saroj Kumar Sahani

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  1. Saroj Kumar Sahani
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Correspondence to Saroj Kumar Sahani .

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Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

    Kusum Deep

  2. Department of Mathematics, South Asian University, New Delhi, India

    Jagdish Chand Bansal

  3. Department of Mathematics, National Institute of Technology, Silchar, Silchar, Assam, India

    Kedar Nath Das

  4. School of Mathematics, Thapar Institute of Engineering and Technology University, Patiala, Punjab, India

    Arvind Kumar Lal

  5. School of Mathematics, Thapar Institute of Engineering and Technology University, Patiala, Punjab, India

    Harish Garg

  6. Department of Mathematics and Computer Science, Liverpool Hope University, Liverpool, United Kingdom

    Atulya K. Nagar

  7. Department of Paper Technology, Indian Institute of Technology Roorkee, Roorkee, India

    Millie Pant

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Sahani, S.K. (2017). Effects of Delay and Drug on HIV Infection. In: Deep, K., et al. Proceedings of Sixth International Conference on Soft Computing for Problem Solving. Advances in Intelligent Systems and Computing, vol 547. Springer, Singapore. https://doi.org/10.1007/978-981-10-3325-4_38

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  • DOI: https://doi.org/10.1007/978-981-10-3325-4_38

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