- Volume 33 Issue 5_6
Human Immune Deficiency Virus (or simply HIV) induces a persistent infection that leads to AIDS causing death in almost every infected individual. As HIV affects the immune system directly by attacking the CD4+ T cells, to exterminate the infection, the natural immune system produces virus-specific cytotoxic T lymphocytes(CTLs) that kills the infected CD4+ T cells. The reduced CD4+ T cell count produce reduced amount of cytokines to stimulate the production of CTLs to fight the invaders that weakens the body immunity succeeding to AIDS. In this paper, we introduce a mathematical model with discrete time-delay to represent this cell dynamics between CD4+ T cells and the CTLs under HIV infection. A modified functional form has been considered to describe the infection mechanism. Characteristics of the system are studied through mathematical analysis. Numerical simulations are carried out to illustrate the analytical findings.
- S. Sharma and G.P. Samanta, Dynamical behaviour of an HIV/AIDS epidemic model, Differential Equations and Dynamical Systems 22 (2014), 369–395. https://doi.org/10.1007/s12591-013-0173-7
- E. Vergu, A. Mallet and J. Golmard, A modelling approach to the impact of HIV mutations on the immune system, Computers Biol. Med. 35 (2005), 1–24. https://doi.org/10.1016/j.compbiomed.2004.01.001
- P.A. Volberding and S.G. Deeks, Antiretroviral therapy and management of HIV infection, Lancet 3 (2010), 49–62. https://doi.org/10.1016/S0140-6736(10)60676-9
- L. Wang and M.Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T-cells, Math. Biosci. 200 (2006), 44–57. https://doi.org/10.1016/j.mbs.2005.12.026
- D.Wodarz and M. Nowak, Specific therapies could lead to long-term immunological control of HIV, Proc. Natl. Acad. Sci. 96 (1999), 464–469. https://doi.org/10.1073/pnas.96.25.14464
- A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard and D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582–1586. https://doi.org/10.1126/science.271.5255.1582
- P. Nelson, J. Murray and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), 201–215. https://doi.org/10.1016/S0025-5564(99)00055-3
- M. Nowak, S. Bonhoeffer, G. Shaw and R. May, Antiviral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol. 184 (1997), 203–217. https://doi.org/10.1006/jtbi.1996.0307
- A.S. Perelson, D.E. Kirschner and R. DeBoer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci. 114 (1993), 81-125. https://doi.org/10.1016/0025-5564(93)90043-A
- A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), 3–44. https://doi.org/10.1137/S0036144598335107
- G.P. Samanta, Analysis of a nonautonomous HIV/AIDS model. Math. Model. Nat. Phenom. 5 (2010), 70–95. https://doi.org/10.1051/mmnp/20105604
- G.P. Samanta, Analysis of a nonautonomous HIV/AIDS epidemic model with distributed time delay, Math. Model. Anal. 15 (2010), 327–347. https://doi.org/10.3846/1392-6292.2010.15.327-347
- G.P. Samanta, Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay, Nonlinear Anal. Real World Appl. 12 (2011), 1163–1177. https://doi.org/10.1016/j.nonrwa.2010.09.010
- M. Stafford, C. Lawrence, Y. Cao, E. Daar, D. Ho and A. Perelson, Modelling plasma virus concentration during primary HIV infection, J. Theor. Biol. 203 (2000), 285–301. https://doi.org/10.1006/jtbi.2000.1076
- S. Sharma and G.P. Samanta, Dynamical behaviour of a tumor-immune system with chemotherapy and optimal control, J. Nonlinear Dynamics 2013 (2013), Article ID 608598, DOI 10.1155/2013/608598. https://doi.org/10.1155/2013/608598
- A.B. Gumel, C.C. McCluskey and P. van den Driessche, Mathematical study of a stagedprogressive HIV model with imperfect vaccine, Bull. Math. Biol. 68 (2006), 2105–2128. https://doi.org/10.1007/s11538-006-9095-7
- R.V. Culshaw, S. Ruan and R.J. Spiteri, Optimal treatment by maximisining immune response, J. Math. Biol. 48 (2004), 545–562. https://doi.org/10.1007/s00285-003-0245-3
- N. Dalal, D. Greenhalgh and X. Mao, Mathematical modelling of internal HIV dynamics, Discrete and Continuous Dynamical Systems - Series B 12 (2009), 305–321. https://doi.org/10.3934/dcdsb.2009.12.305
- K. Gopalsamy, Stability and Oscillations in Delay-Differential Equations of Population Dynamics, Kluwer, Dordrecht, 1992.
- HIV infection and AIDS, National Institute of Allergy and Infectious Diseases, Bethesda, USA, (2006), www.niaid.nih.gov.
- A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May and M.A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA. 93 (1996), 7247–7251. https://doi.org/10.1073/pnas.93.14.7247
- H.W. Hethcote, M.A. Lewis and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol. 27 (1989), 49–64. https://doi.org/10.1007/BF00276080
- T.W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol. 46 (2003), 17–30. https://doi.org/10.1007/s00285-002-0165-7
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, London, 1993.
- R.M. May and R.M. Anderson, Transmission dynamics of HIV infection, Nature 326 (1987), 137–142. https://doi.org/10.1038/326137a0
- S. Bonhoeffer, G. Shaw, R. May and M. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. 94 (1997), 6971–6976. https://doi.org/10.1073/pnas.94.13.6971
- S. Bajaria, G. Webb and D. Kirschner, Predicting differential responses to structured treatment interruptions during HAART, Bull. Math. Biol. 66 (2004), 1093–1118. https://doi.org/10.1016/j.bulm.2003.11.003
- E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM. J. Math. Anal. 33 (2002), 1144–1165. https://doi.org/10.1137/S0036141000376086
- S. Bonhoeffer, J. Coffin and M. Nowak, Human Immunodeficiency Virus drug therapy and virus load, J. Virol. 71 (1997), 3275–3278.
- G. Birkhoff and G.C. Rota, Ordinary Differential Equations, Ginn and Co., Boston, 1982.
- L.M. Cai, X. Li, M. Ghosh and B. Guo, Stability of an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math. 229 (2009), 313–323. https://doi.org/10.1016/j.cam.2008.10.067
- D. Callaway and A. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol. 64 (2002), 29–64. https://doi.org/10.1006/bulm.2001.0266
- T.W. Chun, D.C. Nickle, J.S. Justement, D. Large, A. Semerjian, M.E. Curlin, M.A. O’Shea, C.W. Hallahan, M. Daucher and other authors, HIV-infected individuals receiving effective antiviral therapy for extended periods of time continually replenish their viral reservoir, J. Clin. Invest. 115 (2005), 3250–3255. https://doi.org/10.1172/JCI26197
- M.C. Connell, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci. 181 (2003), 1–16. https://doi.org/10.1016/S0025-5564(02)00149-9
- R.V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci. ,165 (2000), 27–39. https://doi.org/10.1016/S0025-5564(00)00006-7
- M. Bachar and A. Dorfmayr, HIV treatment models with time delay, C. R. Biol. 327 (2004), 983–994. https://doi.org/10.1016/j.crvi.2004.08.007
- AIDS epidemic update, UNAIDS, (2005), www.unaids.org.
- R.M. Anderson, The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS, J. Acquir. Immune Defic. Syndr. 1 (1988), 241–256.
- R.A. Arnaout, M.A. Nowak and D. Wodarz, HIV-1 dynamics revisited: biphasic decay by cytotoxic T lymphocyte killing? Proc. Roy. Soc. Lond. B 265 (2000), 1347?1354. https://doi.org/10.1098/rspb.2000.1149