\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A singularly perturbed HIV model with treatment and antigenic variation

Abstract Related Papers Cited by
  • We study the long term dynamics and the multiscale aspects of a within-host HIV model that takes into account both mutation and treatment with enzyme inhibitors. This model generalizes a number of other models that have been extensively used to describe the HIV dynamics. Since the free virus dynamics occur on a much faster time-scale than cell dynamics, the model has two intrinsic time scales and should be viewed as a singularly perturbed system. Using Tikhonov's theorem we prove that the model can be approximated by a lower dimensional nonlinear model. Furthermore, we show that this reduced system is globally asymptotically stable by using Lyapunov's stability theory.
    Mathematics Subject Classification: Primary: 92C50; Secondary: 34E13, 92C60.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    R. M. Anderson and R. M. May, Epidemiological parameters of HIV transmission, Nature, 333 (1988), 514-519.

    [2]

    B. Asquith and C. R. M. Bangham, Review. An introduction to lymphocyte and viral dynamics: The power and limitations of mathematical analysis, Proceedings of the Royal Society of London. Series B: Biological Sciences, 270 (2003), 1651-1657.doi: 10.1098/rspb.2003.2386.

    [3]

    J. Banasiak, E. K. Phongi and M. Lachowicz, A singularly perturbed sis model with age structure, Mathematical Biosciences and Engineering, 10 (2013), 499-521.doi: 10.3934/mbe.2013.10.499.

    [4]

    N. Bobko, Estabilidade de Lyapunov e propriedades globais para modelos de dinâmica viral, (2010).

    [5]

    S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proceedings of the National Academy of Sciences, 94 (1997), 6971-6976.doi: 10.1073/pnas.94.13.6971.

    [6]

    L. N. Cooper, Theory of an immune system retrovirus, Proceedings of the National Academy of Sciences, 83 (1986), 9159-9163.doi: 10.1073/pnas.83.23.9159.

    [7]

    M. L. B. M. F. E. S. Sumikawa, L. R. da Motta and O. da C. F. Junior, Manual Técnico Para O Diagnóstico da Infecção Pelo HIV, Ministério da Saúde, 2013.

    [8]

    N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.doi: 10.1016/0022-0396(79)90152-9.

    [9]

    S. D. W. Frost and A. R. McLean, Germinal centre destruction as a major pathway of HIV pathogenesis, JAIDS Journal of Acquired Immune Deficiency Syndromes, 7 (1994), 236-244.

    [10]

    J. B. Gilmore, A. D. Kelleher, D. A. Cooper and J. M. Murray, Explaining the determinants of first phase HIV decay dynamics through the effects of stage-dependent drug action, PLoS computational biology, 9 (2013), e1002971, 12 pp.doi: 10.1371/journal.pcbi.1002971.

    [11]

    N. Gulzar and K. F. T. Copeland, Cd8+ T-cells: Function and response to HIV infection, Current HIV research, 2 (2004), 23-37.doi: 10.2174/1570162043485077.

    [12]

    D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, et al., Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.doi: 10.1038/373123a0.

    [13]

    J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.doi: 10.1017/CBO9781139173179.

    [14]

    J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, vol. 114, Springer New York, 1996.doi: 10.1007/978-1-4612-3968-0.

    [15]

    D. Kirschner, Using mathematics to understand HIV immune dynamics, AMS notices, 43 (1996), 191-202.

    [16]

    A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66 (2004), 879-883.doi: 10.1016/j.bulm.2004.02.001.

    [17]

    A. L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: Limitations of studies of viral load data, Proceedings of the Royal Society of London. Series B: Biological Sciences, 268 (2001), 847-854.doi: 10.1098/rspb.2000.1572.

    [18]

    J. M. McCune, M. B. Hanley, D. Cesar, R. Halvorsen, R. Hoh, D. Schmidt, E. Wieder, S. Deeks, S. Siler and R. Neese, et al., Factors influencing T-cell turnover in HIV-1 seropositive patients, Journal of Clinical Investigation, 105 (2000), R1-R8.doi: 10.1172/JCI8647.

    [19]

    M. L. Munier and A. D. Kelleher, Acutely dysregulated, chronically disabled by the enemy within: T-cell responses to HIV-1 infection, Immunology and cell biology, 85 (2006), 6-15.doi: 10.1038/sj.icb.7100015.

    [20]

    P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Mathematical Biosciences, 179 (2002), 73-94.doi: 10.1016/S0025-5564(02)00099-8.

    [21]

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

    [22]

    M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.doi: 10.1126/science.272.5258.74.

    [23]

    Joint United Nations Programme on HIV/AIDS (UNAIDS), Global Report: Unaids Report on the Global AIDS Epidemic 2012, (2012).

    [24]

    S. A. Orszag and C. M. Bender, Advanced Mathematical Methods for Scientists and Engineers, Mac Graw Hill, 1978.

    [25]

    D. H. Pastore, A Dinamica do HIV no Sistema Imunológico na Presença de Mutação, Ph.D. thesis, IMPA, 2005.

    [26]

    A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Mathematical biosciences, 114 (1993), 81-125.doi: 10.1016/0025-5564(93)90043-A.

    [27]

    A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM review, 41 (1999), 3-44.doi: 10.1137/S0036144598335107.

    [28]

    T. C. Quinn, HIV viral load, The Hopkins HIV Report, 8 (1996), no. 3.

    [29]

    A. Rambaut, D. Posada, K. A. Crandall and E. C. Holmes, The causes and consequences of HIV evolution, Nature Reviews Genetics, 5 (2004), 52-61.doi: 10.1038/nrg1246.

    [30]

    D. L. Robertson, B. H. Hahn and P. M. Sharp, Recombination in AIDS viruses, Journal of molecular evolution, 40 (1995), 249-259.doi: 10.1007/BF00163230.

    [31]

    M. A. Sande and P. A. Volberding, et al., The Medical Management of AIDS, no. Ed. 4, WB Saunders, 1995.

    [32]

    N. Siewe, The Tikhonov Theorem in Multiscale Modelling: An Application to the SIRS Epidemic Model, Ph.D. thesis, AIMS, 2012.

    [33]

    V. Simon and D. D. Ho, HIV-1 dynamics in vivo: Implications for therapy, Nature Reviews Microbiology, 1 (2003), 181-190.doi: 10.1038/nrmicro772.

    [34]

    H. L. Smith and P. D. Leenheer, Virus dynamics: A global analysis, SIAM Journal on Applied Mathematics, 63 (2003), 1313-1327.doi: 10.1137/S0036139902406905.

    [35]

    M. Somasundaran and H. L. Robinson, Unexpectedly high levels of HIV-1 RNA and protein synthesis in a cytocidal infection, Science, 242 (1988), 1554-1557.doi: 10.1126/science.3201245.

    [36]

    M. O. Souza, Multiscale analysis for a vector-borne epidemic model, Journal of mathematical biology, 68 (2014), 1269-1293.doi: 10.1007/s00285-013-0666-6.

    [37]

    M. O Souza and J. P. Zubelli, Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625.doi: 10.1007/s11538-010-9543-2.

    [38]

    M. A Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, Journal of Theoretical Biology, 203 (2000), 285-301.doi: 10.1006/jtbi.2000.1076.

    [39]

    P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, Journal of differential equations, 92 (1991), 252-281.doi: 10.1016/0022-0396(91)90049-F.

    [40]

    A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, Differential Equations, Springer-Verlag Berlin, 1984.

    [41]

    A. B. Vasileva and V. F. Butuzov, Asimptoticheskie Metody v Teorii Singulyarnykh Vozmushchenij, Moskva: Vysshaya Shkola, 1990 (Russian).

    [42]

    X. Wang and X. Song, Global properties of a model of immune effector responses to viral infections, Advances in Complex Systems, 10 (2007), 495-503.doi: 10.1142/S0219525907001252.

    [43]

    W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover Publications, Inc., New York, 1987.

    [44]

    World Health Organization (WHO), Global Update on HIV Treatment 2013: Results, Impact and Opportunities, (2013).

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(62) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return