2015, 12(1): 1-21. doi: 10.3934/mbe.2015.12.1

A singularly perturbed HIV model with treatment and antigenic variation

1. 

Instituto Nacional de Matemática Pura e Aplicada, Rio do Janeiro, RJ 22460-320, Brazil, Brazil

Received  July 2014 Revised  October 2014 Published  December 2014

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.
Citation: Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1
References:
[1]

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

[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. doi: 10.1098/rspb.2003.2386. Google Scholar

[3]

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

[4]

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

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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. doi: 10.1073/pnas.94.13.6971. Google Scholar

[6]

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

[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). Google Scholar

[8]

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

[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. Google Scholar

[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). doi: 10.1371/journal.pcbi.1002971. Google Scholar

[11]

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

[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. doi: 10.1038/373123a0. Google Scholar

[13]

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

[14]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods,, vol. 114, (1996). doi: 10.1007/978-1-4612-3968-0. Google Scholar

[15]

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

[16]

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

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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. doi: 10.1098/rspb.2000.1572. Google Scholar

[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). doi: 10.1172/JCI8647. Google Scholar

[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. doi: 10.1038/sj.icb.7100015. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

D. H. Pastore, A Dinamica do HIV no Sistema Imunológico na Presença de Mutação,, Ph.D. thesis, (2005). Google Scholar

[26]

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

[27]

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

[28]

T. C. Quinn, HIV viral load,, The Hopkins HIV Report, 8 (1996). Google Scholar

[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. doi: 10.1038/nrg1246. Google Scholar

[30]

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

[31]

M. A. Sande and P. A. Volberding, et al., The Medical Management of AIDS,, no. Ed. 4, (1995). Google Scholar

[32]

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

[33]

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

[34]

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

[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. doi: 10.1126/science.3201245. Google Scholar

[36]

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

[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. doi: 10.1007/s11538-010-9543-2. Google Scholar

[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. doi: 10.1006/jtbi.2000.1076. Google Scholar

[39]

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

[40]

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

[41]

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

[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. doi: 10.1142/S0219525907001252. Google Scholar

[43]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations,, Dover Publications, (1987). Google Scholar

[44]

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

show all references

References:
[1]

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

[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. doi: 10.1098/rspb.2003.2386. Google Scholar

[3]

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

[4]

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

[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. doi: 10.1073/pnas.94.13.6971. Google Scholar

[6]

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

[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). Google Scholar

[8]

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

[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. Google Scholar

[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). doi: 10.1371/journal.pcbi.1002971. Google Scholar

[11]

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

[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. doi: 10.1038/373123a0. Google Scholar

[13]

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

[14]

J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods,, vol. 114, (1996). doi: 10.1007/978-1-4612-3968-0. Google Scholar

[15]

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

[16]

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

[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. doi: 10.1098/rspb.2000.1572. Google Scholar

[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). doi: 10.1172/JCI8647. Google Scholar

[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. doi: 10.1038/sj.icb.7100015. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

D. H. Pastore, A Dinamica do HIV no Sistema Imunológico na Presença de Mutação,, Ph.D. thesis, (2005). Google Scholar

[26]

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

[27]

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

[28]

T. C. Quinn, HIV viral load,, The Hopkins HIV Report, 8 (1996). Google Scholar

[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. doi: 10.1038/nrg1246. Google Scholar

[30]

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

[31]

M. A. Sande and P. A. Volberding, et al., The Medical Management of AIDS,, no. Ed. 4, (1995). Google Scholar

[32]

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

[33]

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

[34]

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

[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. doi: 10.1126/science.3201245. Google Scholar

[36]

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

[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. doi: 10.1007/s11538-010-9543-2. Google Scholar

[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. doi: 10.1006/jtbi.2000.1076. Google Scholar

[39]

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

[40]

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

[41]

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

[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. doi: 10.1142/S0219525907001252. Google Scholar

[43]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations,, Dover Publications, (1987). Google Scholar

[44]

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

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