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August  2014, 19(6): 1783-1800. doi: 10.3934/dcdsb.2014.19.1783

Permanence and extinction of a non-autonomous HIV-1 model with time delays

1. 

School of Mathematics and Information Sciences, Shaanxi Normal University, Xi'an, 710062, China

2. 

Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

3. 

Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, United States

Received  May 2013 Revised  November 2013 Published  June 2014

The environment of HIV-1 infection and treatment could be non-periodically time-varying. The purposes of this paper are to investigate the effects of time-dependent coefficients on the dynamics of a non-autonomous and non-periodic HIV-1 infection model with two delays, and to provide explicit estimates of the lower and upper bounds of the viral load. We established sufficient conditions for the permanence and extinction of the non-autonomous system based on two positive constants $R^{\ast}$ and $R_{\ast}$ ($R^{\ast}\geq R_{\ast}$) that could be precisely expressed by the coefficients of the system: (i) If $R^{\ast}<1$, then the infection-free steady state is globally attracting; (ii) if $R_{\ast}>1$, then the system is permanent. When the system is permanent, we further obtained detailed estimates of both the lower and upper bounds of the viral load. The results show that both $R^{\ast}$ and $R_{\ast}$ reduce to the basic reproduction ratio of the corresponding autonomous model when all the coefficients become constants. Numerical simulations have been performed to verify/extend our analytical results. We also provided some numerical results showing that both permanence and extinction are possible when $R_{\ast }< 1 < R^{\ast}$ holds.
Citation: Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783
References:
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N. Grassly and C. Fraser, Seasonal infectious disease epidemiology,, Proceedings of the Royal Society B: Biological Sciences, 273 (2006), 2541. doi: 10.1098/rspb.2006.3604. Google Scholar

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A. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, Proceedings of the National Academy of Sciences, 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar

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H. Hethcote, Asymptotic behavior in a deterministic epidemic model,, Bulletin of Mathematical Biology, 36 (1973), 607. Google Scholar

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D. Ho and Y. Huang, The HIV-1 vaccine race,, Cell, 110 (2002), 135. doi: 10.1016/S0092-8674(02)00832-2. Google Scholar

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Y. Huang, S. Rosenkranz and H. Wu, Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity,, Mathematical Biosciences, 184 (2003), 165. doi: 10.1016/S0025-5564(03)00058-0. Google Scholar

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S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Mathematical biosciences and engineering, 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. Google Scholar

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Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model,, Discrete and Continuous Dynamical Systems. Series B, 12 (2009), 169. doi: 10.3934/dcdsb.2009.12.169. Google Scholar

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Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population,, SIAM Journal on Applied Mathematics, 70 (2010), 2023. doi: 10.1137/080744438. Google Scholar

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J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects,, Journal of Mathematical Biology, 65 (2012), 623. doi: 10.1007/s00285-011-0474-9. Google Scholar

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Z. Ma, M. Stone, M. Piatak, B. Schweighardt, N. L. Haigwood, D. Montefiori, J. D. Lifson, M. Busch and C. J. Miller, High specific infectivity of plasma virus from the pre-ramp and ramp up stages of acute simian immunodeficiency virus infection,, Journal of Virology, 83 (2009), 3288. doi: 10.1128/JVI.02423-08. Google Scholar

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M. Martcheva, A non-autonomous multi-strain SIS epidemic model,, Journal of Biological Dynamics, 3 (2009), 235. doi: 10.1080/17513750802638712. Google Scholar

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P. Nelson, J. Murray and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Mathematical Biosciences, 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

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P. Nelson, J. Mittler and A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV-1 viral dynamic parameters,, Journal of Acquired Immune Deficiency Syndromes, 26 (2001), 405. Google Scholar

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P. Nelson and A. 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

[20]

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

[21]

K. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Mathematical Biosciences,, Mathematical Biosciences, 235 (2012), 98. doi: 10.1016/j.mbs.2011.11.002. Google Scholar

[22]

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

[23]

A. Perelson, D. 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

[24]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582. Google Scholar

[25]

L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment,, Bulletin of Mathematical Biology, 69 (2007), 2027. doi: 10.1007/s11538-007-9203-3. Google Scholar

[26]

L. Rong, Z. Feng and A. Perelson, Mathematical Analysis of Age-Structured HIV-1 Dynamics with Combination Antiretroviral Therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731. doi: 10.1137/060663945. Google Scholar

[27]

G. Samanta, Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay,, Nonlinear Analysis: Real World Applications, 12 (2011), 1163. doi: 10.1016/j.nonrwa.2010.09.010. Google Scholar

[28]

H. L. Smith, Multiple stable subharmonics for a periodic epidemic model,, Journal of Mathematical Biology, 17 (1983), 179. doi: 10.1007/BF00305758. Google Scholar

[29]

H. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proceedings of the American Mathematical Society, 127 (1999), 2395. doi: 10.1090/S0002-9939-99-05034-0. Google Scholar

[30]

H. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Mathematical Biosciences, 166 (2000), 173. doi: 10.1016/S0025-5564(00)00018-3. Google Scholar

[31]

N. Vaidya, R. Ribeiro, C. Miller and A. Perelson, Viral dynamics during primary simian immunodeficiency virus infection: effect of time-dependent virus infectivity,, Journal of Virology, 84 (2010), 4302. doi: 10.1128/JVI.02284-09. Google Scholar

[32]

K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons & Fractals, 28 (2006), 90. doi: 10.1016/j.chaos.2005.05.003. Google Scholar

[33]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, Journal of Dynamics and Differential Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar

[34]

X. Wang and Y. Tao, Lyapunov function and global properties of virus dynamics with CTL immune response,, International Journal of Biomathematics, 1 (2008), 443. doi: 10.1142/S1793524508000382. Google Scholar

[35]

Y. Yang, Y. Xiao and J. Wu, Pulse HIV vaccination: Feasibility for virus eradication and optimal vaccination schedule,, Bulletin of Mathematical Biology, 75 (2013), 725. doi: 10.1007/s11538-013-9831-8. Google Scholar

[36]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology,, Bulletin of Mathematical Biology, 69 (2007), 2537. doi: 10.1007/s11538-007-9231-z. Google Scholar

show all references

References:
[1]

C. Browne and S. Pilyugin, Periodic multidrug therapy in a within-host virus model,, Bulletin of Mathematical Biology, 74 (2012), 562. doi: 10.1007/s11538-011-9677-x. Google Scholar

[2]

K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctuations,, in Mathematical Models in Medicine, 11 (1976), 1. doi: 10.1007/978-3-642-93048-5_1. Google Scholar

[3]

S. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases,, Emerging Infectious Diseases, 7 (2001), 369. Google Scholar

[4]

P. De Leenheer, Within-host virus models with periodic antiviral therapy,, Bulletin of Mathematical Biology, 71 (2009), 189. doi: 10.1007/s11538-008-9359-5. Google Scholar

[5]

N. Grassly and C. Fraser, Seasonal infectious disease epidemiology,, Proceedings of the Royal Society B: Biological Sciences, 273 (2006), 2541. doi: 10.1098/rspb.2006.3604. Google Scholar

[6]

A. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, Proceedings of the National Academy of Sciences, 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar

[7]

H. Hethcote, Asymptotic behavior in a deterministic epidemic model,, Bulletin of Mathematical Biology, 36 (1973), 607. Google Scholar

[8]

D. Ho and Y. Huang, The HIV-1 vaccine race,, Cell, 110 (2002), 135. doi: 10.1016/S0092-8674(02)00832-2. Google Scholar

[9]

Y. Huang, S. Rosenkranz and H. Wu, Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity,, Mathematical Biosciences, 184 (2003), 165. doi: 10.1016/S0025-5564(03)00058-0. Google Scholar

[10]

T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance,, Proceedings of the National Academy of Sciences, 95 (1998), 11514. doi: 10.1073/pnas.95.20.11514. Google Scholar

[11]

S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Mathematical biosciences and engineering, 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. Google Scholar

[12]

Y. Lou and X.-Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model,, Discrete and Continuous Dynamical Systems. Series B, 12 (2009), 169. doi: 10.3934/dcdsb.2009.12.169. Google Scholar

[13]

Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population,, SIAM Journal on Applied Mathematics, 70 (2010), 2023. doi: 10.1137/080744438. Google Scholar

[14]

J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects,, Journal of Mathematical Biology, 65 (2012), 623. doi: 10.1007/s00285-011-0474-9. Google Scholar

[15]

Z. Ma, M. Stone, M. Piatak, B. Schweighardt, N. L. Haigwood, D. Montefiori, J. D. Lifson, M. Busch and C. J. Miller, High specific infectivity of plasma virus from the pre-ramp and ramp up stages of acute simian immunodeficiency virus infection,, Journal of Virology, 83 (2009), 3288. doi: 10.1128/JVI.02423-08. Google Scholar

[16]

M. Martcheva, A non-autonomous multi-strain SIS epidemic model,, Journal of Biological Dynamics, 3 (2009), 235. doi: 10.1080/17513750802638712. Google Scholar

[17]

P. Nelson, J. Murray and A. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Mathematical Biosciences, 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

[18]

P. Nelson, J. Mittler and A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV-1 viral dynamic parameters,, Journal of Acquired Immune Deficiency Syndromes, 26 (2001), 405. Google Scholar

[19]

P. Nelson and A. 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

[20]

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

[21]

K. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Mathematical Biosciences,, Mathematical Biosciences, 235 (2012), 98. doi: 10.1016/j.mbs.2011.11.002. Google Scholar

[22]

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

[23]

A. Perelson, D. 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

[24]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582. Google Scholar

[25]

L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment,, Bulletin of Mathematical Biology, 69 (2007), 2027. doi: 10.1007/s11538-007-9203-3. Google Scholar

[26]

L. Rong, Z. Feng and A. Perelson, Mathematical Analysis of Age-Structured HIV-1 Dynamics with Combination Antiretroviral Therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731. doi: 10.1137/060663945. Google Scholar

[27]

G. Samanta, Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay,, Nonlinear Analysis: Real World Applications, 12 (2011), 1163. doi: 10.1016/j.nonrwa.2010.09.010. Google Scholar

[28]

H. L. Smith, Multiple stable subharmonics for a periodic epidemic model,, Journal of Mathematical Biology, 17 (1983), 179. doi: 10.1007/BF00305758. Google Scholar

[29]

H. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proceedings of the American Mathematical Society, 127 (1999), 2395. doi: 10.1090/S0002-9939-99-05034-0. Google Scholar

[30]

H. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Mathematical Biosciences, 166 (2000), 173. doi: 10.1016/S0025-5564(00)00018-3. Google Scholar

[31]

N. Vaidya, R. Ribeiro, C. Miller and A. Perelson, Viral dynamics during primary simian immunodeficiency virus infection: effect of time-dependent virus infectivity,, Journal of Virology, 84 (2010), 4302. doi: 10.1128/JVI.02284-09. Google Scholar

[32]

K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response,, Chaos Solitons & Fractals, 28 (2006), 90. doi: 10.1016/j.chaos.2005.05.003. Google Scholar

[33]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, Journal of Dynamics and Differential Equations, 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar

[34]

X. Wang and Y. Tao, Lyapunov function and global properties of virus dynamics with CTL immune response,, International Journal of Biomathematics, 1 (2008), 443. doi: 10.1142/S1793524508000382. Google Scholar

[35]

Y. Yang, Y. Xiao and J. Wu, Pulse HIV vaccination: Feasibility for virus eradication and optimal vaccination schedule,, Bulletin of Mathematical Biology, 75 (2013), 725. doi: 10.1007/s11538-013-9831-8. Google Scholar

[36]

T. Zhang and Z. Teng, On a nonautonomous SEIRS model in epidemiology,, Bulletin of Mathematical Biology, 69 (2007), 2537. doi: 10.1007/s11538-007-9231-z. Google Scholar

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