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2016, 13(1): 135-157. doi: 10.3934/mbe.2016.13.135

## A delayed HIV-1 model with virus waning term

 1 Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada 3 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080

Received  April 2015 Revised  July 2015 Published  October 2015

In this paper, we propose and analyze a delayed HIV-1 model with CTL immune response and virus waning. The two discrete delays stand for the time for infected cells to produce viruses after viral entry and for the time for CD$8^+$ T cell immune response to emerge to control viral replication. We obtain the positiveness and boundedness of solutions and find the basic reproduction number $R_0$. If $R_0<1$, then the infection-free steady state is globally asymptotically stable and the infection is cleared from the T-cell population; whereas if $R_0>1$, then the system is uniformly persistent and the viral concentration maintains at some constant level. The global dynamics when $R_0>1$ is complicated. We establish the local stability of the infected steady state and show that Hopf bifurcation can occur. Both analytical and numerical results indicate that if, in the initial infection stage, the effect of delays on HIV-1 infection is ignored, then the risk of HIV-1 infection (if persists) will be underestimated. Moreover, the viral load differs from that without virus waning. These results highlight the important role of delays and virus waning on HIV-1 infection.
Citation: Bing Li, Yuming Chen, Xuejuan Lu, Shengqiang Liu. A delayed HIV-1 model with virus waning term. Mathematical Biosciences & Engineering, 2016, 13 (1) : 135-157. doi: 10.3934/mbe.2016.13.135
##### References:
 [1] , Possible transfusion-associated acquired immune deficiency syndrome (AIDS)- California,, MMWR Morb Mortal Wkly Rep, 31 (1982), 652. [2] , Unexplained immunodeficiency and opportunistic infections infants-New York,, New Jersey, 31 (1982), 665. [3] , Immunodeficiency among female sexual partners males with acquired immune deficiency syndrome(AIDS)-New York,, MMWR Morb Mortal Wkly Rep, 31 (1983), 697. [4] D. Burg, L. Rong, A. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection, J. Theor. Biol., 259 (2009), 751-759. doi: 10.1016/j.jtbi.2009.04.010. [5] M. Ciupe, B. Bivort, D. Bortz and P. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27. doi: 10.1016/j.mbs.2005.12.006. [6] V. R. Culshaw, S. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3. [7] P. de Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [8] A. S. Fauci, HIV and AIDS: 20 years of science, Nature Medicine, 9 (2003), 839-843. doi: 10.1038/nm0703-839. [9] R. C. Gallo, Historical essay. The early years of HIV/AIDS, Science, 298 (2002), 1728-1730. doi: 10.1126/science.1078050. [10] J. K. Hale and S. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] J. M. Heffernan and L. M. Wahl, Monte carlo estimates of natural variation in HIV infection, J. Theor. Biol., 236 (2005), 137-153. doi: 10.1016/j.jtbi.2005.03.002. [12] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [13] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals of delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [14] A. Kent and M. D. Sepkowitz, AIDS-The First 20 Years, The New England Journal of Medicine, 344 (2001), 1764-1772. [15] B. Li and S. Liu, A delayed HIV-1 model with multiple target cells and general nonlinear incidence rate, Journal of Biological Systems, 21 (2013), 1340012, 20pp. doi: 10.1142/S0218339013400123. [16] M. Y. Li and H. Shu, Impact of intercellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322. [17] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intercellular delays and a combination therapy, Mathematical Biosciences and Engineering, 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [18] X. Lu, L. Hui, S. Liu and J. Li, A Mathematical model of HTLV-1 infection with two time delays, Mathematical Biosciences and Engineering, 12 (2015), 431-449. doi: 10.3934/mbe.2015.12.431. [19] L. Montagnier, Historical essay. A history of HIV discovery, Science, 298 (2002), 1727-1728. doi: 10.1126/science.1079027. [20] M. A. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74. [21] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [22] K. A. 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, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [23] J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: Continuous versus burst productionofvirions, PLoS Comput Biol., 7 (2011), e1001058, 17 pp. doi: 10.1371/journal.pcbi.1001058. [24] 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. doi: 10.1126/science.271.5255.1582. [25] R. Ribeiro, L. Qin, L. Chavez, D. Li, S. Self and A. Perelson, Estimation of the Initial Viral Growth Rate and Basic Reproductive Number during Acute HIV-1 Infection, Journal of Virology, 84 (2010), 6096-6102. doi: 10.1128/JVI.00127-10. [26] L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology, 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3. [27] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302. doi: 10.1137/120896463. [28] H. L. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [29] H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response, Mathematical biosciences and engineering, 12 (2015), 185-208. doi: 10.3934/mbe.2015.12.xx. [30] 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, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. [31] X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Mathematical Methods in the Applied Science, 36 (2013), 125-142. doi: 10.1002/mma.2576. [32] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3. [33] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D, 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3. [34] X. Zhao, Dynamical Systems in Population Biology, Springer, Berlin, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

##### References:
 [1] , Possible transfusion-associated acquired immune deficiency syndrome (AIDS)- California,, MMWR Morb Mortal Wkly Rep, 31 (1982), 652. [2] , Unexplained immunodeficiency and opportunistic infections infants-New York,, New Jersey, 31 (1982), 665. [3] , Immunodeficiency among female sexual partners males with acquired immune deficiency syndrome(AIDS)-New York,, MMWR Morb Mortal Wkly Rep, 31 (1983), 697. [4] D. Burg, L. Rong, A. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection, J. Theor. Biol., 259 (2009), 751-759. doi: 10.1016/j.jtbi.2009.04.010. [5] M. Ciupe, B. Bivort, D. Bortz and P. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27. doi: 10.1016/j.mbs.2005.12.006. [6] V. R. Culshaw, S. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562. doi: 10.1007/s00285-003-0245-3. [7] P. de Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [8] A. S. Fauci, HIV and AIDS: 20 years of science, Nature Medicine, 9 (2003), 839-843. doi: 10.1038/nm0703-839. [9] R. C. Gallo, Historical essay. The early years of HIV/AIDS, Science, 298 (2002), 1728-1730. doi: 10.1126/science.1078050. [10] J. K. Hale and S. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] J. M. Heffernan and L. M. Wahl, Monte carlo estimates of natural variation in HIV infection, J. Theor. Biol., 236 (2005), 137-153. doi: 10.1016/j.jtbi.2005.03.002. [12] V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [13] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals of delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [14] A. Kent and M. D. Sepkowitz, AIDS-The First 20 Years, The New England Journal of Medicine, 344 (2001), 1764-1772. [15] B. Li and S. Liu, A delayed HIV-1 model with multiple target cells and general nonlinear incidence rate, Journal of Biological Systems, 21 (2013), 1340012, 20pp. doi: 10.1142/S0218339013400123. [16] M. Y. Li and H. Shu, Impact of intercellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322. [17] S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intercellular delays and a combination therapy, Mathematical Biosciences and Engineering, 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [18] X. Lu, L. Hui, S. Liu and J. Li, A Mathematical model of HTLV-1 infection with two time delays, Mathematical Biosciences and Engineering, 12 (2015), 431-449. doi: 10.3934/mbe.2015.12.431. [19] L. Montagnier, Historical essay. A history of HIV discovery, Science, 298 (2002), 1727-1728. doi: 10.1126/science.1079027. [20] M. A. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74. [21] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [22] K. A. 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, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [23] J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: Continuous versus burst productionofvirions, PLoS Comput Biol., 7 (2011), e1001058, 17 pp. doi: 10.1371/journal.pcbi.1001058. [24] 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. doi: 10.1126/science.271.5255.1582. [25] R. Ribeiro, L. Qin, L. Chavez, D. Li, S. Self and A. Perelson, Estimation of the Initial Viral Growth Rate and Basic Reproductive Number during Acute HIV-1 Infection, Journal of Virology, 84 (2010), 6096-6102. doi: 10.1128/JVI.00127-10. [26] L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology, 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3. [27] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302. doi: 10.1137/120896463. [28] H. L. Smith and X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2. [29] H. Song, W. Jiang and S. Liu, Virus dynamics model with intracellular delays and immune response, Mathematical biosciences and engineering, 12 (2015), 185-208. doi: 10.3934/mbe.2015.12.xx. [30] 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, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. [31] X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Mathematical Methods in the Applied Science, 36 (2013), 125-142. doi: 10.1002/mma.2576. [32] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3. [33] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D, 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3. [34] X. Zhao, Dynamical Systems in Population Biology, Springer, Berlin, 2003. doi: 10.1007/978-0-387-21761-1.
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