# American Institute of Mathematical Sciences

June  2014, 19(4): 1087-1103. doi: 10.3934/dcdsb.2014.19.1087

## Complete classification of global dynamics of a virus model with immune responses

 1 Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

Received  April 2013 Revised  July 2013 Published  April 2014

A virus dynamics model for HIV or HBV is studied, which incorporates saturation effects of immune responses and an intracellular time delay. With the aid of persistence theory and Liapunov method, it is shown that the global stability of the model is totally determined by the reproductive numbers for viral infection, for CTL immune response, for antibody immune response, for antibody invasion and for CTL immune invasion. The results preclude the complicated behaviors such as the backward bifurcations and Hopf bifurcations which may be induced by saturation factors and a time delay.
Citation: Cuicui Jiang, Wendi Wang. Complete classification of global dynamics of a virus model with immune responses. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1087-1103. doi: 10.3934/dcdsb.2014.19.1087
##### References:
 [1] L. Cai and X. Li, Stability and Hopf bifurcation in a delayed model for HIV infection of $CD4^{+}T$ cells, Chaos, Solitons and Fractals, 42 (2009), 1-11. doi: 10.1016/j.chaos.2008.04.048. [2] R. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^{+}T$ cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [3] R. De Boer, Which of our modeling predictions are robust? Plos Computational Biology, 8 (2012), e10002593. doi: 10.1371/journal.pcbi.1002593. [4] O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [5] T. Gao, W. Wang and X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Mathematics and Computers in Simulation, 82 (2011), 653-665. doi: 10.1016/j.matcom.2011.10.007. [6] J. K. Hale and S. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [7] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [8] G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japanese J. Indust. Appl. Math., 28 (2011), 383-411. doi: 10.1007/s13160-011-0045-x. [9] S. Iwami, T. Miura, S. Nakaoka and Y.Takeuchi, Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds, J. Theor. Biol., 260 (2009), 490-501. doi: 10.1016/j.jtbi.2009.06.023. [10] S. Iwami, S. Nakaoka and Y. Takeuchi, Viral diversity limits immune diversity in asymptomatic phase of HIV infection, Theoretical Population Biology, 73 (2008), 332-341. doi: 10.1016/j.tpb.2008.01.003. [11] T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis: Real World Applications, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011. [12] T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci. USA, 95 (1998), 11514-11519. doi: 10.1073/pnas.95.20.11514. [13] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [14] D. Li and W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [15] J. Li, Y. Xiao , F. Zhang and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Analysis: Real World Applications, 13 (2012), 2006-2016. doi: 10.1016/j.nonrwa.2011.12.022. [16] J. Li, Y. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Analysis: Real World Applications, 12 (2011), 2163-2173. doi: 10.1016/j.nonrwa.2010.12.030. [17] M. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of Mathematical Biology, 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [18] M. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of $CD4^{+}T$ cells with delayed CTL response, Nonlinear Analysis: Real World Applications, 13 (2012), 1080-1092. doi: 10.1016/j.nonrwa.2011.02.026. [19] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. [20] P. Nelson, J. Mittler and A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, Journal of Aids, 26 (2001), 405-412. doi: 10.1097/00126334-200104150-00002. [21] M. Nowak and R. May, Viral Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [22] M. Nowak, S. Bonhoeffer, A. Hill, R. Boehme, H. Thomas and H. Mcdade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398. [23] A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [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), 1582-1586. doi: 10.1126/science.271.5255.1582. [25] H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796-799. [26] 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, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [27] T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamic model for HIV-1 therapy, Math. Biosci., 185 (2003), 191-203. doi: 10.1016/S0025-5564(03)00091-9. [28] B. Reddy and J. Yin, Quantitative intracellular kinetics of HIV type 1, AIDS Res. Hum. Retrovir., 15 (1999), 273-283. doi: 10.1089/088922299311457. [29] L. Rong and A. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. [30] L. Rong, M. Gilchristb, Z. Feng and A. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, Journal of Theoretical Biology, 4 (2007), 804-818. doi: 10.1016/j.jtbi.2007.04.014. [31] L. Rong, Z. Feng and A. 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. [32] H. Shu and L. Wang, Role of CD4+T-cell proliferation in HIV infection under antiretroviral therapy, J. Math. Anal. Appl., 394 (2012), 529-544. doi: 10.1016/j.jmaa.2012.05.027. [33] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [34] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [35] K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020. [36] K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response, Chaos Solutions Fractals, 28 (2006), 90-99. doi: 10.1016/j.chaos.2005.05.003. [37] K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208. doi: 10.1016/j.physd.2006.12.001. [38] L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of $CD4^{+}$ T cells, Mathematical Biosciences, 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026. [39] X. Wang and W. Wang, An HIV infection model based on a vectored immunoprophylaxis experiment, Journal of Theoretical Biology, 313 (2012), 127-135. doi: 10.1016/j.jtbi.2012.08.023. [40] X. Wang, W. Wang and P. Liu, Global properties of an HIV dynamic model with latent infection and CTL immune responses, Journal of Southwest university (Nature Science Edition), 7 (2013), 1-7. [41] Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences, 219 (2009), 104-112. doi: 10.1016/j.mbs.2009.03.003. [42] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0. [43] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune response and delay, Discrete and continutious dynamical systems series B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401. [44] Y. Yang and Y. Xiao, Threshold dynamics for an HIV model in periodic environments, J. Math. Anal. Appl., 361 (2010), 59-68. doi: 10.1016/j.jmaa.2009.09.012. [45] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [46] X. Zhou, X. Song and X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Applied Mathematics and Computation, 199 (2008), 23-38. doi: 10.1016/j.amc.2007.09.030. [47] H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Computers and Mathematics with Applications, 62 (2011), 3091-3102. doi: 10.1016/j.camwa.2011.08.022.

show all references

##### References:
 [1] L. Cai and X. Li, Stability and Hopf bifurcation in a delayed model for HIV infection of $CD4^{+}T$ cells, Chaos, Solitons and Fractals, 42 (2009), 1-11. doi: 10.1016/j.chaos.2008.04.048. [2] R. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^{+}T$ cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [3] R. De Boer, Which of our modeling predictions are robust? Plos Computational Biology, 8 (2012), e10002593. doi: 10.1371/journal.pcbi.1002593. [4] O. Diekmann, J. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [5] T. Gao, W. Wang and X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Mathematics and Computers in Simulation, 82 (2011), 653-665. doi: 10.1016/j.matcom.2011.10.007. [6] J. K. Hale and S. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [7] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025. [8] G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japanese J. Indust. Appl. Math., 28 (2011), 383-411. doi: 10.1007/s13160-011-0045-x. [9] S. Iwami, T. Miura, S. Nakaoka and Y.Takeuchi, Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds, J. Theor. Biol., 260 (2009), 490-501. doi: 10.1016/j.jtbi.2009.06.023. [10] S. Iwami, S. Nakaoka and Y. Takeuchi, Viral diversity limits immune diversity in asymptomatic phase of HIV infection, Theoretical Population Biology, 73 (2008), 332-341. doi: 10.1016/j.tpb.2008.01.003. [11] T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis: Real World Applications, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011. [12] T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci. USA, 95 (1998), 11514-11519. doi: 10.1073/pnas.95.20.11514. [13] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [14] D. Li and W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [15] J. Li, Y. Xiao , F. Zhang and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Analysis: Real World Applications, 13 (2012), 2006-2016. doi: 10.1016/j.nonrwa.2011.12.022. [16] J. Li, Y. Yang and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Analysis: Real World Applications, 12 (2011), 2163-2173. doi: 10.1016/j.nonrwa.2010.12.030. [17] M. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bulletin of Mathematical Biology, 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [18] M. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of $CD4^{+}T$ cells with delayed CTL response, Nonlinear Analysis: Real World Applications, 13 (2012), 1080-1092. doi: 10.1016/j.nonrwa.2011.02.026. [19] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. [20] P. Nelson, J. Mittler and A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, Journal of Aids, 26 (2001), 405-412. doi: 10.1097/00126334-200104150-00002. [21] M. Nowak and R. May, Viral Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [22] M. Nowak, S. Bonhoeffer, A. Hill, R. Boehme, H. Thomas and H. Mcdade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398. [23] A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [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), 1582-1586. doi: 10.1126/science.271.5255.1582. [25] H. Pang, W. Wang and K. Wang, Global properties of virus dynamics model with immune response, Journal of Southwest China Normal University (Natural Science), 30 (2005), 796-799. [26] 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, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [27] T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamic model for HIV-1 therapy, Math. Biosci., 185 (2003), 191-203. doi: 10.1016/S0025-5564(03)00091-9. [28] B. Reddy and J. Yin, Quantitative intracellular kinetics of HIV type 1, AIDS Res. Hum. Retrovir., 15 (1999), 273-283. doi: 10.1089/088922299311457. [29] L. Rong and A. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. [30] L. Rong, M. Gilchristb, Z. Feng and A. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, Journal of Theoretical Biology, 4 (2007), 804-818. doi: 10.1016/j.jtbi.2007.04.014. [31] L. Rong, Z. Feng and A. 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. [32] H. Shu and L. Wang, Role of CD4+T-cell proliferation in HIV infection under antiretroviral therapy, J. Math. Anal. Appl., 394 (2012), 529-544. doi: 10.1016/j.jmaa.2012.05.027. [33] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026. [34] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [35] K. Wang, W. Wang and X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses, Comput. Math. Appl., 51 (2006), 1593-1610. doi: 10.1016/j.camwa.2005.07.020. [36] K. Wang, W. Wang and X. Liu, Viral infection model with periodic lytic immune response, Chaos Solutions Fractals, 28 (2006), 90-99. doi: 10.1016/j.chaos.2005.05.003. [37] K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208. doi: 10.1016/j.physd.2006.12.001. [38] L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of $CD4^{+}$ T cells, Mathematical Biosciences, 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026. [39] X. Wang and W. Wang, An HIV infection model based on a vectored immunoprophylaxis experiment, Journal of Theoretical Biology, 313 (2012), 127-135. doi: 10.1016/j.jtbi.2012.08.023. [40] X. Wang, W. Wang and P. Liu, Global properties of an HIV dynamic model with latent infection and CTL immune responses, Journal of Southwest university (Nature Science Edition), 7 (2013), 1-7. [41] Y. Wang, Y. Zhou, J. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences, 219 (2009), 104-112. doi: 10.1016/j.mbs.2009.03.003. [42] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0. [43] Y. Yan and W. Wang, Global stability of a five-dimensional model with immune response and delay, Discrete and continutious dynamical systems series B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401. [44] Y. Yang and Y. Xiao, Threshold dynamics for an HIV model in periodic environments, J. Math. Anal. Appl., 361 (2010), 59-68. doi: 10.1016/j.jmaa.2009.09.012. [45] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [46] X. Zhou, X. Song and X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Applied Mathematics and Computation, 199 (2008), 23-38. doi: 10.1016/j.amc.2007.09.030. [47] H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Computers and Mathematics with Applications, 62 (2011), 3091-3102. doi: 10.1016/j.camwa.2011.08.022.
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