American Institute of Mathematical Sciences

Advanced
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

Cuicui Jiang 1, and  Wendi Wang 1,

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.
Keywords: saturation effect, time delay., viral dynamics, Global stability, Lyapunov functional.
Mathematics Subject Classification: Primary: 92D30; Secondary: 34K2.
Citation: Cuicui Jiang, Wendi Wang. Complete classification of global dynamics of a virus model with immune responses. Discrete & 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.  Google Scholar

[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.  Google Scholar

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R. De Boer, Which of our modeling predictions are robust? Plos Computational Biology, 8 (2012), e10002593. doi: 10.1371/journal.pcbi.1002593.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. K. Hale and S. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[7]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  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), 1582-1586. doi: 10.1126/science.271.5255.1582.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. De Boer, Which of our modeling predictions are robust? Plos Computational Biology, 8 (2012), e10002593. doi: 10.1371/journal.pcbi.1002593.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. K. Hale and S. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.  Google Scholar

[7]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

A. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  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), 1582-1586. doi: 10.1126/science.271.5255.1582.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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