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2017, 14(5-6): 1233-1246. doi: 10.3934/mbe.2017063

Global dynamics of a delay virus model with recruitment and saturation effects of immune responses

Department of Mathematics School of Biomedical Engineering, Third Military Medical University Chongqing 400038, China

* Corresponding authorr: Kaifa Wang

Received  July 27, 2016 Revised  October 2016 Published  May 2017

In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number $R_0$. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

Citation: Cuicui Jiang, Kaifa Wang, Lijuan Song. Global dynamics of a delay virus model with recruitment and saturation effects of immune responses. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1233-1246. doi: 10.3934/mbe.2017063
References:
[1]

I. Al-Darabsah, Y. Yuan, A time-delayed epidemic model for Ebola disease transmission, Appl. Math. Comput., 290 (2016), 307-325. doi: 10.1016/j.amc.2016.05.043.

[2]

G. Bocharov, B. Ludewig, A. Bertoletti, P. Klenerman, T. Junt, P. Krebs, T. Luzyanina, C. Fraser, R. Anderson, Underwhelming the immune response: Effect of slow virus growth on CD8+T lymphocytes responses, J. Virol., 78 (2004), 2247-2254. doi: 10.1128/JVI.78.5.2247-2254.2004.

[3]

S. Chen, C. Cheng, Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. doi: 10.1016/j.jmaa.2016.05.003.

[4]

R. Culshaw, 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.

[5]

R. De Boer, Which of our modeling predictions are robust? PLoS Comput. Biol. , 8 (2012), e10002593, 5pp.

[6]

O. Diekmann, J. Heesterbeek, 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.

[7]

T. Gao, W. Wang, X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Math. Comput. Simulation, 82 (2011), 653-665. doi: 10.1016/j.matcom.2011.10.007.

[8]

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

[9]

M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Parageorgopolous, E. Wieder, D. Schmidt, R. Hoh, R. Neese, D. Macallan, S. Deeks, J. M. McCune, Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans, Nat. Med., 5 (1999), 83-89. doi: 10.1038/4772.

[10]

M. Hirsh, H. Hanisch, P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pur. Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.

[11]

Y. Ji, L. Liu, Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 133-149. doi: 10.3934/dcdsb.2016.21.133.

[12]

C. Jiang, W. Wang, Complete classification of global dynamics of a virus model with immune responses, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 1087-1103. doi: 10.3934/dcdsb.2014.19.1087.

[13]

T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci. , USA, 95 (1998), 11514–11519.

[14]

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

[15]

J. Li, Y. Yang, Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl., 12 (2011), 2163-2173. doi: 10.1016/j.nonrwa.2010.12.030.

[16]

B. Li, Y. Chen, X. Lu, S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157. doi: 10.3934/mbe.2016.13.135.

[17]

J. Luo, W. Wang, H. Chen, R. Fu, Bifurcations of a mathematical model for HIV dynamics, J. Math. Anal. Appl., 434 (2016), 837-857. doi: 10.1016/j.jmaa.2015.09.048.

[18]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[19]

P. Nelson, J. Mittler, A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, J. AIDS, 26 (2001), 405-412. doi: 10.1097/00126334-200104150-00002.

[20]

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

[21]

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.

[22]

K. Pawelek, S. Liu, F. Pahlevani, L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002.

[23]

J. Pang, J. Cui, J. Hui, The importance of immune rsponses in a model of hepatitis B virus, Nonlinear Dyn., 67 (2012), 727-734. doi: 10.1007/s11071-011-0022-6.

[24]

H. Pang, W. Wang, K. Wang, Global properties of virus dynamics model with immune response, J. Southeast Univ. Nat. Sci., 30 (2005), 796-799.

[25]

A. Perelson, P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[26]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard, 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.

[27]

L. Rong, A. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011.

[28]

L. Rong, M. Gilchristb, Z. Feng, A. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, J. Theoret. Biol., 247 (2007), 804-818. doi: 10.1016/j.jtbi.2007.04.014.

[29]

H. Shu, 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.

[30]

H. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.

[31]

H. Smith, X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[32]

X. Song, S. Wang, J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J. Math. Anal. Appl., 373 (2011), 345-355. doi: 10.1016/j.jmaa.2010.04.010.

[33]

M. Stafford, L. Corey, Y. Cao, E. Daar, D. Ho, A. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theoret. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076.

[34]

P. van den Driessche, 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, 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, X. Liu, Viral infection model with periodic lytic immune response, Chaos Solitons Fractals, 28 (2006), 90-99. doi: 10.1016/j.chaos.2005.05.003.

[37]

X. Wang, W. Wang, An HIV infection model based on a vectored immunoprophylaxis experiment, J. Theoret. Biol., 313 (2012), 127-135. doi: 10.1016/j.jtbi.2012.08.023.

[38]

Y. Wang, Y. Zhou, J. Wu, J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104-112. doi: 10.1016/j.mbs.2009.03.003.

[39]

K. Wang, Y. Jin, A. Fan, The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 3379-3396. doi: 10.3934/dcdsb.2014.19.3379.

[40]

Z. Wang, R. Xu, Stability and Hopf bifurcation in a viral infection modelwith nonlinear incidence rate and delayed immune response, Commun Nonlinear Sci Numer Simulat., 17 (2012), 964-978. doi: 10.1016/j.cnsns.2011.06.024.

[41]

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.

[42]

Y. Yan, W. Wang, Global stability of a five-dimensional model with immune response and delay, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

[43]

Y. Yang, 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.

[44]

Y. Yang, L. Zou, S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. doi: 10.1016/j.mbs.2015.05.001.

[45]

Y. Yang, Y. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington -DeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930. doi: 10.1016/j.camwa.2016.01.009.

[46]

J. Zack, S. Arrigo, S. Weitsman, A. Go, A. Haislip, I. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viral structure, Cell, 61 (1990), 213-222. doi: 10.1016/0092-8674(90)90802-L.

[47]

X. Zhou, X. Song, X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput., 199 (2008), 23-38. doi: 10.1016/j.amc.2007.09.030.

show all references

References:
[1]

I. Al-Darabsah, Y. Yuan, A time-delayed epidemic model for Ebola disease transmission, Appl. Math. Comput., 290 (2016), 307-325. doi: 10.1016/j.amc.2016.05.043.

[2]

G. Bocharov, B. Ludewig, A. Bertoletti, P. Klenerman, T. Junt, P. Krebs, T. Luzyanina, C. Fraser, R. Anderson, Underwhelming the immune response: Effect of slow virus growth on CD8+T lymphocytes responses, J. Virol., 78 (2004), 2247-2254. doi: 10.1128/JVI.78.5.2247-2254.2004.

[3]

S. Chen, C. Cheng, Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. doi: 10.1016/j.jmaa.2016.05.003.

[4]

R. Culshaw, 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.

[5]

R. De Boer, Which of our modeling predictions are robust? PLoS Comput. Biol. , 8 (2012), e10002593, 5pp.

[6]

O. Diekmann, J. Heesterbeek, 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.

[7]

T. Gao, W. Wang, X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Math. Comput. Simulation, 82 (2011), 653-665. doi: 10.1016/j.matcom.2011.10.007.

[8]

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

[9]

M. Hellerstein, M. Hanley, D. Cesar, S. Siler, C. Parageorgopolous, E. Wieder, D. Schmidt, R. Hoh, R. Neese, D. Macallan, S. Deeks, J. M. McCune, Directly measured kinetics of circulating T lymphocytes in normal and HIV-1-infected humans, Nat. Med., 5 (1999), 83-89. doi: 10.1038/4772.

[10]

M. Hirsh, H. Hanisch, P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pur. Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.

[11]

Y. Ji, L. Liu, Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate, Discrete Contin. Dyn. Syst. Ser. B., 21 (2016), 133-149. doi: 10.3934/dcdsb.2016.21.133.

[12]

C. Jiang, W. Wang, Complete classification of global dynamics of a virus model with immune responses, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 1087-1103. doi: 10.3934/dcdsb.2014.19.1087.

[13]

T. Kepler and A. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci. , USA, 95 (1998), 11514–11519.

[14]

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

[15]

J. Li, Y. Yang, Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl., 12 (2011), 2163-2173. doi: 10.1016/j.nonrwa.2010.12.030.

[16]

B. Li, Y. Chen, X. Lu, S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157. doi: 10.3934/mbe.2016.13.135.

[17]

J. Luo, W. Wang, H. Chen, R. Fu, Bifurcations of a mathematical model for HIV dynamics, J. Math. Anal. Appl., 434 (2016), 837-857. doi: 10.1016/j.jmaa.2015.09.048.

[18]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[19]

P. Nelson, J. Mittler, A. Perelson, Effect of drug efficacy and the eclipse phase of the viral life cycle on the estimates of HIV viral dynamic parameters, J. AIDS, 26 (2001), 405-412. doi: 10.1097/00126334-200104150-00002.

[20]

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

[21]

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.

[22]

K. Pawelek, S. Liu, F. Pahlevani, L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002.

[23]

J. Pang, J. Cui, J. Hui, The importance of immune rsponses in a model of hepatitis B virus, Nonlinear Dyn., 67 (2012), 727-734. doi: 10.1007/s11071-011-0022-6.

[24]

H. Pang, W. Wang, K. Wang, Global properties of virus dynamics model with immune response, J. Southeast Univ. Nat. Sci., 30 (2005), 796-799.

[25]

A. Perelson, P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[26]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard, 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.

[27]

L. Rong, A. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011.

[28]

L. Rong, M. Gilchristb, Z. Feng, A. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, J. Theoret. Biol., 247 (2007), 804-818. doi: 10.1016/j.jtbi.2007.04.014.

[29]

H. Shu, 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.

[30]

H. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.

[31]

H. Smith, X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179. doi: 10.1016/S0362-546X(01)00678-2.

[32]

X. Song, S. Wang, J. Dong, Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, J. Math. Anal. Appl., 373 (2011), 345-355. doi: 10.1016/j.jmaa.2010.04.010.

[33]

M. Stafford, L. Corey, Y. Cao, E. Daar, D. Ho, A. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theoret. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076.

[34]

P. van den Driessche, 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, 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, X. Liu, Viral infection model with periodic lytic immune response, Chaos Solitons Fractals, 28 (2006), 90-99. doi: 10.1016/j.chaos.2005.05.003.

[37]

X. Wang, W. Wang, An HIV infection model based on a vectored immunoprophylaxis experiment, J. Theoret. Biol., 313 (2012), 127-135. doi: 10.1016/j.jtbi.2012.08.023.

[38]

Y. Wang, Y. Zhou, J. Wu, J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104-112. doi: 10.1016/j.mbs.2009.03.003.

[39]

K. Wang, Y. Jin, A. Fan, The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst. Ser. B., 19 (2014), 3379-3396. doi: 10.3934/dcdsb.2014.19.3379.

[40]

Z. Wang, R. Xu, Stability and Hopf bifurcation in a viral infection modelwith nonlinear incidence rate and delayed immune response, Commun Nonlinear Sci Numer Simulat., 17 (2012), 964-978. doi: 10.1016/j.cnsns.2011.06.024.

[41]

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.

[42]

Y. Yan, W. Wang, Global stability of a five-dimensional model with immune response and delay, Discrete Contin. Dyn. Syst. Ser. B., 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

[43]

Y. Yang, 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.

[44]

Y. Yang, L. Zou, S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. doi: 10.1016/j.mbs.2015.05.001.

[45]

Y. Yang, Y. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington -DeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930. doi: 10.1016/j.camwa.2016.01.009.

[46]

J. Zack, S. Arrigo, S. Weitsman, A. Go, A. Haislip, I. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viral structure, Cell, 61 (1990), 213-222. doi: 10.1016/0092-8674(90)90802-L.

[47]

X. Zhou, X. Song, X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput., 199 (2008), 23-38. doi: 10.1016/j.amc.2007.09.030.

Figure 1.  Illustration of the proportion of infected cells ($I_1$) and virus load ($V_1$) at the endemic equilibrium $E_{1}$. Here parameters are $\lambda=50, \beta=5\times 10^{-7}, d_1=0.008, d_2=0.8,$$ d_3=3, d_4=0.05, d_5=0.1,$$ p=0.05, r=2500, q=0.2, k_1=0.12, h_1=1200, k_2=1.5, s=0.001, \tau=1.5$
Table 1.  Parameter definitions and values used in numerical simulations
Par.ValueDescriptionRef.
$\lambda$0-50 cells ml-day$^{-1}$Recruitment rate of healthy cells[33,38]
$d_1$ $0.007-0.1$day $^{-1}$Death rate of healthy cells[38]
$\beta$ $5\times10^{-7}-0.5$ ml virion-day$^{-1}$Infection rate of target cells by virus[33,38]
$d_2$ $0.2-0.8$ day$^{-1}$Death rate of infected cells[41,46]
$r$ $10-2500$ virions/cellBurst size of virus[38]
$d_3$ $2.4-3$ day$^{-1}$Clearance rate of free virus[38]
$p$$0.05-1$ day$^{-1}$Killing rate of CTL cells[41,40]
$q$ $0.1-1$ day$^{-1}$Neutralizing rate of antibody[41]
$k_1$ $0.1-0.12$ day$^{-1}$Proliferation rate of CTL response[2,41]
$k_2$ $1.5$ day$^{-1}$Production rate of antibody response[41]
$d_4$ $0.05-2$ day$^{-1}$Mortality rate of CTL response[2,40]
$d_5$ $0.1$ day$^{-1}$Clearance rate of antibody[41]
$s$ $0.001-1.4$1/s is the average time[32,47]
$\tau$ $0-2$ daysVirus replication time[38]
$h_1$1200Saturation constantAssumed
$h_2$1500Saturation constantAssumed
$\lambda_1$VariedRate of CTL export from thymus[9]
$\lambda_2$VariedRecruitment rate of antibody[9]
Par.ValueDescriptionRef.
$\lambda$0-50 cells ml-day$^{-1}$Recruitment rate of healthy cells[33,38]
$d_1$ $0.007-0.1$day $^{-1}$Death rate of healthy cells[38]
$\beta$ $5\times10^{-7}-0.5$ ml virion-day$^{-1}$Infection rate of target cells by virus[33,38]
$d_2$ $0.2-0.8$ day$^{-1}$Death rate of infected cells[41,46]
$r$ $10-2500$ virions/cellBurst size of virus[38]
$d_3$ $2.4-3$ day$^{-1}$Clearance rate of free virus[38]
$p$$0.05-1$ day$^{-1}$Killing rate of CTL cells[41,40]
$q$ $0.1-1$ day$^{-1}$Neutralizing rate of antibody[41]
$k_1$ $0.1-0.12$ day$^{-1}$Proliferation rate of CTL response[2,41]
$k_2$ $1.5$ day$^{-1}$Production rate of antibody response[41]
$d_4$ $0.05-2$ day$^{-1}$Mortality rate of CTL response[2,40]
$d_5$ $0.1$ day$^{-1}$Clearance rate of antibody[41]
$s$ $0.001-1.4$1/s is the average time[32,47]
$\tau$ $0-2$ daysVirus replication time[38]
$h_1$1200Saturation constantAssumed
$h_2$1500Saturation constantAssumed
$\lambda_1$VariedRate of CTL export from thymus[9]
$\lambda_2$VariedRecruitment rate of antibody[9]
[1]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[2]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

[3]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[4]

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