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## Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment

 1 Department of Mathematics, Hubei Minzu University, Enshi, 445000, China 2 School of Information and Mathematics, Yangtze University, Jingzhou, 434023, China 3 Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, United States

* Corresponding author: Zhijun Liu

Received  December 2018 Revised  April 2019 Published  September 2019

Fund Project: The work is supported by National Natural Science Foundation of China (No.11871201) and Youth Talents Project of Science and Technology Research Plan of Hubei Provincial Education Department (No.Q20171904)

It is well known that CTL (cytotoxic T lymphocyte) immune response could be broadly classified into lytic and nonlytic components, nonlinear functions can better reproduce saturated effects in the interaction processes between cell and viral populations, and distributed intracellular delay can realistically reflect the stochastic element in the delay effects. For these reasons, we develop an HTLV-I (Human T-cell leukemia virus type I) infection model with nonlinear lytic and nonlytic CTL immune responses, nonlinear incidence rate, distributed intracellular delay and immune impairment. Through conducting complete analysis, it is revealed that all these factors influence the concentration level of infected T-cells at the chronic-infection equilibrium, whereas intracellular distributed delay and nonlinear incidence rate may change the expression of the basic reproduction number $\mathfrak{R}_0$ in the context where the model proposed still preserves the threshold dynamics. Our analysis results obtained may improve several existing works by comparison. We also perform global sensitivity analysis for $\mathfrak{R}_0$ in order to explore the effective strategies of lowering the concentration level of infected T-cells.

Citation: Lianwen Wang, Zhijun Liu, Yong Li, Dashun Xu. Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019196
##### References:
 [1] E. Avila-Vales, N. Chan-Chí and G. García-Almeida, Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate, Chaos Solitons Fractals, 69 (2014), 1-9.  doi: 10.1016/j.chaos.2014.08.009.  Google Scholar [2] C. Bartholdy, J. P. Christensen, D. Wodarz and A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with lymphocytic choriomeningitis virus, J. Virol., 74 (2000), 10304-10311.   Google Scholar [3] A. Carpentier, et al., Modes of human T cell leukemia virus type 1 transmission, replication and persistence, Viruses, 7 (2015), 3603–3624. Google Scholar [4] P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.   Google Scholar [5] R. V. Culshaw, S. G. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5.  Google Scholar [6] R. J. De Boer, Which of our modeling predictions are robust?, PLoS Comput. Biol., 8 (2012), e10002593, 5 pp. doi: 10.1371/journal.pcbi.1002593.  Google Scholar [7] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209.   Google Scholar [8] A. Gessain and O. Cassar, Epidemiological aspects and world distribution of HTLV-1 infection, Front. Microbiol., 3 (2012), e388. Google Scholar [9] H. Gómez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4$^+$T cells, Bull. Math. Biol., 67 (2005), 101-114.  doi: 10.1016/j.bulm.2004.06.004.  Google Scholar [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci. 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equation with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar [12] Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations With Infinite Delay, Lecture Notes in Math., 1473. Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [13] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.   Google Scholar [14] S. Jacobson, Immunopathogenesis of HTLV-I associated neurological disease, J. Infect. Dis., 186 (2002), 187-192.   Google Scholar [15] Y. Ji and 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.  Google Scholar [16] J. P. La Salle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. Google Scholar [17] X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Math. Anal., 74 (2014), 898-917.  doi: 10.1137/130930145.  Google Scholar [18] B. R. Levin, M. Lipsitch and S. Bonhoeffer, Population biology, evolution, and infectious disease: Convergence and synthesis, Science, 283 (1999), 806-809.  doi: 10.1126/science.283.5403.806.  Google Scholar [19] S. M. Li and Y. C. Zhou, Backward bifurcation of an HTLV-I model with immune response, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 863-881.  doi: 10.3934/dcdsb.2016.21.863.  Google Scholar [20] M. Y. Li and H. Y. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793.  doi: 10.1007/s11538-010-9591-7.  Google Scholar [21] M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4$^+$ T cells with delayed CTL response, Nonlinear Anal. RWA, 13 (2012), 1080-1092.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar [22] X. J. Lu, L. L. Hui, S. Q. Liu and J. Li, A mathematical model of HTLV-I infection with two time delays, Math. Biosci. Eng., 12 (2015), 431-449.  doi: 10.3934/mbe.2015.12.431.  Google Scholar [23] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Spring-Verlag, Heidelberg, 1978. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [24] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar [25] J. E. Mittler, B. Sulzer ac, A. U. Neumann ade and A. S. Perelson a, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163.  doi: 10.1016/S0025-5564(98)10027-5.  Google Scholar [26] Y. Nakata, Y. Enatsu and Y. Muroya, Complete global dynamics of a delayed viral infection model with lytic and nonlytic effectors, SeMA J., (2012), 27–50.  Google Scholar [27] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74.  Google Scholar [28] C. M. Rooney, C. Y. C. Ng, S. Loftin, C. A. Smith, C. Li, R. A. Krance, M. K. Brenner, H. E. Heslop, C. M. Rooney, M. K. Brenner, M. K. Brenner, R. A. Krance and H. E. Heslop, Use of gene-modified virus-specific T lymphocytes to control Epstein-Barr-virus-related lymphoproliferation, Lancet, 345 (1995), 9-13.  doi: 10.1016/S0140-6736(95)91150-2.  Google Scholar [29] H. Y. 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.  Google Scholar [30] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689.  doi: 10.1016/j.amc.2014.06.020.  Google Scholar [31] N. I. Stilianakis and J. Seydel, Modeling the T-cell dynamics and pathogenesis of HTLV-I infection, Bull. Math. Biol., 61 (1999), 935-947.  doi: 10.1006/bulm.1999.0117.  Google Scholar [32] C. Vargas-De-León, Global properties for a virus dynamics model with lytic and nonlytic immune responses and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449-462.  doi: 10.1142/S021833901450017X.  Google Scholar [33] J. L. Wang, M. Guo, X. N. Liu and Z. T. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149-161.  doi: 10.1016/j.amc.2016.06.032.  Google Scholar [34] K. F. Wang, W. D. Wang and X. N. 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 [35] S. L. Wang, X. Y. Song and Z. H. Ge, Dynamics analysis of a delayed viral infection model with immune impairment, Appl. Math. Model., 35 (2011), 4877-4885.  doi: 10.1016/j.apm.2011.03.043.  Google Scholar [36] Y. Wang, J. Liu and J. M. Heffernan, Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay, J. Math. Anal. Appl., 459 (2018), 506-527.  doi: 10.1016/j.jmaa.2017.10.027.  Google Scholar [37] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trend in Immunology, 23 (2002), 194-200.  doi: 10.1016/S1471-4906(02)02189-0.  Google Scholar [38] D. Wodarz and M. A. Nowak, Immune response and viral phenotype: Do replication rate and cytopathogenicity influence virus load?, Comput. Math. Method. M., 2 (2000), 113-127.  doi: 10.1080/10273660008833041.  Google Scholar

show all references

##### References:
 [1] E. Avila-Vales, N. Chan-Chí and G. García-Almeida, Analysis of a viral infection model with immune impairment, intracellular delay and general non-linear incidence rate, Chaos Solitons Fractals, 69 (2014), 1-9.  doi: 10.1016/j.chaos.2014.08.009.  Google Scholar [2] C. Bartholdy, J. P. Christensen, D. Wodarz and A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with lymphocytic choriomeningitis virus, J. Virol., 74 (2000), 10304-10311.   Google Scholar [3] A. Carpentier, et al., Modes of human T cell leukemia virus type 1 transmission, replication and persistence, Viruses, 7 (2015), 3603–3624. Google Scholar [4] P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.   Google Scholar [5] R. V. Culshaw, S. G. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5.  Google Scholar [6] R. J. De Boer, Which of our modeling predictions are robust?, PLoS Comput. Biol., 8 (2012), e10002593, 5 pp. doi: 10.1371/journal.pcbi.1002593.  Google Scholar [7] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209.   Google Scholar [8] A. Gessain and O. Cassar, Epidemiological aspects and world distribution of HTLV-1 infection, Front. Microbiol., 3 (2012), e388. Google Scholar [9] H. Gómez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4$^+$T cells, Bull. Math. Biol., 67 (2005), 101-114.  doi: 10.1016/j.bulm.2004.06.004.  Google Scholar [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci. 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [11] J. K. Hale and J. Kato, Phase space for retarded equation with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar [12] Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations With Infinite Delay, Lecture Notes in Math., 1473. Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [13] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.   Google Scholar [14] S. Jacobson, Immunopathogenesis of HTLV-I associated neurological disease, J. Infect. Dis., 186 (2002), 187-192.   Google Scholar [15] Y. Ji and 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.  Google Scholar [16] J. P. La Salle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. Google Scholar [17] X. L. Lai and X. F. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Math. Anal., 74 (2014), 898-917.  doi: 10.1137/130930145.  Google Scholar [18] B. R. Levin, M. Lipsitch and S. Bonhoeffer, Population biology, evolution, and infectious disease: Convergence and synthesis, Science, 283 (1999), 806-809.  doi: 10.1126/science.283.5403.806.  Google Scholar [19] S. M. Li and Y. C. Zhou, Backward bifurcation of an HTLV-I model with immune response, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 863-881.  doi: 10.3934/dcdsb.2016.21.863.  Google Scholar [20] M. Y. Li and H. Y. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793.  doi: 10.1007/s11538-010-9591-7.  Google Scholar [21] M. Y. Li and H. Y. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4$^+$ T cells with delayed CTL response, Nonlinear Anal. RWA, 13 (2012), 1080-1092.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar [22] X. J. Lu, L. L. Hui, S. Q. Liu and J. Li, A mathematical model of HTLV-I infection with two time delays, Math. Biosci. Eng., 12 (2015), 431-449.  doi: 10.3934/mbe.2015.12.431.  Google Scholar [23] N. MacDonald, Time Lags in Biological Models, Lecture Notes in Biomathematics, 27. Spring-Verlag, Heidelberg, 1978. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [24] S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theoret. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar [25] J. E. Mittler, B. Sulzer ac, A. U. Neumann ade and A. S. Perelson a, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163.  doi: 10.1016/S0025-5564(98)10027-5.  Google Scholar [26] Y. Nakata, Y. Enatsu and Y. Muroya, Complete global dynamics of a delayed viral infection model with lytic and nonlytic effectors, SeMA J., (2012), 27–50.  Google Scholar [27] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74.  Google Scholar [28] C. M. Rooney, C. Y. C. Ng, S. Loftin, C. A. Smith, C. Li, R. A. Krance, M. K. Brenner, H. E. Heslop, C. M. Rooney, M. K. Brenner, M. K. Brenner, R. A. Krance and H. E. Heslop, Use of gene-modified virus-specific T lymphocytes to control Epstein-Barr-virus-related lymphoproliferation, Lancet, 345 (1995), 9-13.  doi: 10.1016/S0140-6736(95)91150-2.  Google Scholar [29] H. Y. 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.  Google Scholar [30] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689.  doi: 10.1016/j.amc.2014.06.020.  Google Scholar [31] N. I. Stilianakis and J. Seydel, Modeling the T-cell dynamics and pathogenesis of HTLV-I infection, Bull. Math. Biol., 61 (1999), 935-947.  doi: 10.1006/bulm.1999.0117.  Google Scholar [32] C. Vargas-De-León, Global properties for a virus dynamics model with lytic and nonlytic immune responses and nonlinear immune attack rates, J. Biol. Syst., 22 (2014), 449-462.  doi: 10.1142/S021833901450017X.  Google Scholar [33] J. L. Wang, M. Guo, X. N. Liu and Z. T. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149-161.  doi: 10.1016/j.amc.2016.06.032.  Google Scholar [34] K. F. Wang, W. D. Wang and X. N. 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 [35] S. L. Wang, X. Y. Song and Z. H. Ge, Dynamics analysis of a delayed viral infection model with immune impairment, Appl. Math. Model., 35 (2011), 4877-4885.  doi: 10.1016/j.apm.2011.03.043.  Google Scholar [36] Y. Wang, J. Liu and J. M. Heffernan, Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay, J. Math. Anal. Appl., 459 (2018), 506-527.  doi: 10.1016/j.jmaa.2017.10.027.  Google Scholar [37] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trend in Immunology, 23 (2002), 194-200.  doi: 10.1016/S1471-4906(02)02189-0.  Google Scholar [38] D. Wodarz and M. A. Nowak, Immune response and viral phenotype: Do replication rate and cytopathogenicity influence virus load?, Comput. Math. Method. M., 2 (2000), 113-127.  doi: 10.1080/10273660008833041.  Google Scholar
The long-term and short-term behaviors for infected T-cells $y(t)$ of model (3) with the weak or strong kernel and the same initial condition are respectively shown as follows: (a) the global dynamics of $y(t)$; (b) the short-term dynamics of $y(t)$
PRCC values for $\mathfrak{R}_0$ in the cases where input parameters respectively follow (a) uniform and (b) normal distributions
Simulations for the concentrations of infected T-cells of model (22) when (a) $\sigma$ varies by 0.75 and 0.5, (b) $a$ varies by 1.5 and 2 times, (c) $\alpha_1$ varies by 1.5 and 2 times and (d) $\beta$ varies by 0.75 and 0.5 of their baseline values in Table 1, respectively
Biological description of parameters in models (1) and (3)
 Para.(Unit) Description Value Range Ref. $\lambda$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The recruitment rate of healthy T-cells $105$ [10,200] [20,21,22,34,37] $d$ ($\mbox{d}^{-1}$) The death rate of healthy T-cells $0.11$ [0.01, 0.2] [20,21,22,34,37] $\beta$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) Viral infectivity rate $0.026$ [0.001, 0.05] [20,21,22,34,37] $\rho$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The death rate of infected not productive cells $0.11$ [0.01, 0.2] [36] $\tau$ ($\mbox{d}$) The intracellular latent delay $5$ [0, 10] [1] $a$ ($\mbox{d}^{-1}$) The sum of the released rate of viral particles and the death rate of infected T-cells $1.005$ [0.01, 0.2] [20,21,22,34,37] $q_0$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The efficacy of NL-CTL response $0.5$ [0, 1] [34,37] $p_0$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The strength of L-CTL response $0.1$ [0, 1] [20,21,22,34,37] $c$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The proliferation rate of CTLs $0.2$ [0, 1] [20,21,22,34,37] $b$ ($\mbox{d}^{-1}$) The decay rate of CTLs $0.4$ [0, 1] [20,21,22,34,37] $\sigma$ (-) A fraction of cells newly infected by contacts that survive the antibody immune response $0.5$ [0, 1] [9,22] $\alpha_1$ ($\mu \mbox{l}$) The inhibitory rate from healthy T-cells $0.003$ [0, 10] Estimated $\alpha_2$ ($\mu \mbox{l}$) The inhibitory rate from infected T-cells $0.005$ [0, 10] Estimated $\omega$ ($\mu \mbox{l}$) The inhibitory rate from NL-CTL response $10$ [0, 10] [32] $m$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) Immune impairment rate of virus $0.01$ [0, 1] [1]
 Para.(Unit) Description Value Range Ref. $\lambda$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The recruitment rate of healthy T-cells $105$ [10,200] [20,21,22,34,37] $d$ ($\mbox{d}^{-1}$) The death rate of healthy T-cells $0.11$ [0.01, 0.2] [20,21,22,34,37] $\beta$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) Viral infectivity rate $0.026$ [0.001, 0.05] [20,21,22,34,37] $\rho$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The death rate of infected not productive cells $0.11$ [0.01, 0.2] [36] $\tau$ ($\mbox{d}$) The intracellular latent delay $5$ [0, 10] [1] $a$ ($\mbox{d}^{-1}$) The sum of the released rate of viral particles and the death rate of infected T-cells $1.005$ [0.01, 0.2] [20,21,22,34,37] $q_0$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The efficacy of NL-CTL response $0.5$ [0, 1] [34,37] $p_0$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The strength of L-CTL response $0.1$ [0, 1] [20,21,22,34,37] $c$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) The proliferation rate of CTLs $0.2$ [0, 1] [20,21,22,34,37] $b$ ($\mbox{d}^{-1}$) The decay rate of CTLs $0.4$ [0, 1] [20,21,22,34,37] $\sigma$ (-) A fraction of cells newly infected by contacts that survive the antibody immune response $0.5$ [0, 1] [9,22] $\alpha_1$ ($\mu \mbox{l}$) The inhibitory rate from healthy T-cells $0.003$ [0, 10] Estimated $\alpha_2$ ($\mu \mbox{l}$) The inhibitory rate from infected T-cells $0.005$ [0, 10] Estimated $\omega$ ($\mu \mbox{l}$) The inhibitory rate from NL-CTL response $10$ [0, 10] [32] $m$ ($\mu \mbox{l}^{-1}\mbox{d}^{-1}$) Immune impairment rate of virus $0.01$ [0, 1] [1]
PRCC values for $\mathfrak{R}_0$
 Para. Distribution1 PRCC1 p-value1 Distribution2 PRCC2 p-value2 Rank $\lambda$ U(10, 200) $-0.0193$ $0.2548$ N(105, 30) $0.0230$ $0.1732$ 7 $d$ U(0.01, 0.2) $-0.0180$ $0.2880$ N(0.11, 0.03) $0.0149$ $0.3779$ 8 $\beta$ U(0.001, 0.05) $0.6968$ $0$ N(0.026, 0.005) $0.5697$ $0$ 4 $\alpha_1$ U(0, 10) $-0.7167$ $0$ N(5, 1.667) $-0.7789$ $0$ 3 $\sigma$ U(0, 1) $0.7411$ $0$ N(0.5, 0.167) $0.7885$ $0$ 1 $\rho$ U(0.01, 0.2) $-0.3574$ $0$ N(0.11, 0.03) $-0.4692$ $0$ 6 $\tau$ U(0, 10) $-0.3753$ $0$ N(5, 1.667) $-0.5536$ $0$ 5 $a$ U(0.01, 0.2) $-0.7211$ $0$ N(1.005, 0.332) $-0.7811$ $0$ 2
 Para. Distribution1 PRCC1 p-value1 Distribution2 PRCC2 p-value2 Rank $\lambda$ U(10, 200) $-0.0193$ $0.2548$ N(105, 30) $0.0230$ $0.1732$ 7 $d$ U(0.01, 0.2) $-0.0180$ $0.2880$ N(0.11, 0.03) $0.0149$ $0.3779$ 8 $\beta$ U(0.001, 0.05) $0.6968$ $0$ N(0.026, 0.005) $0.5697$ $0$ 4 $\alpha_1$ U(0, 10) $-0.7167$ $0$ N(5, 1.667) $-0.7789$ $0$ 3 $\sigma$ U(0, 1) $0.7411$ $0$ N(0.5, 0.167) $0.7885$ $0$ 1 $\rho$ U(0.01, 0.2) $-0.3574$ $0$ N(0.11, 0.03) $-0.4692$ $0$ 6 $\tau$ U(0, 10) $-0.3753$ $0$ N(5, 1.667) $-0.5536$ $0$ 5 $a$ U(0.01, 0.2) $-0.7211$ $0$ N(1.005, 0.332) $-0.7811$ $0$ 2
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