American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020212

Viral dynamics of HIV-1 with CTL immune response

Aiping Wang 1, and  Michael Y. Li 2,,

1. 

School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China
2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

* Corresponding author: Michael Y. Li

Dedicated to Professor Sze-Bi Hsu on the occasion of his retirement.

Received  June 2019 Revised  November 2019 Published  July 2020

In this paper, we investigate an in-host model for the viral dynamics of HIV-1 infection and its interaction with the CTL immune response. The model is sufficiently general to allow nonlinear forms for both viral infection and CTL response. Threshold parameters are identified that completely determine the global dynamics and outcomes of the virus-target cell-CTL interactions. Impacts of key parameter values for CTL functions and viral budding rate on the HIV-1 viral load and CD4 count are investigated using numerical simulations. Results support clinical evidence for important differences between HIV-1 nonprogressors and progressors.

Keywords: HIV-1 infection, CTL immune response, viral dynamics, global stability, HIV nonprogressors.
Mathematics Subject Classification: Primary: 34C60, 34D23; Secondary: 92C60.
Citation: Aiping Wang, Michael Y. Li. Viral dynamics of HIV-1 with CTL immune response. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020212
References:
[1]

J. B. Alimonti, T. B. Ball and K. R. Fowke, Mechanisms of CD$4^+$ T lymphocyte cell death in human immunodeficiency virus infection and AIDS, J. Gen. Virol., 84 (2003), 1649-1661.  doi: 10.1099/vir.0.19110-0.  Google Scholar

[2]

R. A. Arnaout, M. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic T lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.  doi: 10.1098/rspb.2000.1149.  Google Scholar

[3]

G. J. Butler, S. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math., 45 (1983), 435-449.  doi: 10.1137/0145025.  Google Scholar

[4]

J. M. Conway and R. M. Ribeiro, Modeling the immune response to HIV infection, Curr. Opin. Syst. Biol., 12 (2018), 61-69.   Google Scholar

[5]

J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112.  Google Scholar

[6]

T. Dumrongpokaphan, Y. Lenbury, R. Ouncharoen and Y. S. Xu, An intracellular delay-differential equation model of the HIV infection and immune control, Math. Model. Nat. Phenom. Epidemiol., 2 (2007), 75-99.  doi: 10.1051/mmnp:2008012.  Google Scholar

[7]

H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.  doi: 10.1007/BF02218848.  Google Scholar

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Y. Gao, P. F. McKay and J. F. S. Mann, Advances in HIV-1 vaccine development, Viruses, 10 (2018), 167. doi: 10.3390/v10040167.  Google Scholar

[9]

J. C. Gea-Banacloche, S. A. Migueles and L. Martino, Maintenance of large numbers of virus-specific CD8 + T cells in HIV-infected progressors and long-term nonprogressors, J. Immunol., 165 (2000), 1082-1092.  doi: 10.4049/jimmunol.165.2.1082.  Google Scholar

[10]

B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1997), 135-143.  doi: 10.1086/283144.  Google Scholar

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H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multi-stability In a model for CTL response to HTLV-1 infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z.  Google Scholar

[12]

R. A. Gruters, C. A. van Baalen and A. D. M. E. Osterhaus, The advantage of early recognition of HIV-infected cells by cytotoxic T-lymphocytes, Vaccine, 20 (2002), 2011-2015.  doi: 10.1016/S0264-410X(02)00089-0.  Google Scholar

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H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284.   Google Scholar

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H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

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S. M. Hammer, M. E. Sobieszczyk and H. Janes, Efficacy trial of a DNA/rAd5 HIV-1 preventive vaccine, N. Engl. J. Med., 369 (2013), 2083-2092.  doi: 10.1056/NEJMoa1310566.  Google Scholar

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S. B. Hsu, On global stability of a predator-prey systems, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

[18]

S. Imlach, S. McBreen and T. Shirafuji, Activated peripheral CD8 lymphocytes express CD4 in vivo and are targets for infection by human immunodeficiency virus type 1, J. Virol., 75 (2001), 11555-11564.  doi: 10.1128/JVI.75.23.11555-11564.2001.  Google Scholar

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[20]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[21]

J. P. LaSalle, The Stability of Dynamical System, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[22]

J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199.  doi: 10.1007/s00285-011-0455-z.  Google Scholar

[23]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[24]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[25]

M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.  doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[26]

M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonl. Anal. RWA, 17 (2014), 147-160.  doi: 10.1016/j.nonrwa.2013.11.002.  Google Scholar

[27]

J. Lin, R. Xu and X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Engin., 16 (2018), 292-319.  doi: 10.3934/mbe.2019015.  Google Scholar

[28]

S. A. Migueles, A. C. Laborico and W. L Shupert, HIV-specific CD8+ T cell proliferation is coupled to perforin expression and is maintained in nonprogressors, Nature Immunol., 3 (2002), 1061-1068.  doi: 10.1038/ni845.  Google Scholar

[29]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar

[30]

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

[31] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.   Google Scholar
[32]

G. S. Ogg and X. Jin, Quantitation of HIV-1-specific cytotoxic T lymphocytes and plasma load of viral RNA, Science, 279 (1998), 2103-2106.  doi: 10.1126/science.279.5359.2103.  Google Scholar

[33]

A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD$4^+$ T cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[34]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[35]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96. doi: 10.1186/1741-7007-11-96.  Google Scholar

[36]

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

[37]

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.  Google Scholar

[38]

V. Simon and D. D. Ho, HIV-1 dynamics in vivo: Implications for therapy, Nat. Rev. Microbiol., 1 (2003), 181-190.   Google Scholar

[39]

G. D. Tomarasa and B. F. Haynes, HIV-1-specific antibody responses during acute and chronic HIV-1 infection, Curr. Opin. HIV AIDS, 4 (2009), 373-379.  doi: 10.1097/COH.0b013e32832f00c0.  Google Scholar

[40]

J. Overbaugh and L. Morris, The Antibody Response against HIV-1, Cold Spring Harb. Perspect. Med., 2 (2012), a007039. doi: 10.1101/cshperspect.a007039.  Google Scholar

[41]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

[42]

Y. Wang, Y. C. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.   Google Scholar

[43]

J. Weber, The pathogenesis of HIV-1 infection, Br. Med. Bull., 58 (2001), 61-72.  doi: 10.1093/bmb/58.1.61.  Google Scholar

[44]

D. Wodarz and M. A. Nowak, Correlates of cytotoxic t-lymphocyte-mediated virus control: Implications for immunosuppressive infections and their treatment, Phil. Trans. R. Soc. Lond. B, 355 (2000), 1059-1070.  doi: 10.1098/rstb.2000.0643.  Google Scholar

[45]

H. Y. Zhu and X. F. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, DCDS B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

show all references

References:
[1]

J. B. Alimonti, T. B. Ball and K. R. Fowke, Mechanisms of CD$4^+$ T lymphocyte cell death in human immunodeficiency virus infection and AIDS, J. Gen. Virol., 84 (2003), 1649-1661.  doi: 10.1099/vir.0.19110-0.  Google Scholar

[2]

R. A. Arnaout, M. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic T lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.  doi: 10.1098/rspb.2000.1149.  Google Scholar

[3]

G. J. Butler, S. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math., 45 (1983), 435-449.  doi: 10.1137/0145025.  Google Scholar

[4]

J. M. Conway and R. M. Ribeiro, Modeling the immune response to HIV infection, Curr. Opin. Syst. Biol., 12 (2018), 61-69.   Google Scholar

[5]

J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112.  Google Scholar

[6]

T. Dumrongpokaphan, Y. Lenbury, R. Ouncharoen and Y. S. Xu, An intracellular delay-differential equation model of the HIV infection and immune control, Math. Model. Nat. Phenom. Epidemiol., 2 (2007), 75-99.  doi: 10.1051/mmnp:2008012.  Google Scholar

[7]

H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.  doi: 10.1007/BF02218848.  Google Scholar

[8]

Y. Gao, P. F. McKay and J. F. S. Mann, Advances in HIV-1 vaccine development, Viruses, 10 (2018), 167. doi: 10.3390/v10040167.  Google Scholar

[9]

J. C. Gea-Banacloche, S. A. Migueles and L. Martino, Maintenance of large numbers of virus-specific CD8 + T cells in HIV-infected progressors and long-term nonprogressors, J. Immunol., 165 (2000), 1082-1092.  doi: 10.4049/jimmunol.165.2.1082.  Google Scholar

[10]

B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1997), 135-143.  doi: 10.1086/283144.  Google Scholar

[11]

H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multi-stability In a model for CTL response to HTLV-1 infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z.  Google Scholar

[12]

R. A. Gruters, C. A. van Baalen and A. D. M. E. Osterhaus, The advantage of early recognition of HIV-infected cells by cytotoxic T-lymphocytes, Vaccine, 20 (2002), 2011-2015.  doi: 10.1016/S0264-410X(02)00089-0.  Google Scholar

[13]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284.   Google Scholar

[14]

H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar

[15]

J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, 1969.  Google Scholar

[16]

S. M. Hammer, M. E. Sobieszczyk and H. Janes, Efficacy trial of a DNA/rAd5 HIV-1 preventive vaccine, N. Engl. J. Med., 369 (2013), 2083-2092.  doi: 10.1056/NEJMoa1310566.  Google Scholar

[17]

S. B. Hsu, On global stability of a predator-prey systems, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.  Google Scholar

[18]

S. Imlach, S. McBreen and T. Shirafuji, Activated peripheral CD8 lymphocytes express CD4 in vivo and are targets for infection by human immunodeficiency virus type 1, J. Virol., 75 (2001), 11555-11564.  doi: 10.1128/JVI.75.23.11555-11564.2001.  Google Scholar

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.  Google Scholar

[20]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.  Google Scholar

[21]

J. P. LaSalle, The Stability of Dynamical System, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[22]

J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199.  doi: 10.1007/s00285-011-0455-z.  Google Scholar

[23]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[24]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[25]

M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.  doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[26]

M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonl. Anal. RWA, 17 (2014), 147-160.  doi: 10.1016/j.nonrwa.2013.11.002.  Google Scholar

[27]

J. Lin, R. Xu and X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Engin., 16 (2018), 292-319.  doi: 10.3934/mbe.2019015.  Google Scholar

[28]

S. A. Migueles, A. C. Laborico and W. L Shupert, HIV-specific CD8+ T cell proliferation is coupled to perforin expression and is maintained in nonprogressors, Nature Immunol., 3 (2002), 1061-1068.  doi: 10.1038/ni845.  Google Scholar

[29]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.  Google Scholar

[30]

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

[31] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.   Google Scholar
[32]

G. S. Ogg and X. Jin, Quantitation of HIV-1-specific cytotoxic T lymphocytes and plasma load of viral RNA, Science, 279 (1998), 2103-2106.  doi: 10.1126/science.279.5359.2103.  Google Scholar

[33]

A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD$4^+$ T cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[34]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[35]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96. doi: 10.1186/1741-7007-11-96.  Google Scholar

[36]

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

[37]

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.  Google Scholar

[38]

V. Simon and D. D. Ho, HIV-1 dynamics in vivo: Implications for therapy, Nat. Rev. Microbiol., 1 (2003), 181-190.   Google Scholar

[39]

G. D. Tomarasa and B. F. Haynes, HIV-1-specific antibody responses during acute and chronic HIV-1 infection, Curr. Opin. HIV AIDS, 4 (2009), 373-379.  doi: 10.1097/COH.0b013e32832f00c0.  Google Scholar

[40]

J. Overbaugh and L. Morris, The Antibody Response against HIV-1, Cold Spring Harb. Perspect. Med., 2 (2012), a007039. doi: 10.1101/cshperspect.a007039.  Google Scholar

[41]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

[42]

Y. Wang, Y. C. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.   Google Scholar

[43]

J. Weber, The pathogenesis of HIV-1 infection, Br. Med. Bull., 58 (2001), 61-72.  doi: 10.1093/bmb/58.1.61.  Google Scholar

[44]

D. Wodarz and M. A. Nowak, Correlates of cytotoxic t-lymphocyte-mediated virus control: Implications for immunosuppressive infections and their treatment, Phil. Trans. R. Soc. Lond. B, 355 (2000), 1059-1070.  doi: 10.1098/rstb.2000.0643.  Google Scholar

[45]

H. Y. Zhu and X. F. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, DCDS B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.  Google Scholar

Figure 1.  Transfer-infection diagram of model (2). Solid lines indicate cell transfer and dotted lines indicated virus-cell or cell-cell interaction
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Figure 2.  Graphs of functions $ F(v) $ and $ G(v) $ and a geometric demonstration of existence of chronic-infection equilibria $ P_1 $ and $ P_2 $
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Figure 3.  Effects of CTL response parameter $ c $ on HIV-1 viral load and CD4 count
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Figure 4.  Effects of CTL killing rate $ \alpha $ on HIV-1 viral load and CD4 count
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Figure 5.  Effects of viral budding number $ N $ on HIV-1 viral load and CD4 count
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