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May  2016, 21(3): 863-881. doi: 10.3934/dcdsb.2016.21.863

## Backward bifurcation of an HTLV-I model with immune response

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 2 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049

Received  May 2015 Revised  September 2015 Published  January 2016

Human T-cell Lymphotropic virus type 1(HTLV-I) causes HAM/T SP and other illnesses. HTLV-I mainly infects $CD4^+$ T cells and activates HTLV-I-specific immune response. In this paper, we formulate a mathematical model of HTLV-I to investigate the role of selective mitotic transmission, Tax expression, and CTL response in vivo. We define two parameters ($R_0$ and $R_1$) to study the model dynamics. The unique infection-free equilibrium $P_0$ is globally asymptomatic stable if $R_0<1$. There exists the chronic-infection equilibrium $P_1$ if $R_1 < 1 < R_0$. There exists a unique chronic-infection equilibrium $P_2$ if $R_1 > 1$. There is a backward bifurcation of chronic-infection equilibria with CTL response if $R_1 < 1 < R_0$. The numerical simulations shown that the existence of backward bifurcation may lead to the existence of periodic solutions.
Citation: Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863
##### References:
 [1] B. Asquith and C. R. M. Bangham, How does HTLV-I persist despite a strong cell-mediated immune response?, Trends in Immunology, 29 (2008), 4-11. doi: 10.1016/j.it.2007.09.006. [2] C. R. M. Bangham, HTLV-I infections, Journal of Clinical Pathology, 53 (2000), 581-586. [3] R. C. Gallo, The discovery of the first human retrovirus: HTLV-1 and HTLV-2, Rrtrovirology, 2 (2005), 17-23. [4] U. Tomaru, Y. Yamano and S. Jacobson, HTLV-I Infection and the Nervous System In Clinical Neuroimmunology, $2^{nd}$ edition, Oxford University Press, Oxford, 2005. [5] L. B. Cook, M. Elemans, A. G. Rowan and B. Asquith, HTLV-I: Persistence and pathogenesis, Virology, 435 (2013), 131-140. doi: 10.1016/j.virol.2012.09.028. [6] F. A. Proietti, A. B. F.Carneiro-Proietti, B. C. Catalan-Soares and E. L. MURPHY, Global epidemiology of HTLV-I infection and associated deseases, Oncogene, 24 (2005), 6058-6068. [7] D. Wodarz and C. R. M. Bangham, Evolutionary dynamics of HTLV-I, Journal of Molecular Evolution, 50 (2000), 448-455. [8] J. E. Kaplan, M. Osame, H. Kubota, A. Igata, H. Nishitani, Y. Maeda, R. F. Khabbaz and R. S. Janssen, The risk of development of HTLV-I-associted myelopathy/tropocal spastic paraparesis among persons infected with HTLV-I, Journal of Acquired Immune Deficiency Syndromes, 3 (1990), 1096-1101. [9] C. Pique and K. S. Jones, Pathways of cell-cell transmission of HTLV-I, Frontiers in Microbiology, 3 (2012), p378. doi: 10.3389/fmicb.2012.00378. [10] M. Nagai, K. Usuku, W. Matsumoto, D. Kodama, N. Takenouchi, T. Moritoyo, S. Hashiguchi, M. Ichinose, R. M. Bangham, S. Izumo and M. Osame, Analysis of HTLV-I proviral load in 202 HAM/TSP patients and 243 asymptomatic HTLV-i carriers: high proviral load strongly predisposes to HAM/TSP, Journal of NeuroVirology, 4 (1998), 586-593. [11] Y. Ina and T. Gogobori, Molecular evoluton of human T-cell leukemia virus, Journal of Molecular Evolution, 31 (1990), 493-499. [12] R. Kubota, T. Fujiyoshi, S. Izumo, S. Yashiki, I. Maruyama, M. Osame and S. Sonoda, Fluctuation of HTLV-I proviral DNA in peripheral blood mononuclear cells of HTLV-I-associated myelopathy, Journal of NeuroVirology, 42 (1993), 147-154. doi: 10.1016/0165-5728(93)90004-I. [13] H. GÓmez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of $CD4^+$ T cells, Bulletin of Mathematical Biology, 67 (2005), 101-114. doi: 10.1016/j.bulm.2004.06.004. [14] F. Mortreux, A. S. Gabet and E. Wattel, Molecular and cellular aspects of HTLV-I associated leukemogenesis in vivo, Leukemia, 17 (2003), 26-38. doi: 10.1038/sj.leu.2402777. [15] P. Hollsberg, Mechanisms of T-cell activation by human T-cell lymphotropic virus type 1, Microbiology and Molecular Biology Reviews, 63 (1999), 308-333. [16] J. Mesnard and C. Devaux, Multiple control levels of cell proliferation by human T-cell leukemia virus type 1 Tax protein, Virology, 257 (1999), 277-284. doi: 10.1006/viro.1999.9685. [17] F. Bex and R. Gaynor, Regulation of gene expression by HTLV-I Tax protein, Methods, 16 (1998), 83-94. doi: 10.1006/meth.1998.0646. [18] M. Yoshida, Multiple viral strategies of HTLV-I for dysregulation of cell growth control, Annual Review of Immunology, 19 (2001), 475-496. [19] B. Asquith, A. J. Mosley, A. Heaps, Y. Tanaka, G. P. Taylor, A. R. Mclean and C. R. M Bangham, Quantification of the virus-host interaction in human T lymphotropic virus 1 infection, Retrovirology, 75 (2005), 1-9. [20] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74. [21] D. Wodarz, M. A. Nowak and C. R. M. Bangham, The dynamics of HTLV-I and the CTL response, Immunology Today, 20 (1999), 220-227. doi: 10.1016/S0167-5699(99)01446-2. [22] M. Y. Li and A. G. Lim, Modelling the role of tax expression in HTLV-I persistence in vivo, Bulletin of Mathematical Biology, 73 (2011), 3008-3029. doi: 10.1007/s11538-011-9657-1. [23] S. Li and Y. Zhou, Global dynamics of an HTLV-I model with cell-to-cell infection and mitosis, Abstract and Applied Analysis, 2014 (2014), Art. ID 132781, 12 pp. doi: 10.1155/2014/132781. [24] H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multistability in a model for CTL Response to HTLV-I infection and its implications to HAM/TSP decelopment and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. [25] M. Y. 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. [26] M. Y. Li and H. Shu, Multiple stable peridic oscillations in a mathematical model of CTL response to HTLVE-I infection, Bulletin of Mathematical Biology, 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7. [27] J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, Mathematical Biology, 65 (2012), 181-199. doi: 10.1007/s00285-011-0455-z. [28] A. G. Lim and P. K. Maini, HTLV-I infection: A dynamica struggle between viral persistence and host immunity, Journal of Theoretical Biology, 352 (2014), 92-108. doi: 10.1016/j.jtbi.2014.02.022. [29] B. Asquith, Y. Zhang, A. J. Mosley, C. M. d.Lara, D. L. Wallace, A. Worth, L. Kaftantzi, K. Meekings, G. E. Griffin, Y. Tanaka, D. F. Tough, P. C. Beverley, G. P. Taylor, D. C. Macallan and C. R. Bangham, In vivo T lymphocyte dynamics in humans and the impact of human T-lymphotropic virus 1 infection, Proceedings of the National Academy of Sciences, 104 (2007), 8035-8040. doi: 10.1073/pnas.0608832104. [30] X. Yang and L. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers and Mathematics with Applications, 32 (1996), 109-116. doi: 10.1016/0898-1221(96)00129-0. [31] P. v. d. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical. Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [32] J. P. LaSalle, The stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, 1976. [33] A. Melamed, D. J. Laydon, N. A. Gillet, Y. Tanaka, G. P. Taylor and C. R. Bangham, Genome-wide determinants of proviral targeting, clonal abundance and expression in natural HTLV-I infection, PLoS pathogens, 9 (2013), e1003271. doi: 10.1371/journal.ppat.1003271.

show all references

##### References:
 [1] B. Asquith and C. R. M. Bangham, How does HTLV-I persist despite a strong cell-mediated immune response?, Trends in Immunology, 29 (2008), 4-11. doi: 10.1016/j.it.2007.09.006. [2] C. R. M. Bangham, HTLV-I infections, Journal of Clinical Pathology, 53 (2000), 581-586. [3] R. C. Gallo, The discovery of the first human retrovirus: HTLV-1 and HTLV-2, Rrtrovirology, 2 (2005), 17-23. [4] U. Tomaru, Y. Yamano and S. Jacobson, HTLV-I Infection and the Nervous System In Clinical Neuroimmunology, $2^{nd}$ edition, Oxford University Press, Oxford, 2005. [5] L. B. Cook, M. Elemans, A. G. Rowan and B. Asquith, HTLV-I: Persistence and pathogenesis, Virology, 435 (2013), 131-140. doi: 10.1016/j.virol.2012.09.028. [6] F. A. Proietti, A. B. F.Carneiro-Proietti, B. C. Catalan-Soares and E. L. MURPHY, Global epidemiology of HTLV-I infection and associated deseases, Oncogene, 24 (2005), 6058-6068. [7] D. Wodarz and C. R. M. Bangham, Evolutionary dynamics of HTLV-I, Journal of Molecular Evolution, 50 (2000), 448-455. [8] J. E. Kaplan, M. Osame, H. Kubota, A. Igata, H. Nishitani, Y. Maeda, R. F. Khabbaz and R. S. Janssen, The risk of development of HTLV-I-associted myelopathy/tropocal spastic paraparesis among persons infected with HTLV-I, Journal of Acquired Immune Deficiency Syndromes, 3 (1990), 1096-1101. [9] C. Pique and K. S. Jones, Pathways of cell-cell transmission of HTLV-I, Frontiers in Microbiology, 3 (2012), p378. doi: 10.3389/fmicb.2012.00378. [10] M. Nagai, K. Usuku, W. Matsumoto, D. Kodama, N. Takenouchi, T. Moritoyo, S. Hashiguchi, M. Ichinose, R. M. Bangham, S. Izumo and M. Osame, Analysis of HTLV-I proviral load in 202 HAM/TSP patients and 243 asymptomatic HTLV-i carriers: high proviral load strongly predisposes to HAM/TSP, Journal of NeuroVirology, 4 (1998), 586-593. [11] Y. Ina and T. Gogobori, Molecular evoluton of human T-cell leukemia virus, Journal of Molecular Evolution, 31 (1990), 493-499. [12] R. Kubota, T. Fujiyoshi, S. Izumo, S. Yashiki, I. Maruyama, M. Osame and S. Sonoda, Fluctuation of HTLV-I proviral DNA in peripheral blood mononuclear cells of HTLV-I-associated myelopathy, Journal of NeuroVirology, 42 (1993), 147-154. doi: 10.1016/0165-5728(93)90004-I. [13] H. GÓmez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of $CD4^+$ T cells, Bulletin of Mathematical Biology, 67 (2005), 101-114. doi: 10.1016/j.bulm.2004.06.004. [14] F. Mortreux, A. S. Gabet and E. Wattel, Molecular and cellular aspects of HTLV-I associated leukemogenesis in vivo, Leukemia, 17 (2003), 26-38. doi: 10.1038/sj.leu.2402777. [15] P. Hollsberg, Mechanisms of T-cell activation by human T-cell lymphotropic virus type 1, Microbiology and Molecular Biology Reviews, 63 (1999), 308-333. [16] J. Mesnard and C. Devaux, Multiple control levels of cell proliferation by human T-cell leukemia virus type 1 Tax protein, Virology, 257 (1999), 277-284. doi: 10.1006/viro.1999.9685. [17] F. Bex and R. Gaynor, Regulation of gene expression by HTLV-I Tax protein, Methods, 16 (1998), 83-94. doi: 10.1006/meth.1998.0646. [18] M. Yoshida, Multiple viral strategies of HTLV-I for dysregulation of cell growth control, Annual Review of Immunology, 19 (2001), 475-496. [19] B. Asquith, A. J. Mosley, A. Heaps, Y. Tanaka, G. P. Taylor, A. R. Mclean and C. R. M Bangham, Quantification of the virus-host interaction in human T lymphotropic virus 1 infection, Retrovirology, 75 (2005), 1-9. [20] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74. [21] D. Wodarz, M. A. Nowak and C. R. M. Bangham, The dynamics of HTLV-I and the CTL response, Immunology Today, 20 (1999), 220-227. doi: 10.1016/S0167-5699(99)01446-2. [22] M. Y. Li and A. G. Lim, Modelling the role of tax expression in HTLV-I persistence in vivo, Bulletin of Mathematical Biology, 73 (2011), 3008-3029. doi: 10.1007/s11538-011-9657-1. [23] S. Li and Y. Zhou, Global dynamics of an HTLV-I model with cell-to-cell infection and mitosis, Abstract and Applied Analysis, 2014 (2014), Art. ID 132781, 12 pp. doi: 10.1155/2014/132781. [24] H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multistability in a model for CTL Response to HTLV-I infection and its implications to HAM/TSP decelopment and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. [25] M. Y. 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. [26] M. Y. Li and H. Shu, Multiple stable peridic oscillations in a mathematical model of CTL response to HTLVE-I infection, Bulletin of Mathematical Biology, 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7. [27] J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, Mathematical Biology, 65 (2012), 181-199. doi: 10.1007/s00285-011-0455-z. [28] A. G. Lim and P. K. Maini, HTLV-I infection: A dynamica struggle between viral persistence and host immunity, Journal of Theoretical Biology, 352 (2014), 92-108. doi: 10.1016/j.jtbi.2014.02.022. [29] B. Asquith, Y. Zhang, A. J. Mosley, C. M. d.Lara, D. L. Wallace, A. Worth, L. Kaftantzi, K. Meekings, G. E. Griffin, Y. Tanaka, D. F. Tough, P. C. Beverley, G. P. Taylor, D. C. Macallan and C. R. Bangham, In vivo T lymphocyte dynamics in humans and the impact of human T-lymphotropic virus 1 infection, Proceedings of the National Academy of Sciences, 104 (2007), 8035-8040. doi: 10.1073/pnas.0608832104. [30] X. Yang and L. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers and Mathematics with Applications, 32 (1996), 109-116. doi: 10.1016/0898-1221(96)00129-0. [31] P. v. d. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical. Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [32] J. P. LaSalle, The stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, 1976. [33] A. Melamed, D. J. Laydon, N. A. Gillet, Y. Tanaka, G. P. Taylor and C. R. Bangham, Genome-wide determinants of proviral targeting, clonal abundance and expression in natural HTLV-I infection, PLoS pathogens, 9 (2013), e1003271. doi: 10.1371/journal.ppat.1003271.
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