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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

Sumei Li 1, and  Yicang Zhou 2,

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.
Keywords: HTLV-I, tax expression, CTL immune response, periodic solutions., backward bifurcation.
Mathematics Subject Classification: Primary: 34K18, 34K60; Secondary: 34K20, 92D3.
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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