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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 & 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.  doi: 10.1016/j.it.2007.09.006.  Google Scholar

[2]

C. R. M. Bangham, HTLV-I infections,, Journal of Clinical Pathology, 53 (2000), 581.   Google Scholar

[3]

R. C. Gallo, The discovery of the first human retrovirus: HTLV-1 and HTLV-2,, Rrtrovirology, 2 (2005), 17.   Google Scholar

[4]

U. Tomaru, Y. Yamano and S. Jacobson, HTLV-I Infection and the Nervous System In Clinical Neuroimmunology,, $2^{nd}$ edition, (2005).   Google Scholar

[5]

L. B. Cook, M. Elemans, A. G. Rowan and B. Asquith, HTLV-I: Persistence and pathogenesis,, Virology, 435 (2013), 131.  doi: 10.1016/j.virol.2012.09.028.  Google Scholar

[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.   Google Scholar

[7]

D. Wodarz and C. R. M. Bangham, Evolutionary dynamics of HTLV-I,, Journal of Molecular Evolution, 50 (2000), 448.   Google Scholar

[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.   Google Scholar

[9]

C. Pique and K. S. Jones, Pathways of cell-cell transmission of HTLV-I,, Frontiers in Microbiology, 3 (2012).  doi: 10.3389/fmicb.2012.00378.  Google Scholar

[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.   Google Scholar

[11]

Y. Ina and T. Gogobori, Molecular evoluton of human T-cell leukemia virus,, Journal of Molecular Evolution, 31 (1990), 493.   Google Scholar

[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.  doi: 10.1016/0165-5728(93)90004-I.  Google Scholar

[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.  doi: 10.1016/j.bulm.2004.06.004.  Google Scholar

[14]

F. Mortreux, A. S. Gabet and E. Wattel, Molecular and cellular aspects of HTLV-I associated leukemogenesis in vivo,, Leukemia, 17 (2003), 26.  doi: 10.1038/sj.leu.2402777.  Google Scholar

[15]

P. Hollsberg, Mechanisms of T-cell activation by human T-cell lymphotropic virus type 1,, Microbiology and Molecular Biology Reviews, 63 (1999), 308.   Google Scholar

[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.  doi: 10.1006/viro.1999.9685.  Google Scholar

[17]

F. Bex and R. Gaynor, Regulation of gene expression by HTLV-I Tax protein,, Methods, 16 (1998), 83.  doi: 10.1006/meth.1998.0646.  Google Scholar

[18]

M. Yoshida, Multiple viral strategies of HTLV-I for dysregulation of cell growth control,, Annual Review of Immunology, 19 (2001), 475.   Google Scholar

[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.   Google Scholar

[20]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses,, Science, 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar

[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.  doi: 10.1016/S0167-5699(99)01446-2.  Google Scholar

[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.  doi: 10.1007/s11538-011-9657-1.  Google Scholar

[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).  doi: 10.1155/2014/132781.  Google Scholar

[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.  doi: 10.1007/s11538-009-9465-z.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar

[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.  doi: 10.1007/s11538-010-9591-7.  Google Scholar

[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.  doi: 10.1007/s00285-011-0455-z.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2014.02.022.  Google Scholar

[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.  doi: 10.1073/pnas.0608832104.  Google Scholar

[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.  doi: 10.1016/0898-1221(96)00129-0.  Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[32]

J. P. LaSalle, The stability of Dynamical systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[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).  doi: 10.1371/journal.ppat.1003271.  Google Scholar

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.  doi: 10.1016/j.it.2007.09.006.  Google Scholar

[2]

C. R. M. Bangham, HTLV-I infections,, Journal of Clinical Pathology, 53 (2000), 581.   Google Scholar

[3]

R. C. Gallo, The discovery of the first human retrovirus: HTLV-1 and HTLV-2,, Rrtrovirology, 2 (2005), 17.   Google Scholar

[4]

U. Tomaru, Y. Yamano and S. Jacobson, HTLV-I Infection and the Nervous System In Clinical Neuroimmunology,, $2^{nd}$ edition, (2005).   Google Scholar

[5]

L. B. Cook, M. Elemans, A. G. Rowan and B. Asquith, HTLV-I: Persistence and pathogenesis,, Virology, 435 (2013), 131.  doi: 10.1016/j.virol.2012.09.028.  Google Scholar

[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.   Google Scholar

[7]

D. Wodarz and C. R. M. Bangham, Evolutionary dynamics of HTLV-I,, Journal of Molecular Evolution, 50 (2000), 448.   Google Scholar

[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.   Google Scholar

[9]

C. Pique and K. S. Jones, Pathways of cell-cell transmission of HTLV-I,, Frontiers in Microbiology, 3 (2012).  doi: 10.3389/fmicb.2012.00378.  Google Scholar

[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.   Google Scholar

[11]

Y. Ina and T. Gogobori, Molecular evoluton of human T-cell leukemia virus,, Journal of Molecular Evolution, 31 (1990), 493.   Google Scholar

[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.  doi: 10.1016/0165-5728(93)90004-I.  Google Scholar

[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.  doi: 10.1016/j.bulm.2004.06.004.  Google Scholar

[14]

F. Mortreux, A. S. Gabet and E. Wattel, Molecular and cellular aspects of HTLV-I associated leukemogenesis in vivo,, Leukemia, 17 (2003), 26.  doi: 10.1038/sj.leu.2402777.  Google Scholar

[15]

P. Hollsberg, Mechanisms of T-cell activation by human T-cell lymphotropic virus type 1,, Microbiology and Molecular Biology Reviews, 63 (1999), 308.   Google Scholar

[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.  doi: 10.1006/viro.1999.9685.  Google Scholar

[17]

F. Bex and R. Gaynor, Regulation of gene expression by HTLV-I Tax protein,, Methods, 16 (1998), 83.  doi: 10.1006/meth.1998.0646.  Google Scholar

[18]

M. Yoshida, Multiple viral strategies of HTLV-I for dysregulation of cell growth control,, Annual Review of Immunology, 19 (2001), 475.   Google Scholar

[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.   Google Scholar

[20]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses,, Science, 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar

[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.  doi: 10.1016/S0167-5699(99)01446-2.  Google Scholar

[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.  doi: 10.1007/s11538-011-9657-1.  Google Scholar

[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).  doi: 10.1155/2014/132781.  Google Scholar

[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.  doi: 10.1007/s11538-009-9465-z.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2011.02.026.  Google Scholar

[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.  doi: 10.1007/s11538-010-9591-7.  Google Scholar

[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.  doi: 10.1007/s00285-011-0455-z.  Google Scholar

[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.  doi: 10.1016/j.jtbi.2014.02.022.  Google Scholar

[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.  doi: 10.1073/pnas.0608832104.  Google Scholar

[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.  doi: 10.1016/0898-1221(96)00129-0.  Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[32]

J. P. LaSalle, The stability of Dynamical systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[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).  doi: 10.1371/journal.ppat.1003271.  Google Scholar

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