# American Institute of Mathematical Sciences

2015, 12(3): 431-449. doi: 10.3934/mbe.2015.12.431

## A mathematical model of HTLV-I infection with two time delays

 1 Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, 3041#, 2 Yi-Kuang street, Harbin, 150080, China, China 2 Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080 3 Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received  December 2013 Revised  December 2014 Published  January 2015

In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxic T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection, where one is the intracellular infection delay and the other is the immune delay to account for a series of immunological events leading to the CTL response. We show that the global dynamics of the model system are determined by two threshold values $R_0$, the corresponding reproductive number of a viral infection, and $R_1$, the corresponding reproductive number of a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globally asymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-free equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no persistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-I infection becomes chronic with a persistent CTL response. Moreover, we show that the immune delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical simulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize the HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the stability of the HAM/TSP equilibrium may generate rich dynamics combining the stabilizing" effects from the intracellular delay with those destabilizing" influences from immune delay.
Citation: Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li. A mathematical model of HTLV-I infection with two time delays. Mathematical Biosciences & Engineering, 2015, 12 (3) : 431-449. doi: 10.3934/mbe.2015.12.431
##### References:
 [1] B. Asquit and C. R. M. Bangham, Quantifying HTLV-I dynamics, Immunol. Cell Biol., 85 (2007), 280-286. doi: 10.1038/sj.icb.7100050. [2] A. J. Cann and I. S. Y. Chen, Human T-cell leukemia virus type I and II, in Fields (eds. B.N. Knipe, D.M. Howley, P.M.), Lippincott-Raven Publishers, (1996), 1849-1880. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [4] A. Gessain, F. Barin, J. C. Vernant, O. Gout, L. Maurs, A. Calender and G. de Thé, Antibodies to human T-lymphotropic virus type-I in patient with tropical spastic paraparesis, Lancet, 326 (1985), 407-410. doi: 10.1016/S0140-6736(85)92734-5. [5] T. Greten and J. Slansky et al, Direct visualization of antigen-specific T cells: HTLV-I Tax 11-19-specific CD8$^+$ cells are activated in peripheral blood and accumulate in cerebrospinal fluid from HAM/TSP patients, Proc Natl Acad Sci USA, 95 (1998), 7568-7573. [6] H. Gómez-Acevedo, M. Y. Li and S. Jacobson, Multistability in a model for CLT response to HTLV-I infection and its implications to HAM/TSP development and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. [7] K. Gu, S. I. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253. doi: 10.1016/j.jmaa.2005.02.034. [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-associated myelopathy/tropical spastic paraparesis among persons infected with HTLV-I, J. Acquir. Immune Defic. Syndr., 3 (1990), 1096-1011. [9] 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. [10] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961. [11] M. Y. Li and H. 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. [12] 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. [13] S. Liu and L. Wang, Global Stability of an HIV-1 Model with Distributed Intracellular Delays and a Combination Therapy, Mathematical Biosciences and Engineering, 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [14] Y. Muroya, Y. Enatsu and H. Li, Global stability of a delayed HTLV-I infection model with a class of nonlinear incidence rates and CTLs immune response, Applied Mathematics and Computation, 219 (2013), 10559-10573. doi: 10.1016/j.amc.2013.03.081. [15] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-I pathogenesis that includes an intracellular delay, Math.Biosci, 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [16] M. A. Nowak and R. M. May, Virus Dynamics:Mathematical Principles of Immunology and Virology, Oxford University Press, London, 2000. [17] K. Okochi, H. Sato and Y. Hinuma, A retrospective study on transmission of adult T-cell leukemia virus by blood transfusion:seroconversion in recipients, Vox Sang, 46 (1984), 245-253. doi: 10.1111/j.1423-0410.1984.tb00083.x. [18] M. Osame, K. Usuku, S. Izumo, N. Ijichi, H. Aminati, A. Igata, M. Matsumoto and M. Tara, HTLV-I-associated myelopathy: A new clinical entity, Lancet, 327 (1986), 1031-1032. doi: 10.1016/S0140-6736(86)91298-5. [19] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math.Biosci, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [20] A. S. Perelson, Modeling the interaction of the immune system with HIV, In Castillo-Chavez,C.(Ed),Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, 83 (1989), 350-370, Springer, Berlin. doi: 10.1007/978-3-642-93454-4_17. [21] V. Rebecca, W. Culsha and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cell, Mathematical Biosciences, 165 (2000), 27-39. [22] J. H. Richardson, A. J. Edwards, J. K. Cruickshank, P. Rudge and A. G. Dalgleish, In vivo cellular tropism of human T-cell leukemia virus type 1, J.Virol, 64 (1990), 5682-5687. [23] H. Shiraki, Y. Sagara and Y. Inoue, Cell-to-cell transmission of HTLV-I, in Two Decades of Adult T-cell Leukemia and HTLV-I Research (eds. K. Sugamura, T. Uehiyam, M. Matsuoka and M. Kannagi), Japan Scientific Societies, Tokyo, (2003), 303-316. [24] H. Song, W. Jiang and S. Liu, Virus Dynamics model with intracellular delays and immune response, Mathematical Biosciences and Engineering, 12 (2015), 185-208. [25] X. Sun and J. Wei, Global dynamics of a HTLV-I infection model with CTL response, Electronic Journal of Qualitative Theory of Differential Equations, 40 (2013), 1-15. [26] 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. [27] X. Wang, Y. Chen, S. Liu and X. Song, A class of delayed virus dynamics models with multiple target cells, Computational and Applied Mathematics, 32 (2013), 211-229. doi: 10.1007/s40314-013-0004-z. [28] Y. Yamano, M. Nagai, M. Brennan, C. A. Mora, S. S. Soldan, U. Tomaru, N. Takenouchi, S. Izumo, M. Osame and S. Jacobson, Correlation of human T-cell lymphotropic virus type 1(HTLV-1) mRNA with proviral DNA load, virus-specific $CD8^+$ T cells and disease severity in HTLV-1-associated myelopathy(HAM/TSP), Blood, 99 (2002), 88-94. doi: 10.1182/blood.V99.1.88. [29] X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.

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##### References:
 [1] B. Asquit and C. R. M. Bangham, Quantifying HTLV-I dynamics, Immunol. Cell Biol., 85 (2007), 280-286. doi: 10.1038/sj.icb.7100050. [2] A. J. Cann and I. S. Y. Chen, Human T-cell leukemia virus type I and II, in Fields (eds. B.N. Knipe, D.M. Howley, P.M.), Lippincott-Raven Publishers, (1996), 1849-1880. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of diseases transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [4] A. Gessain, F. Barin, J. C. Vernant, O. Gout, L. Maurs, A. Calender and G. de Thé, Antibodies to human T-lymphotropic virus type-I in patient with tropical spastic paraparesis, Lancet, 326 (1985), 407-410. doi: 10.1016/S0140-6736(85)92734-5. [5] T. Greten and J. Slansky et al, Direct visualization of antigen-specific T cells: HTLV-I Tax 11-19-specific CD8$^+$ cells are activated in peripheral blood and accumulate in cerebrospinal fluid from HAM/TSP patients, Proc Natl Acad Sci USA, 95 (1998), 7568-7573. [6] H. Gómez-Acevedo, M. Y. Li and S. Jacobson, Multistability in a model for CLT response to HTLV-I infection and its implications to HAM/TSP development and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z. [7] K. Gu, S. I. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253. doi: 10.1016/j.jmaa.2005.02.034. [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-associated myelopathy/tropical spastic paraparesis among persons infected with HTLV-I, J. Acquir. Immune Defic. Syndr., 3 (1990), 1096-1011. [9] 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. [10] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, Academic Press, New York, 1961. [11] M. Y. Li and H. 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. [12] 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. [13] S. Liu and L. Wang, Global Stability of an HIV-1 Model with Distributed Intracellular Delays and a Combination Therapy, Mathematical Biosciences and Engineering, 7 (2010), 675-685. doi: 10.3934/mbe.2010.7.675. [14] Y. Muroya, Y. Enatsu and H. Li, Global stability of a delayed HTLV-I infection model with a class of nonlinear incidence rates and CTLs immune response, Applied Mathematics and Computation, 219 (2013), 10559-10573. doi: 10.1016/j.amc.2013.03.081. [15] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-I pathogenesis that includes an intracellular delay, Math.Biosci, 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [16] M. A. Nowak and R. M. May, Virus Dynamics:Mathematical Principles of Immunology and Virology, Oxford University Press, London, 2000. [17] K. Okochi, H. Sato and Y. Hinuma, A retrospective study on transmission of adult T-cell leukemia virus by blood transfusion:seroconversion in recipients, Vox Sang, 46 (1984), 245-253. doi: 10.1111/j.1423-0410.1984.tb00083.x. [18] M. Osame, K. Usuku, S. Izumo, N. Ijichi, H. Aminati, A. Igata, M. Matsumoto and M. Tara, HTLV-I-associated myelopathy: A new clinical entity, Lancet, 327 (1986), 1031-1032. doi: 10.1016/S0140-6736(86)91298-5. [19] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math.Biosci, 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [20] A. S. Perelson, Modeling the interaction of the immune system with HIV, In Castillo-Chavez,C.(Ed),Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, 83 (1989), 350-370, Springer, Berlin. doi: 10.1007/978-3-642-93454-4_17. [21] V. Rebecca, W. Culsha and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cell, Mathematical Biosciences, 165 (2000), 27-39. [22] J. H. Richardson, A. J. Edwards, J. K. Cruickshank, P. Rudge and A. G. Dalgleish, In vivo cellular tropism of human T-cell leukemia virus type 1, J.Virol, 64 (1990), 5682-5687. [23] H. Shiraki, Y. Sagara and Y. Inoue, Cell-to-cell transmission of HTLV-I, in Two Decades of Adult T-cell Leukemia and HTLV-I Research (eds. K. Sugamura, T. Uehiyam, M. Matsuoka and M. Kannagi), Japan Scientific Societies, Tokyo, (2003), 303-316. [24] H. Song, W. Jiang and S. Liu, Virus Dynamics model with intracellular delays and immune response, Mathematical Biosciences and Engineering, 12 (2015), 185-208. [25] X. Sun and J. Wei, Global dynamics of a HTLV-I infection model with CTL response, Electronic Journal of Qualitative Theory of Differential Equations, 40 (2013), 1-15. [26] 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. [27] X. Wang, Y. Chen, S. Liu and X. Song, A class of delayed virus dynamics models with multiple target cells, Computational and Applied Mathematics, 32 (2013), 211-229. doi: 10.1007/s40314-013-0004-z. [28] Y. Yamano, M. Nagai, M. Brennan, C. A. Mora, S. S. Soldan, U. Tomaru, N. Takenouchi, S. Izumo, M. Osame and S. Jacobson, Correlation of human T-cell lymphotropic virus type 1(HTLV-1) mRNA with proviral DNA load, virus-specific $CD8^+$ T cells and disease severity in HTLV-1-associated myelopathy(HAM/TSP), Blood, 99 (2002), 88-94. doi: 10.1182/blood.V99.1.88. [29] X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications, Can. Appl. Math. Q., 3 (1995), 473-495.
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