American Institute of Mathematical Sciences

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

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

Xuejuan Lu 1, , Lulu Hui 1, , Shengqiang Liu 2, and  Jia Li 3,

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.
Keywords: Hopf bifurcation., Epidemic threshold, HTLV-I infection, Lyapunov functional, time delay.
Mathematics Subject Classification: Primary: 34K20, 34K60; Secondary: 34K1.
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.

show all references

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.

[1]

Lianwen Wang, Zhijun Liu, Yong Li, Dashun Xu. Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 917-933. doi: 10.3934/dcdsb.2019196
[2]

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
[3]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[4]

Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069
[5]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[6]

Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044
[7]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[8]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[9]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[10]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169
[11]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
[12]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[13]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[14]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[15]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[16]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[17]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[18]

Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006
[19]

Masaki Sekiguchi, Emiko Ishiwata, Yukihiko Nakata. Dynamics of an ultra-discrete SIR epidemic model with time delay. Mathematical Biosciences & Engineering, 2018, 15 (3) : 653-666. doi: 10.3934/mbe.2018029
[20]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy