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.  doi: 10.1038/sj.icb.7100050.  Google Scholar

[2]

A. J. Cann and I. S. Y. Chen, Human T-cell leukemia virus type I and II,, in Fields (eds. B.N. Knipe, (1996), 1849.   Google Scholar

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

[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.  doi: 10.1016/S0140-6736(85)92734-5.  Google Scholar

[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.   Google Scholar

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

[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.  doi: 10.1016/j.jmaa.2005.02.034.  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-associated myelopathy/tropical spastic paraparesis among persons infected with HTLV-I,, J. Acquir. Immune Defic. Syndr., 3 (1990), 1096.   Google Scholar

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

[10]

J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method,, Academic Press, (1961).   Google Scholar

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

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

[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.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

[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.  doi: 10.1016/j.amc.2013.03.081.  Google Scholar

[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.  doi: 10.1016/S0025-5564(99)00055-3.  Google Scholar

[16]

M. A. Nowak and R. M. May, Virus Dynamics:Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[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.  doi: 10.1111/j.1423-0410.1984.tb00083.x.  Google Scholar

[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.  doi: 10.1016/S0140-6736(86)91298-5.  Google Scholar

[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.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[20]

A. S. Perelson, Modeling the interaction of the immune system with HIV,, In Castillo-Chavez, 83 (1989), 350.  doi: 10.1007/978-3-642-93454-4_17.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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, (2003), 303.   Google Scholar

[24]

H. Song, W. Jiang and S. Liu, Virus Dynamics model with intracellular delays and immune response,, Mathematical Biosciences and Engineering, 12 (2015), 185.   Google Scholar

[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.   Google Scholar

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

[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.  doi: 10.1007/s40314-013-0004-z.  Google Scholar

[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.  doi: 10.1182/blood.V99.1.88.  Google Scholar

[29]

X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications,, Can. Appl. Math. Q., 3 (1995), 473.   Google Scholar

show all references

References:
[1]

B. Asquit and C. R. M. Bangham, Quantifying HTLV-I dynamics,, Immunol. Cell Biol., 85 (2007), 280.  doi: 10.1038/sj.icb.7100050.  Google Scholar

[2]

A. J. Cann and I. S. Y. Chen, Human T-cell leukemia virus type I and II,, in Fields (eds. B.N. Knipe, (1996), 1849.   Google Scholar

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

[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.  doi: 10.1016/S0140-6736(85)92734-5.  Google Scholar

[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.   Google Scholar

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

[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.  doi: 10.1016/j.jmaa.2005.02.034.  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-associated myelopathy/tropical spastic paraparesis among persons infected with HTLV-I,, J. Acquir. Immune Defic. Syndr., 3 (1990), 1096.   Google Scholar

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

[10]

J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method,, Academic Press, (1961).   Google Scholar

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

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

[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.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

[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.  doi: 10.1016/j.amc.2013.03.081.  Google Scholar

[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.  doi: 10.1016/S0025-5564(99)00055-3.  Google Scholar

[16]

M. A. Nowak and R. M. May, Virus Dynamics:Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[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.  doi: 10.1111/j.1423-0410.1984.tb00083.x.  Google Scholar

[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.  doi: 10.1016/S0140-6736(86)91298-5.  Google Scholar

[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.  doi: 10.1016/j.mbs.2011.11.002.  Google Scholar

[20]

A. S. Perelson, Modeling the interaction of the immune system with HIV,, In Castillo-Chavez, 83 (1989), 350.  doi: 10.1007/978-3-642-93454-4_17.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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, (2003), 303.   Google Scholar

[24]

H. Song, W. Jiang and S. Liu, Virus Dynamics model with intracellular delays and immune response,, Mathematical Biosciences and Engineering, 12 (2015), 185.   Google Scholar

[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.   Google Scholar

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

[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.  doi: 10.1007/s40314-013-0004-z.  Google Scholar

[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.  doi: 10.1182/blood.V99.1.88.  Google Scholar

[29]

X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflow with applications,, Can. Appl. Math. Q., 3 (1995), 473.   Google Scholar

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