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January  2016, 21(1): 103-119. doi: 10.3934/dcdsb.2016.21.103

Global behavior of delay differential equations model of HIV infection with apoptosis

1. 

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received  February 2015 Revised  May 2015 Published  November 2015

In this paper, a class of delay differential equations model of HIV infection dynamics with nonlinear transmissions and apoptosis induced by infected cells is proposed, and then the global properties of the model are considered. It shows that the infection-free equilibrium of the model is globally asymptotically stable if the basic reproduction number $R_{0}<1$, and globally attractive if $R_{0}=1$. The positive equilibrium of the model is locally asymptotically stable if $R_{0}>1$. Furthermore, it also shows that the model is permanent, and some explicit expressions for the eventual lower bounds of positive solutions of the model are given.
Citation: Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103
References:
[1]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts,, Philos. T. R. Soc. B, 291 (1981), 451. doi: 10.1098/rstb.1981.0005.

[2]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics,, Math. Biosci., 183 (2003), 63. doi: 10.1016/S0025-5564(02)00218-3.

[3]

A. L. Cunningham, H. Donaghy, A. N. Harman, M. Kim and S. G. Turville, Manipulation of dendritic cell function by viruses,, Curr. Opin. Microbiol., 13 (2010), 524. doi: 10.1016/j.mib.2010.06.002.

[4]

M. Carbonari, M. Cibati, A. M. Pesce, D. Sbarigia, P. Grossi, G. D'Offizi, G. Luzi and M. Fiorilli, Frequency of provirus-bearing CD4$^+$ cells in HIV type 1 infection correlates with extent of in vitro apoptosis of CD8$^+$ but not of CD4$^+$ cells,, AIDS Res. Hum. Retrov., 11 (1995), 789.

[5]

L. Conti, G. Rainaldi, P. Matarrese, B. Varano, R. Rivabene, S. Columba, A. Sato, F. Belardelli, W. Malorni and S. Gessani, The HIV-1 vpr protein acts as a negative regulator of apoptosis in a human lymphoblastoid T cell line: Possible implications for the pathogenesis of AIDS,, J. Exp. Med., 187 (1998), 403. doi: 10.1084/jem.187.3.403.

[6]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay,, J. Math. Biol., 46 (2003), 425. doi: 10.1007/s00285-002-0191-5.

[7]

W. Cheng, W. Ma and S. Guo, A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis,, Commun. Pur. Appl. Anal., ().

[8]

O. Diekmann, S. A. van Oils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2.

[9]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[10]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[11]

J. Embretson, M. Zupancic, J. L. Ribas, A. Burke, P. Racz, K. T.-Racz and A. T. Haase, Massive covert infection of helper T lymphocytes and macrophages by HIV during the incubation period of AIDS,, Nature, 362 (1993), 359. doi: 10.1038/362359a0.

[12]

B. Ensoli, G. Barillari, S. Z. Salahuddin, R. C. Gallo and F. W.-Staal, Tat protein of HIV-1 stimulates growth of cells derived from Kaposi's sarcoma lesions of AIDS patients,, Nature, 345 (1990), 84. doi: 10.1038/345084a0.

[13]

Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model,, Nonlinear Anal.-Real, 13 (2012), 2120. doi: 10.1016/j.nonrwa.2012.01.007.

[14]

A. M. Elaiw and N. H. AlShamrani, Global properties of nonlinear humoral immunity viral infection models,, Int. J. Biomath., 8 (2015). doi: 10.1142/S1793524515500588.

[15]

H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations,, J. Differ. Equations, 115 (1995), 173. doi: 10.1006/jdeq.1995.1011.

[16]

Z. Feng and L. Rong, The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens,, DIMACS Series in Discrete Math. Theor., 71 (2006), 161.

[17]

R. Fan, Y. Dong, G. Huang and Y. Takeuchi, Apoptosis in virus infection dynamics models,, J. Biol. Dyn., 8 (2014), 20. doi: 10.1080/17513758.2014.895433.

[18]

H. Garg, J. Mohl and A. Joshi, HIV-1 induced bystander apoptosis,, Viruses, 4 (2012), 3020. doi: 10.3390/v4113020.

[19]

M.-L. Gougeon, H. Lecoeur, A. Dulioust, M.-G. Enouf, M. Crouvoiser, C. Goujard, T. Debord and L. Montagnier, Programmed cell death in peripheral lymphocytes from HIV-infected persons: increased susceptibility to apoptosis of CD4 and CD8 T cells correlates with lymphocyte activation and with disease progression,, J. Immunol., 156 (1996), 3509.

[20]

S. B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760. doi: 10.1137/0134064.

[21]

M. Heinkelein, S. Sopper and C. Jassoy, Contact of human immunodeficiency virus type 1-infected and uninfected CD4$^+$ T lymphocytes is highly cytolytic for both cells,, J. Virol., 69 (1995), 6925.

[22]

A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, P. Natl. Acad. Sci. USA, 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247.

[23]

G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004.

[24]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. Appl. Math., 70 (2010), 2693. doi: 10.1137/090780821.

[25]

G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics,, Jpn. J. Ind. Appl. Math., 28 (2011), 383. doi: 10.1007/s13160-011-0045-x.

[26]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dyn. Differ. Equ., 13 (2001), 107. doi: 10.1023/A:1009044515567.

[27]

J. K. Hale, P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025.

[28]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7.

[29]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).

[30]

A. Korobeinikov, Global properties of basic virus dynamics models,, B. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001.

[31]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, B. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y.

[32]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, B. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x.

[33]

C. J. Li, D. J. Friedman, C. Wang, V. Metelev and A. B. Pardee, Induction of apoptosis in uninfected lymphocytes by HIV-1 Tat protein,, Science, 268 (1995), 429. doi: 10.1126/science.7716549.

[34]

X. Li and S. Fu, Global stability of the virus dynamics model with intracellular delay and Crowley-Martin functional response,, Math. Method. Appl. Sci., 37 (2014), 1405. doi: 10.1002/mma.2895.

[35]

X. Lai and X. Zou, Modeling HIV-1 Virus Dynamics with Both Virus-to-Cell Infection and Cell-to-Cell Transmission,, SIAM J. Appl. Math., 74 (2014), 898. doi: 10.1137/130930145.

[36]

X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,, J. Math. Anal. Appl., 426 (2015), 563. doi: 10.1016/j.jmaa.2014.10.086.

[37]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Eng., 7 (2010), 837. doi: 10.3934/mbe.2010.7.837.

[38]

B. Nardelli, C. J. Gonzalez, M. Schechter and F. T. Valentine, CD4$^+$ blood lymphocytes are rapidly killed in vitro by contact with autologous human immunodeficiency virus-infected cells,, P. Natl. Acad. Sci. USA, 92 (1995), 7312. doi: 10.1073/pnas.92.16.7312.

[39]

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

[40]

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

[41]

P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3.

[42]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8.

[43]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107.

[44]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945.

[45]

N. Selliah and T. H. Finkel, Biochemical mechanisms of HIV induced T cell apoptosis,, Cell Death Differ., 8 (2001), 127. doi: 10.1038/sj.cdd.4400822.

[46]

H. Shu, L. Wang and J. Watmough, Global Stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses,, SIAM J. Appl. Math., 73 (2013), 1280. doi: 10.1137/120896463.

[47]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal.-Theor., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2.

[48]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026.

[49]

J. Wu and X.-Q. Zhao, Permanence and convergence in multi-species competition systems with delay,, P. Am. Math. Soc., 126 (1998), 1709. doi: 10.1090/S0002-9939-98-04522-5.

[50]

W. Wang, Global behavior of an SEIRS epidemic model with time delays,, Appl. Math. Lett., 15 (2002), 423. doi: 10.1016/S0893-9659(01)00153-7.

[51]

X. Wang, S. Liu and X. Song, Dynamics of a non-autonomous HIV-1 infection model with delays,, Int. J. Biomath., 6 (2013). doi: 10.1142/S1793524513500307.

[52]

R. A. Weiss, How does HIV cause AIDS?,, Science, 260 (1993), 1273. doi: 10.1126/science.8493571.

[53]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay,, J. Math. Anal. Appl., 375 (2011), 75. doi: 10.1016/j.jmaa.2010.08.055.

[54]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts,, Philos. T. R. Soc. B, 291 (1981), 451. doi: 10.1098/rstb.1981.0005.

[2]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics,, Math. Biosci., 183 (2003), 63. doi: 10.1016/S0025-5564(02)00218-3.

[3]

A. L. Cunningham, H. Donaghy, A. N. Harman, M. Kim and S. G. Turville, Manipulation of dendritic cell function by viruses,, Curr. Opin. Microbiol., 13 (2010), 524. doi: 10.1016/j.mib.2010.06.002.

[4]

M. Carbonari, M. Cibati, A. M. Pesce, D. Sbarigia, P. Grossi, G. D'Offizi, G. Luzi and M. Fiorilli, Frequency of provirus-bearing CD4$^+$ cells in HIV type 1 infection correlates with extent of in vitro apoptosis of CD8$^+$ but not of CD4$^+$ cells,, AIDS Res. Hum. Retrov., 11 (1995), 789.

[5]

L. Conti, G. Rainaldi, P. Matarrese, B. Varano, R. Rivabene, S. Columba, A. Sato, F. Belardelli, W. Malorni and S. Gessani, The HIV-1 vpr protein acts as a negative regulator of apoptosis in a human lymphoblastoid T cell line: Possible implications for the pathogenesis of AIDS,, J. Exp. Med., 187 (1998), 403. doi: 10.1084/jem.187.3.403.

[6]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay,, J. Math. Biol., 46 (2003), 425. doi: 10.1007/s00285-002-0191-5.

[7]

W. Cheng, W. Ma and S. Guo, A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis,, Commun. Pur. Appl. Anal., ().

[8]

O. Diekmann, S. A. van Oils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2.

[9]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[10]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[11]

J. Embretson, M. Zupancic, J. L. Ribas, A. Burke, P. Racz, K. T.-Racz and A. T. Haase, Massive covert infection of helper T lymphocytes and macrophages by HIV during the incubation period of AIDS,, Nature, 362 (1993), 359. doi: 10.1038/362359a0.

[12]

B. Ensoli, G. Barillari, S. Z. Salahuddin, R. C. Gallo and F. W.-Staal, Tat protein of HIV-1 stimulates growth of cells derived from Kaposi's sarcoma lesions of AIDS patients,, Nature, 345 (1990), 84. doi: 10.1038/345084a0.

[13]

Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model,, Nonlinear Anal.-Real, 13 (2012), 2120. doi: 10.1016/j.nonrwa.2012.01.007.

[14]

A. M. Elaiw and N. H. AlShamrani, Global properties of nonlinear humoral immunity viral infection models,, Int. J. Biomath., 8 (2015). doi: 10.1142/S1793524515500588.

[15]

H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations,, J. Differ. Equations, 115 (1995), 173. doi: 10.1006/jdeq.1995.1011.

[16]

Z. Feng and L. Rong, The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens,, DIMACS Series in Discrete Math. Theor., 71 (2006), 161.

[17]

R. Fan, Y. Dong, G. Huang and Y. Takeuchi, Apoptosis in virus infection dynamics models,, J. Biol. Dyn., 8 (2014), 20. doi: 10.1080/17513758.2014.895433.

[18]

H. Garg, J. Mohl and A. Joshi, HIV-1 induced bystander apoptosis,, Viruses, 4 (2012), 3020. doi: 10.3390/v4113020.

[19]

M.-L. Gougeon, H. Lecoeur, A. Dulioust, M.-G. Enouf, M. Crouvoiser, C. Goujard, T. Debord and L. Montagnier, Programmed cell death in peripheral lymphocytes from HIV-infected persons: increased susceptibility to apoptosis of CD4 and CD8 T cells correlates with lymphocyte activation and with disease progression,, J. Immunol., 156 (1996), 3509.

[20]

S. B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760. doi: 10.1137/0134064.

[21]

M. Heinkelein, S. Sopper and C. Jassoy, Contact of human immunodeficiency virus type 1-infected and uninfected CD4$^+$ T lymphocytes is highly cytolytic for both cells,, J. Virol., 69 (1995), 6925.

[22]

A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, P. Natl. Acad. Sci. USA, 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247.

[23]

G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004.

[24]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. Appl. Math., 70 (2010), 2693. doi: 10.1137/090780821.

[25]

G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara and T. Sasaki, Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics,, Jpn. J. Ind. Appl. Math., 28 (2011), 383. doi: 10.1007/s13160-011-0045-x.

[26]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems,, J. Dyn. Differ. Equ., 13 (2001), 107. doi: 10.1023/A:1009044515567.

[27]

J. K. Hale, P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025.

[28]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7.

[29]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).

[30]

A. Korobeinikov, Global properties of basic virus dynamics models,, B. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001.

[31]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, B. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y.

[32]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, B. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x.

[33]

C. J. Li, D. J. Friedman, C. Wang, V. Metelev and A. B. Pardee, Induction of apoptosis in uninfected lymphocytes by HIV-1 Tat protein,, Science, 268 (1995), 429. doi: 10.1126/science.7716549.

[34]

X. Li and S. Fu, Global stability of the virus dynamics model with intracellular delay and Crowley-Martin functional response,, Math. Method. Appl. Sci., 37 (2014), 1405. doi: 10.1002/mma.2895.

[35]

X. Lai and X. Zou, Modeling HIV-1 Virus Dynamics with Both Virus-to-Cell Infection and Cell-to-Cell Transmission,, SIAM J. Appl. Math., 74 (2014), 898. doi: 10.1137/130930145.

[36]

X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,, J. Math. Anal. Appl., 426 (2015), 563. doi: 10.1016/j.jmaa.2014.10.086.

[37]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Eng., 7 (2010), 837. doi: 10.3934/mbe.2010.7.837.

[38]

B. Nardelli, C. J. Gonzalez, M. Schechter and F. T. Valentine, CD4$^+$ blood lymphocytes are rapidly killed in vitro by contact with autologous human immunodeficiency virus-infected cells,, P. Natl. Acad. Sci. USA, 92 (1995), 7312. doi: 10.1073/pnas.92.16.7312.

[39]

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

[40]

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

[41]

P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3.

[42]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8.

[43]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107.

[44]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945.

[45]

N. Selliah and T. H. Finkel, Biochemical mechanisms of HIV induced T cell apoptosis,, Cell Death Differ., 8 (2001), 127. doi: 10.1038/sj.cdd.4400822.

[46]

H. Shu, L. Wang and J. Watmough, Global Stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses,, SIAM J. Appl. Math., 73 (2013), 1280. doi: 10.1137/120896463.

[47]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems,, Nonlinear Anal.-Theor., 47 (2001), 6169. doi: 10.1016/S0362-546X(01)00678-2.

[48]

H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model),, SIAM J. Math. Anal., 24 (1993), 407. doi: 10.1137/0524026.

[49]

J. Wu and X.-Q. Zhao, Permanence and convergence in multi-species competition systems with delay,, P. Am. Math. Soc., 126 (1998), 1709. doi: 10.1090/S0002-9939-98-04522-5.

[50]

W. Wang, Global behavior of an SEIRS epidemic model with time delays,, Appl. Math. Lett., 15 (2002), 423. doi: 10.1016/S0893-9659(01)00153-7.

[51]

X. Wang, S. Liu and X. Song, Dynamics of a non-autonomous HIV-1 infection model with delays,, Int. J. Biomath., 6 (2013). doi: 10.1142/S1793524513500307.

[52]

R. A. Weiss, How does HIV cause AIDS?,, Science, 260 (1993), 1273. doi: 10.1126/science.8493571.

[53]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay,, J. Math. Anal. Appl., 375 (2011), 75. doi: 10.1016/j.jmaa.2010.08.055.

[54]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.

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