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Mathematical analysis of a delayed HIV infection model with saturated CTL immune response and immune impairment

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Abstract

In this paper, we develop an HIV infection model with intracellular delay, Beddington–DeAngelis incidence rate, saturated CTL immune response and immune impairment. We begin model analysis with proving the positivity and boundedness of solutions of the model. By calculations, we derive immunity-inactivated and immunity-activated reproduction ratios. By analyzing corresponding characteristic equations, the local stabilities of feasible equilibria are addressed. With the help of suitable Lyapunov functionals and LaSalle’s invariance principle, it is proven that the global dynamics of the system is completely determined by the immunity-inactivated and immunity-activated reproduction ratios: if the immunity-inactivated reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; if the immunity-inactivated reproduction ratio is greater than unity, while the immunity-activated reproduction ratio is less than unity, the immunity-inactivated equilibrium is globally asymptotically stable; if the immunity-activated reproduction ratio is greater than unity, the immunity-activated equilibrium is globally asymptotically stable. Furthermore, sensitivity analysis is carried out to illustrate the effects of parameter values on the two thresholds.

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References

  1. Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996)

    Article  Google Scholar 

  2. Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586 (1996)

    Article  Google Scholar 

  3. Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44 (1999)

    Article  MathSciNet  Google Scholar 

  4. Cai, L.M., Guo, B.Z., Li, X.Z.: Global stability for a delayed HIV-1 infection model with nonlinear incidence of infection. Appl. Math. Comput. 219, 617–623 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Nelson, P.W., Murray, J.D., Perelson, A.S.: A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–215 (2000)

    Article  MathSciNet  Google Scholar 

  6. Wang, K.F., Wang, W.D., Liu, X.N.: Global stability in a viral infection model with lytic and nonlytic immune responses. Comput. Math. Appl. 51, 1593–1610 (2006)

    Article  MathSciNet  Google Scholar 

  7. Wang, X.N., Wang, W.D.: An HIV infection model based on a vectored immunoprophylaxis experiment. J. Theor. Biol. 313, 127–135 (2012)

    Article  MathSciNet  Google Scholar 

  8. Wang, K.F., Jin, Y., Fan, A.J.: The effect of immune responses in viral infections: a mathematical model view. Discrete Contin. Dyn. Syst. Ser. B 19, 3379–3396 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Zhou, X.Y., Shi, X.Y., Zhang, Z.H., Song, X.Y.: Dynamical behavior of a virus dynamics model with CTL immune response. Appl. Math. Comput. 213, 329–347 (2009)

    Article  MathSciNet  Google Scholar 

  10. Huang, G., Ma, W.B., Takeuchi, Y.: Global properties for virus dynamics model with Beddington–DeAngelis functional response. Appl. Math. Lett. 22, 1690–1693 (2009)

    Article  MathSciNet  Google Scholar 

  11. Wang, X., Tao, Y.D., Song, X.Y.: Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response. Nonlinear Dyn. 66, 825–830 (2011)

    Article  MathSciNet  Google Scholar 

  12. Huang, G., Ma, W.B., Takeuchi, Y.: Global analysis for delay virus dynamics model with Beddington–DeAngelis functional response. Appl. Math. Lett. 24, 1199–1203 (2011)

    Article  MathSciNet  Google Scholar 

  13. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Article  Google Scholar 

  14. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for tropic interaction. Ecology 56, 881–892 (1975)

  15. Song, X.Y., Neumann, A.U.: Global stability and periodic solution of the viral dynamics. J. Math. Anal. Appl. 329, 281–297 (2007)

    Article  MathSciNet  Google Scholar 

  16. Xu, R.: Global stability of an HIV-1 infection model with saturation infection and intracellular delay. J. Math. Anal. Appl. 375, 75–81 (2011)

    Article  MathSciNet  Google Scholar 

  17. Herz, A.V.M., Bonhoeffer, S., Anderson, R.M., May, R.M., Nowak, M.A.: Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Natl. Acad. Sci. USA 93, 7247–7251 (1996)

    Article  Google Scholar 

  18. De Boer, R.: Which of our modeling predictions are robust? PLoS Comput. Biol. 8, e1002593 (2012)

    Article  MathSciNet  Google Scholar 

  19. Wang, A.P., Michael, Li. Y.: Viral dynamics of HIV-1 with CTL immune response. Discrete Contin. Dyn. Syst. Ser. B 26, 2257–2272 (2021)

    Article  MathSciNet  Google Scholar 

  20. Iwami, S., Nakaoka, S., Takeuchi, Y., Miura, Y., Miura, T.: Immune impairment thresholds in HIV infection. Immunol. Lett. 123, 149–154 (2009)

    Article  Google Scholar 

  21. Regoes, R.R., Wodarz, D., Nowak, M.A.: Virus dynamics: the effect of target cell limitation and immune responses on virus evolution. J. Theor. Biol. 191, 451–462 (1998)

    Article  Google Scholar 

  22. Wang, S.L., Song, X.Y., Ge, Z.H.: Dynamics analysis of a delayed viral infection model with immune impairment. Appl. Math. Model. 35, 4877–4885 (2011)

    Article  MathSciNet  Google Scholar 

  23. Wang, Z.P., Liu, X.N.: A chronic viral infection model with immune impairment. J. Theor. Biol. 249, 532–542 (2007)

    Article  MathSciNet  Google Scholar 

  24. Hale, J.K., Verduyn, Lunel S.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Book  Google Scholar 

  25. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  26. Ren, J., Xu, R., Li, L.C.: Global stability of an HIV infection model with saturated CTL immune response and intracellular delay. Math. Biosci. Eng. 18, 57–68 (2020)

    Article  MathSciNet  Google Scholar 

  27. Stafford, M.A., Corey, L., Cao, Y., Daar, E.S., Ho, D.D., Perelson, A.S.: Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol. 203, 285–301 (2000)

    Article  Google Scholar 

  28. Wang, J.L., Guo, M., Liu, X.N., Zhao, Z.T.: Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay. Appl. Math. Comput. 291, 149–161 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Marino, S., Hogue, I.B., Ray, C.J., Kirschner, D.E.: A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254, 178–196 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871316, 11801340), and the Natural Science Foundation of Shanxi Province (Grant Nos. 201801D121006, 201801D221007).

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Authors and Affiliations

  1. Complex Systems Research Center, School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, China

    Yan Yang

  2. Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, China

    Rui Xu

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  1. Yan Yang
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  2. Rui Xu
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Correspondence to Rui Xu.

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Yang, Y., Xu, R. Mathematical analysis of a delayed HIV infection model with saturated CTL immune response and immune impairment. J. Appl. Math. Comput. 68, 2365–2380 (2022). https://doi.org/10.1007/s12190-021-01621-x

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