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Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses

1 Institute of Applied Mathematics, Army Engineering University, Shijiazhuang 050003, Hebei, P.R. China
2 Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, P.R. China
3 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, Shanxi, P.R. China

Human specific immunity consists of two branches: humoral immunity and cellular immunity. To protect us from pathogens, cell-mediated and humoral immune responses work together to provide the strongest degree of e cacy. In this paper, we propose an HIV-1 model with cell-mediated and humoral immune responses, in which both virus-to-cell infection and cell-to-cell transmission are considered. Five reproduction ratios, namely, immunity-inactivated reproduction ratio, cellmediated immunity-activated reproduction ratio, humoral immunity-activated reproduction ratio, cellmediated immunity-competed reproduction ratio and humoral immunity-competed reproduction ratio, are calculated and verified to be sharp thresholds determining the local and global properties of the virus model. Numerical simulations are carried out to illustrate the corresponding theoretical results and reveal the e ects of some key parameters on viral dynamics.
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1. R. A. Cangelosi, E. J. Schwartz and D. J. Wollkind, A quasi-steady-state approximation to the basic target-cell-limited viral dynamics model with a non-cytopathic effect, Front. Microbiol., 9 (2018), 54.

2. J. Charles, T. Paul and W. Mark, Immunobiology, 5nd edition, Garland Science, New York, 2001.

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

4. A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal. RWA, 26 (2015), 161–190.

5. A. M. Elaiw, A. A. Raezah and K. Hattaf, Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response, Int. J. Biomath., 10 (2017), 1750070.

6. T. R. Fouts, K. Bagley, I. J. Prado, et. al., Balance of cellular and humoral immunity determines the level of protection by HIV vaccines in rhesus macaque models of HIV infection, Proc. Natl. Acad. Sci., 13 (2015), 992–999.

7. J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

8. K. Hattaf and N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, Int. J. Dynam. Control, 4 (2016), 254.

9. A. Hoare, D. G. Regan and D. P. Wilson, Sampling and sensitivity analyses tools (SaSAT) for computational modelling, Theor. Biol. Med. Model., 5 (2008), 4.

10. G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708.

11. X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563–584.

12. X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-tocell transmission, SIAM J. Appl. Math., 74 (2014), 898–917.

13. F. Li and J. Wang, Analysis of an HIV infection model with logistic target-cell growth and cellto- cell transmission, Chaos Soliton Fract., 81 (2015), 136–145.

14. J. Lin, R. Xu and X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516–530.

15. C. Lv, L. Huang and Z. Yuan, Global stability for an HIV-1 infection model with Beddington- DeAngelis incidence rate and CTL immune response, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 121–127.

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

17. N. Martin and Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion, Curr. Opin. HIV AIDS, 4 (2009), 143–149.

18. A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247–267.

19. Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14–27.

20. M. Nowak, S. Bonhoeffer, G. Shaw and R. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203–217.

21. A. S. Perelson and P. W. Nelson, Mathematical Analysis of HIV-1: Dynamics in Vivo, SIAM Review, 41 (1999), 3–44.

22. R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B, 269 (2002), 271–279.

23. E. J. Schwartz, N. K. Vaidya, K. S. Dorman, S. Carpenter and R. H. Mealey, Dynamics of lentiviral infection in vivo in the absence of adaptive immune responses, Virology, 513 (2018), 108–113.

24. 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), 1280–1302.

25. A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95–98.

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

27. J. Wang, J. Pang, T.Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298–316.

28. S. Wang and D. Zou, Global stability of in-host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313–1322.

29. T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulat., 89 (2013), 13–22.

30. T.Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014), 63–74.

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

32. J. Xu, Y. Geng and Y. Zhou, Global dynamics for an age-structured HIV virus infection model with cellular infection and antiretroviral therapy, Appl. Math. Comput., 305 (2017), 62–83.

33. Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete and Continuous Dynamical Systems - Series B, 17 (2012), 401–416.

34. H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete and Continuous Dynamical Systems - Series B, 12 (2009), 511– 524.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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