Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

Special Issues: Applications of delay differential equations in biology

A HIV virus-to-cell dynamical model with distributed delay and Beddington-DeAngelis functional response is proposed in this paper. Using the characteristic equations and analytical means, the principle reproduction number R0 on the local stability of infection-free and chronic-infection equilibria is established. Furthermore, by constructing suitable Lyapunov functionals and using LaSalle invariance principle, we show that if R0 ≤ 1 the infection-free equilibrium is globally asymptotically stable, while if R0 > 1 the chronic-infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Comparing the effects between discrete and distributed delays on the stability of HIV virus-to-cell dynamical models, we can see that they could be same and different even opposite.
  Article Metrics

Keywords distributed delay; principle reproduction number; Beddington-DeAngelis functional response; Lyapunov functional; globally asymptotical stability

Citation: Xinran Zhou, Long Zhang, Tao Zheng, Hong-li Li, Zhidong Teng. Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay. Mathematical Biosciences and Engineering, 2020, 17(5): 4527-4543. doi: 10.3934/mbe.2020250


  • 1. D. D. Richman, D. M. Margolis, M. Delaney, W. C. Greene, D. Hazuda, R. J. Pomerantz, The challenge of finding a cure for HIV infection, Science, 323 (2009), 1304-1307.
  • 2. V. Leonenko, G. Bobashev, Analyzing influenza outbreaks in Russia using an age-structured dynamic transmission model, Epidemics, 29 (2019), 100-358.
  • 3. Y. Cai, K. Wang, W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.
  • 4. Y. Cai, Z. Ding, B. Yang, Z. Peng, W. Wang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Physica A: Stat. Mech. Appl., 514 (2019), 729-740.
  • 5. E. Grigorieva, E. Khailov, Determination of the opimal controls for an Ebola epidemic model, Disc. Cont. Dynam. Syst. S., 11 (2018), 1071-1101.
  • 6. J. Huang, S. Ruan, X. Wu, X. Zhou, Seasonal transmission dynamics of measles in China, Theor. Biosci., 137 (2018), 185-195.
  • 7. T. Zhang, X. Zhao, Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.
  • 8. M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent virus, Science, 272 (1996), 74-79.
  • 9. R. M. Anderson, R. M. May, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. Ser. B, 291 (1981), 451-524.
  • 10. A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-I: dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
  • 11. X. Lai, X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917.
  • 12. A. L. Hill, D. I. Rosenbloom, M. A. Nowak, R. F. Siliciano, Insight into treatment of HIV infection from viral dynamics models, Immun. Rev., 285 (2018), 9-25.
  • 13. P. Aavani, L. S. Allen, The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection, Appl. Math. Model., 75 (2019), 210-222.
  • 14. D. Olabode, L. Rong, X. Wang, Optimal control in HIV chemotherapy with termination viral load and latent reservoir, Math. Biosci. Eng., 16 (2018), 619-635.
  • 15. X. Wang, L. Rong, HIV low viral load persistence under treatment: Insights from a model of cell-to-cell viral transmission, Appl. Math. Lett., 94 (2019), 44-51.
  • 16. P. W. Nelson, A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.
  • 17. R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.
  • 18. J. Lin, R. Xu, 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.
  • 19. G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.
  • 20. G. Huang, W. Ma, Y. Takeuchi, Global analysis for delay virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.
  • 21. Y. Nakata, Global dynamics of a viral infection model with a latent period and beddingtonDeAngelis response, Nonlinear Anal. TMA, 74 (2011), 2929-2940.
  • 22. H. Xiang, L. Feng, H. Huo, Stability of the virus dynamics model with Beddington-DeAngelis functional response and delays, Appl. Math. Model., 37 (2013), 5414-5423.
  • 23. R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 2799-2805.
  • 24. J. Wang, M. Guo, X. Liu, 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.
  • 25. Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27.


Reader Comments

your name: *   your email: *  

© 2020 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)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved