Export file:


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


  • Citation Only
  • Citation and Abstract

Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response

1 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

Special Issues: Modeling and Complex Dynamics of Populations

In this paper, an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immune response is investigated. We give a rigorous mathematical analysis on some necessary technical materials, including the relative compactness and persistence of the solution semiflow, and existence of a global attractor. By subtle construction and estimates of a Lyapunov functional, we show that the global dynamics is determined by two sharp thresholds, namely, basic reproduction number $\Re_0$ and immune-response reproduction number $\Re_1$. When $\Re_0<1$, the virus-free steady state is globally asymptotically stable, which means that the viruses are cleared and immune-response is not active; when $\Re_1<1<\Re_0$, the immune-inactivated infection steady state exists and is globally asymptotically stable; and when $\Re_1>1$, which implies that $\Re_0>1$, the immune-activated infection steady state exists and is globally asymptotically stable. Numerical simulations are given to support our theoretical results.
  Article Metrics


1. M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.

2. P. Nelson, M. Gilchrist, D. Coombs, J. M. Hyman, A. S. perelson, An age-structured model of HIV infection that allow for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.

3. L. Rong, Z. Feng, A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756.

4. G. Huang, X. Liu, Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38.

5. L. Zou, S. Ruan, W. Zhang, An age-structured model for the transmission dynamics of hepatitis B, SIAM J. Appl. Math., 70 (2010), 3121-3139.

6. J. Wang, R. Zhang, T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313.

7. M. Shen, Y. Xiao, L. Rong, Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics, Math. Biosci., 263 (2015), 37-50.

8. Y. Wang, K. Liu, Y. Lou, An age-structured within-host HIV model with T-cell competition, Nonlinear Anal. Real World Appl., 38 (2017), 1-20.

9. C.-Y. Cheng, Y. Dong, Y. Takeuchi, An age-structured virus model with two routes of infection in heterogeneous environments, Nonlinear Anal. Real World Appl., 39 (2018), 464-491.

10. Z. Liu, Y. Rong, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Sci. China Math., 60 (2017), 1371-1398.

11. J. Wang, X. Dong, Analysis of an HIV infection model incorporating latency age and infection age, Math. Biosci. Eng., 15 (2018), 569-594.

12. J. Pang, J. Chen, Z. Liu, P. Bi, S. Ruan, Local and global stabilities of a viral dynamics model with infection-age and immune response, J. Dyn. Diff. Equat., 31 (2019), 793-813.

13. K. Hattaf, Y. Yang, Global dynamics of an age-structured viral infection model with general incidence function and absorption, Int. J. Biomath., 11 (2018), 1850065.

14. W. Hübner, G. McNerney, P. Chen, B. M. Dale, R. E. Gordon, F. Y. Chuang, et al., Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743-1747.

15. A. Imle, P. Kumberger, N. D. Schnellbächer, J. Fehr, P. Carrillo-Bustamante, J. Ales, et al., Experimental and computational analyses reveal that environmental restrictions shape HIV-1 spread in 3D cultures, Nat. Commun., 10 (2019), 2144.

16. N. Barretto, B. Sainz, S. Hussain, S. L. Uprichard, Determining the involvement and therapeutic implications of host cellular factors in hepatitis C virus cell-to-cell spread, J. Virol., 88 (2014), 5050-5061.

17. F. Merwaiss, C. Czibener, D. E. Alvarez, Cell-to-cell transmission is the main mechanism supporting bovine viral diarrhea virus spread in cell culture, J. Virol., 93 (2019), e01776.

18. G. L. Smith, B. J. Murphy, M. Law, Vaccinia virus motility, Annu. Rev. Microbiol., 57 (2003), 323-342.

19. 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.

20. Y. Yang, L. Zou, S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191.

21. J. Wang, J. Lang, X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.

22. X. Zhang, Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virusto-cell and cell-to-cell transmissions, Int. J. Bifurcation Chaos, 28 (2018), 1850109.

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

24. S. Wang, D. Zou, Global stability of in-host viral models with humoral immunity and intracellular delays, Appl. Math. Model, 51 (2012), 1313-1322.

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

26. T. Kajiwara, T. Sasaki, Y. Otani, Global stability of an age-structured model for pathogen-immune interaction, J. Appl. Math. Comput., 59 (2019), 631-660.

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

28. Y. Wang, Y. Zhou, F. Brauer, J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.

29. C. Browne, Immune response in virus model structured by cell infection-age, Math. Biosci. Eng., 13 (2016), 887-909.

30. X. Duan, S. Yuan, Global dynamics of an age-structured virus model with saturation effects, Math. Methods Appl. Sci., 40 (2017), 1851-1864.

31. X. Wang, S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Methods Appl. Sci., 36 (2013), 125-142.

32. H. Shu, L. Wang, 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.

33. X. Tian, R. Xu, J. Lin, Mathematical analysis of an age-structured HIV-1 infection model with CTL immune response, Math. Biosci. Eng., 16 (2019), 7850-7882.

34. K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 21.

35. X. Tian, R. Xu, Global stability of a delayed HIV-1 infection model with absorption and CTL immune response, IMA J. Appl. Math., 79 (2014), 347-359.

36. P. Magal, S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences Vol. 201, Springer, Cham, 2018.

37. P. Magal, Compact attractors for time-periodic age structured population models, Elect. J. Differ. Eqs., 65 (2001), 65.

38. P. Magal, C. C. McCluskey, G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.

39. P. Magal, H. Thieme, Eventual compactness for a semiflow generated by an age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.

40. H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics Vol. 118, American Mathematical Society, Providence, RI, 2011.

41. C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841.

42. C. Jiang, K. Wang, L. Song, Global dynamics of a delay virus model with recruitment and saturation effects of immune responses, Math. Biosci. Eng., 14 (2017), 1233-1246.

43. H. Dahari, A. Lo, R. M. Ribeiro, A. S. Perelson, Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371-381.

44. Y. Wang, Y. Zhou, J. Wu, J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104-112.

45. J. Reyes-Silveyra, A. R. Mikler, Modeling immune response and its effect on infectious disease outbreak dynamics, Theor. Biol. Med. Model., 13 (2016), 10.

46. R. Qesmi, S. ElSaadany, J. M. Heffernan, J. Wu, A hepatitis B and C virus model with age since infection that exhibits backward bifurcation, SIAM J. Appl. Math., 71 (2011), 1509-1530.

© 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

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved