Export file:

Format

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

Content

  • 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.
  Figure/Table
  Supplementary
  Article Metrics

Keywords age structure; viral infection model; immune response; stability; Lyapunov functional

Citation: Ran Zhang, Shengqiang Liu. Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response. Mathematical Biosciences and Engineering, 2020, 17(2): 1450-1478. doi: 10.3934/mbe.2020075

References

  • 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.

 

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