Skip to main content

Advertisement

SpringerLink
Log in

Local and Global Stabilities of a Viral Dynamics Model with Infection-Age and Immune Response

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper, we construct an infection-age model to study the interaction between viruses and the immune system within the host. In the model, the mortality rate of infected cells, the rate that cytotoxic T lymphocytes (CTL) kill infected cells, the rate that infected cells produce new virus, and the CTL proliferate rate may depend on the infection-age. The basic reproduction number and the threshold for the existence of steady states are obtained. Local stability of both the infection-free and infection steady states is studied by analyzing the linearized systems. Global stability of the infection-free steady state is obtained by investigating a renewal integral equation and global stability of the infection steady state is obtained by constructing a Liapunov functional. Numerical simulations are presented to verify the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Althaus, C.L., De Vos, A.S., De Boer, R.J.: Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, \(\cal{R}_0\). J. Virol. 83, 7659–7667 (2009)

    Article  Google Scholar 

  2. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1991)

    Google Scholar 

  3. Arnaout, R.A., Nowak, M.A., Wodarz, D.: HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing? Proc. R. Soc. Lond. B 265, 1347–1354 (2000)

    Article  Google Scholar 

  4. Browne, C.J.: A multi-strain virus model with infected cell age structure: application to HIV. Nonlin. Anal. Real World Appl. 22, 354–372 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Browne, C.J., Pilyugin, S.S.: Global analysis of age-structured within-host virus model. Discrete Contin. Dyn. Syst. B 18, 1999–2017 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Y., Yang, J., Zhang, F.: The global stability of an SIRS model with infection age. Math. Biosci. Eng. 11, 449–469 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Desch, W., Schappacher, W.: Linearized stability for nonlinear semigroups. In: Favini, A., Obrecht, E. (eds.) Differential Equations in Banach Spaces, Lecture Notes in Mathematics, vol. 1223, pp. 61–73. Springer, Berlin (1986)

    Chapter  Google Scholar 

  8. Gilchrist, M.A., Coombs, D., Perelson, A.S.: Optimizing within-host viral fitness: infected cell lifespan and virion production rate. J. Theor. Biol. 229, 281–288 (2004)

    Article  MathSciNet  Google Scholar 

  9. Gurtin, M.E., MacCamy, R.C.: Non-linear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281–300 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hale, J.K.: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence (1988)

    Google Scholar 

  11. Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20, 388–395 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori e Stampatori, Pisa (1995)

    Google Scholar 

  14. Inaba, H., Sekine, H.: A mathematical model for Chagas disease with infection-age-dependent infectivity. Math. Biosci. 190, 39–69 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Magal, P.: Compact attrators for time-periodic age-structured population models. Electron. J. Differ. Equ. 65, 1–35 (2001)

    Google Scholar 

  16. Magal, P., McCluskey, C.: Two-group infection age model including an application to nosocomial infection. SIAM J. Appl. Math. 73, 1058–1095 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Magal, P., McCluskey, C.C., Webb, G.F.: Liapunov fucntional and global asymptoticalc stability for an infection-age model. Appl. Anal. 89, 1109–1140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Martcheva, M., Thieme, H.R.: Progression age enhanced backward bifurcation in an epidemic model with superinfection. J. Math. Biol. 46, 385–424 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Metz, J.A.J., Diekmann, O.: The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  22. Min, L., Su, Y., Kuang, Y.: Mathematical analysis of a basic model of virus infection with application. Rocky Mt. J. Math. 38, 1573–1585 (2008)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  25. Nowak, M.A., May, R.M.: Viral Dynamics. Oxford University Press, Oxford (2000)

    Google Scholar 

  26. Pang, J., Cui, J.: Analysis of a hepatitis B viral infection model with immune response delay. Int. J. Biomath. 10(2), 1750020 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pereson, A.S.: Modelling viral and immune system dynamics. Nat. Rev. Immunol. 2, 28–36 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Shu, H., Wang, L., Watmough, J.: Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses. SIAM J. Appl. Math. 73, 1280–1302 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thieme, H.R.: Semiflows generated by Lipschitz perturbations of non-densely defined operators. Differ. Integral Equ. 3, 1035–1066 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Thieme, H.R., Castillo-Chavez, C.: How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. 53, 1447–1479 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Webb, G.F.: Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York (1985)

    MATH  Google Scholar 

  36. Wodarz, D.: Killer Cell Dynamics–Mathematical and Computational Approaches to Immunology. Springer, New York (2007)

    MATH  Google Scholar 

  37. Wodarz, D., Christensen, J.P., Thomsen, A.R.: The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23, 194–200 (2002)

    Article  Google Scholar 

  38. Wodarz, D.: Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750 (2003)

    Article  Google Scholar 

  39. Wodarz, D., Nowak, M.A.: Immune responses and viral phenotype: Do replication rate and cytopathogenicity influence virus load? J. Theor. Med. 2, 113–127 (2000)

    Article  MATH  Google Scholar 

  40. Yang, Y., Ruan, S., Xiao, D.: Global stability of an age-structured virus dynamics model with Beddington–DeAngelis infection function. Math. Biosci. Eng. 12, 859–877 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the reviewers for their constructive comments, which greatly improve this paper. This research was partially supported by National Natural Science Foundation of China (11401117, 11401060, 11401217, 11771168), Improvement Project for Young Teachers of Guangxi Province (KY2016YB246), and National Science Foundation (DMS-1412454).

Author information

Authors and Affiliations

  1. School of Science, Guangxi University of Science and Technology, Liuzhou, 545006, China

    Jianhua Pang

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Jing Chen & Shigui Ruan

  3. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Zijian Liu

  4. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, China

    Ping Bi

Authors
  1. Jianhua Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zijian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ping Bi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Shigui Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shigui Ruan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pang, J., Chen, J., Liu, Z. et al. Local and Global Stabilities of a Viral Dynamics Model with Infection-Age and Immune Response. J Dyn Diff Equat 31, 793–813 (2019). https://doi.org/10.1007/s10884-018-9663-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9663-1

Keywords

Search

Navigation