Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Lyapunov functional for virus infection model with diffusion and state-dependent delays

1 Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, P.R. China
2 School of Mathematics and Big Data, Foshan University, Foshan, Guangdong, 528000, P.R. China

Special Issues: Systems biology: Modeling of dynamical diseases and cancer

In this paper, a virus dynamics model with diffusion, state-dependent delays and a general nonlinear functional response is investigated. At first, the dynamical system is constructed on a nonlinear metric space. Then the stability of the interior equilibrium is established by using a novel Lyapunov functional. Further, the proposed algorithm has been extended to the model with logistic growth rate.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Lyapunov functional; state-dependent delays; reaction-diffusion differential equation; virus infection model

Citation: Jitai Liang, Junjie Wei. Lyapunov functional for virus infection model with diffusion and state-dependent delays. Mathematical Biosciences and Engineering, 2019, 16(2): 947-966. doi: 10.3934/mbe.2019044

References

  • 1. V. Doceul, M. Hollinshead, L. V. D. Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010): 873–876.
  • 2. A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington–Deangelis functional response, Math. Method. Appl. Sci., 36 (2013): 383–394.
  • 3. S. Guo, W. Ma and X. Q. Zhao, Global dynamics of a time-delayed Microorganism Flocculation model with saturated functional responses, J. Dyn. Differ. Equ., 30 (2018): 1247–1271.
  • 4. K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 69 (2015): 31–40.
  • 5. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.
  • 6. G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington–Deangelis functional response, Appl. Math. Lett., 24 (2011): 1199–1203.
  • 7. 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.
  • 8. T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete statedependent delay: classical solutions and solution manifold, J. Differ. Equ., 260 (2016): 4454– 4472.
  • 9. X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, B. Math. Biol., 76 (2014): 2806–2833.
  • 10. Y. Lv, R. Yuan and Y. Pei, Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equ., 260 (2016): 6201–6231.
  • 11. J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functionaldi fferential equations with multiple state-dependent time lags, Topol. Method. Nonl. An., 3 (1994): 101–162.
  • 12. R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, T. Am. Math. Soc., 321 (1990): 1–44.
  • 13. C. C. Mccluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real, 25 (2015): 64–78.
  • 14. H. Miao, X. Abdurahman, Z. Teng and L. Zhang, Dynamical analysis of a delayed reactiondi ffusion virus infection model with logistic growth and humoral immune impairment, Chaos Soliton. Fract., 110 (2018): 280–291.
  • 15. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, Berlin, 1983.
  • 16. A. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete Cont. Dyn. S. -B, 22 (2017): 1547–1563.
  • 17. A. Rezounenko, Viral infection model with diffusion and state-dependent delay: stability of classical solutions, preprint, 2017. arXiv:1706.08620.
  • 18. A. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: wellposedness in a metric space, Discrete Cont. Dyn. S., 33 (2013): 819–835.
  • 19. Y. Tian and X. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate, Nonlinear Anal. Real, 16 (2014): 17–26.
  • 20. J. Wang, J. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-tocell transmission, J. Math. Anal. Appl., 444 (2016): 1542–1564.
  • 21. K.Wang andW.Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007): 78–95.
  • 22. S.Wang, J. Zhang, F. Xu and X. Song, Dynamics of virus infection models with density-dependent diffusion, Comput. Math. Appl., 74 (2017): 2403–2422.
  • 23. W.Wang andW. Ma, Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections, Discrete Cont. Dyn. S.-B, 23 (2018): 3213– 3235.
  • 24. W. Wang, W. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. Real, 33 (2017): 253–283.
  • 25. W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: A nonlocal spatial mathematical model, B Math. Biol., 80 (2018): 540–582.
  • 26. Z. Wang, Qualitative analysis for a predator-prey system with nonlinear saturated functional response. J. Biomath., 22 (2007): 215–218.
  • 27. J. Wu, Theory and applications of partial functional differential equations, Springer, New York, 1996.
  • 28. W. Xia, S. Tang, X. Song and L. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dyn., 11 (2017): 455– 483.
  • 29. Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington–Deangelis response, Nonlinear Anal. Real, 15 (2014): 118–139.
  • 30. X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, B. Korean Math. Soc., 48 (2011): 555–574.

 

This article has been cited by

  • 1. A. M. Elaiw, S. F. Alshehaiween, A. D. Hobiny, I. A. Abbas, Global properties of latent virus dynamics with B-cell impairment, AIP Advances, 2019, 9, 9, 095035, 10.1063/1.5108890

Reader Comments

your name: *   your email: *  

© 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)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved