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Global dynamics of an immunosuppressive infection model with stage structure

1 School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China
2 Department of Mathematics, Tongji University, Shanghai 200092, China

Special Issues: Modeling, Analysis and Computation in Mathematical Biology

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.
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Keywords delayed antiviral immune response; Stage structure; Hopf bifurcation; Periodic solutions; Hopf branch

Citation: Hongying Shu, Wanxiao Xu, Zenghui Hao. Global dynamics of an immunosuppressive infection model with stage structure. Mathematical Biosciences and Engineering, 2020, 17(3): 2082-2102. doi: 10.3934/mbe.2020111


  • 1. F. C. Bekkering, C. Stalgis, J. G. McHutchison, J. T. Brouwer, A. S. Perelson, Estimation of early hepatitis C viral clearance in patients receiving daily interferon and ribavirin therapy using a mathematical model, Hepatology, 33 (2001), 419-423.
  • 2. A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden, et al., Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α therapy, Science, 282 (1998), 103-107.
  • 3. S. A. Whalley, J. M. Murray, D. Brown, G. J. M. Webster, V. C. Emery, G. M. Dusheiko, et al., Kinetics of acute hepatitis B virus infection in humans, J. Exp. Med., 193 (2001), 847-854.
  • 4. T. W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. Mican, M. Baseler, et al., Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 13193-13197.
  • 5. A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36.
  • 6. N. L. Komarova, E. Barnes,P. Klenerman, D. Wodarz, Boosting immunity by antiviral drug therapy: a simple relationship among timing, efficacy, and success, Proc. Natl. Acad. Sci. USA, 100 (2003), 1855-1860.
  • 7. E. S. Rosenberg, M. Altfeld, S. H. Poon, M. N. Phillips, B. M. Wilkes, R. L. Eldridge, et al., Immune control of HIV-1 after early treatment of acute infection, Nature, 407 (2000), 523-526.
  • 8. A. Fenton, J. Lello, M. B. Bonsall, Pathogen responses to host immunity: the impact of time delays and memory on the evolution of virulence, Proc. R. Soc. B, 273 (2006), 2083-2090.
  • 9. G. M. Ortiz, J. Hu, J. A. Goldwitz, R. Chandwani, M. Larsson, N. Bhardwaj, et al., Residual viral replication during antiretroviral therapy boosts human immunodeficiency virus type 1-specific CD8+ T-cell responses in subjects treated early after infection, J. Virol., 76 (2002), 411-415.
  • 10. H. Shu, L. Wang, J. Watmough, Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model, J. Math. Biol., 68 (2014), 477-503.
  • 11. J. K. Hale, S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
  • 12. X. Pan, Y. Chen, H. Shu, Rich dynamics in a delayed HTLV-I infection model: stability switch, multiple stable cycles, and torus, J. Math. Anal. Appl., 479 (2019), 2214-2235.
  • 13. H. Shu, X. Hu, L. Wang, J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269-1298.
  • 14. J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, 1988.
  • 15. H. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.
  • 16. 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.
  • 17. E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
  • 18. B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
  • 19. J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
  • 20. K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, K. U. Leuven, Belgium, 2001.
  • 21. K. Engelborghs, T. Luzyanina, D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21.


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