Stability and Hopf bifurcation in a virus model with self-proliferation and delayed activation of immune cells

Huan Kong; Guohong Zhang; Kaifa Wang; Huan Kong; Guohong Zhang; Kaifa Wang

doi:10.3934/mbe.2020242

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4384-4405. doi: 10.3934/mbe.2020242

Previous Article Next Article

Research article Special Issues

Stability and Hopf bifurcation in a virus model with self-proliferation and delayed activation of immune cells

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: 14 April 2020 Accepted: 08 June 2020 Published: 22 June 2020

A new mathematical model was proposed to study the effect of self-proliferation and delayed activation of immune cells in the process of virus infection. The global stability of the boundary equilibria was obtained by constructing appropriate Lyapunov functional. For positive equilibrium, the conditions of stability and Hopf bifurcation were obtained by taking the delay as the bifurcation parameter. Furthermore, the direction and stability of the Hopf bifurcation are derived by using the theory of normal form and center manifold. These results indicate that self-proliferation intensity can significantly affect the kinetics of viral infection, and the delayed activation of immune cells can induce periodic oscillation scenario. Along with the increase of delay time, numerical simulations give the corresponding bifurcation diagrams under different self-proliferation rates, and verify that there exists stability switch phenomenon under some conditions.
- viral infection,
- self-proliferation,
- delayed activation of immune cells,
- global stability,
- Hopf bifurcation,
- stability switch
Citation: Huan Kong, Guohong Zhang, Kaifa Wang. Stability and Hopf bifurcation in a virus model with self-proliferation and delayed activation of immune cells[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4384-4405. doi: 10.3934/mbe.2020242

Related Papers:

Abstract

A new mathematical model was proposed to study the effect of self-proliferation and delayed activation of immune cells in the process of virus infection. The global stability of the boundary equilibria was obtained by constructing appropriate Lyapunov functional. For positive equilibrium, the conditions of stability and Hopf bifurcation were obtained by taking the delay as the bifurcation parameter. Furthermore, the direction and stability of the Hopf bifurcation are derived by using the theory of normal form and center manifold. These results indicate that self-proliferation intensity can significantly affect the kinetics of viral infection, and the delayed activation of immune cells can induce periodic oscillation scenario. Along with the increase of delay time, numerical simulations give the corresponding bifurcation diagrams under different self-proliferation rates, and verify that there exists stability switch phenomenon under some conditions.

References

[1]	J. E. Schmitz, M. J. Kuroda, S. Santra, V. G. Sasseville, M. A. Simon, M. A. Lifton, et al, Control of viremia in simian immunodeficiency virus infection by CD8+ lymphocytes, Science, 283 (1999), 857-860.
[2]	L. C. Wang, M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57.
[3]	X. Y. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.
[4]	R. J. De Boer, A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214.
[5]	Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27.
[6]	J. L. Wang, J. M. Pang, T. Kuniya, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298-316.
[7]	M. A. Nowak, S. Bonhoefier, 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.
[8]	A. Korobeinikov, S. Giles, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.
[9]	M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
[10]	H. Zhu, X. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay,Discr. Cont. Dyn. Syst. Ser. B, 12 (2009), 511-524.
[11]	K. Wang, W. Wang, H. Pang, X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197-208.
[12]	Yukihiko Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27.
[13]	H. Gomez-Acevedo, M. Y. Li, S. Jacobson, Multi-stability in a model for CTL response to HTLVI infection and its consequences in HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696.
[14]	R. M. Anderson, R. M. May, S. Gupta, Non-linear phenomena in host-parasite interactions, Parasitology, 99 (1989), 59-79.
[15]	A. Murase, T. Sasaki, T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267.
[16]	C. Chiyaka, W. Garira, S. Dube, Modelling immune response and drug therapy in human malaria infection, Comput. Math. Method. Med., 9 (2008), 143-163.
[17]	A. S. Perelson, Modelling viral and immune system dynamics, Nature Rev. Immunol., 2 (2002), 28-36.
[18]	A. Korobeinikov, Immune response and within-host viral evolution:Immune response can accelerate evolution, J. Theor. Biol., 456 (2018),74-83.
[19]	H. Q. Zhang, H. Chen, C. C Jiang, K. F. Wang, Effect of explicit dynamics of free virus and intracellular delay, Chaos, Solitons Fractals, 104 (2017), 827-834.
[20]	Y. Wang, J. Liu, J. M. Heffernan, Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay, J. Math. Anal. Appl., 459 (2018), 506-527.
[21]	K. Allali, S. Harroudi, D. F. M. Torre, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci., 12 (2018), 111-127.
[22]	H. J. Liu, J. F. Zhang, Dynamics of two time delays differential equation model to HIV latent infection, Physica A, 514 (2019), 384-395.
[23]	M. Y. Li, H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793.
[24]	D. W. Huang, X. Zhang, Y. F. Guo, H. L. Wang, Analysis of an HIV infection model with treatment sand delayed immune response, Appl. Math. Model., 40 (2016), 3081-3089.
[25]	D. Wodarz, J. P. Christensen, A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends Immunol., 23 (2002), 194-200.
[26]	C. Bartholdy, J. P. Christensen, D. Wodarz, A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infected with lymphocytic chroriomeningitis virus, J. Virology, 74 (2000), 10304-10311.
[27]	K. Wang, Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems, Comm. Pure Appl. Anal., 10 (2011), 1537-1548.
[28]	S. Bonhoeffer, J. M. Coffin, M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virology, 71 (1997), 3275-3278.
[29]	M. Nagumo, Uber die lage der integralkurven gewohnlicher differentialgleichungen, Proc. Phys. Math. Soc., 24 (1942), 551-559.
[30]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
[31]	B. Hassard, D. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press, 1981.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)