Dynamical behavior of a delay virus dynamics model with CTL immune response.

*(English)*Zbl 1204.34110From the abstract: The dynamical behaviour of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on the stability of the equilibria of the CTL immune response model is studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation at the immune-free equilibrium and the endemic equilibrium are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data is carried out to support the analytical findings.

Reviewer: Eva Sanchez (Madrid)

##### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

92C60 | Medical epidemiology |

##### Keywords:

virus dynamics; CTL immune response; delay; asymptotic stability of equilibria; Hopf bifurcation
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\textit{X. Shi} et al., Nonlinear Anal., Real World Appl. 11, No. 3, 1795--1809 (2010; Zbl 1204.34110)

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##### References:

[1] | Liu, Weimin, Nonlinear oscillations in models of immune responses to persistent viruses, Theoretical population biology, 52, 224-230, (1997) · Zbl 0890.92015 |

[2] | Wodarz, D., Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses, Journal of general virology, 84, 1743-1750, (2003) |

[3] | Yafia, Radouane, Hopf bifurcation analysis and numerical simulations in an ODE model of the immune system with positive immune response, Nonlinear analysis: real world applications, 8, 5, 1359-1369, (2007) · Zbl 1128.92020 |

[4] | Leenheer, P.D.; Smith, H.L., Virus dynamics: A global analysis, SIAM journal of applied mathematics, 63, 1313-1327, (2003) · Zbl 1035.34045 |

[5] | Song, Xinyu; Neumann, Avidan U., Global stability and periodic solution of the viral dynamics, Journal of mathematical analysis and applications, 329, 1, 281-297, (2007) · Zbl 1105.92011 |

[6] | Korobeinikov, A., Global properties of basic virus dynamics models, Bulletin of mathematical biology, 66, 879-883, (2004) · Zbl 1334.92409 |

[7] | Nowak, Martin A.; Bangham, Charles R.M., Population dynamics of immune responses to persistent viruses, Science, 272, 74-79, (1996) |

[8] | Culshaw, R.V.; Ruan, S., A delay-differential equation model of HIV infection of CD4^{+} T-cells, Mathematical biosciences, 165, 27-39, (2000) · Zbl 0981.92009 |

[9] | Culshaw, R.V.; Ruan, S.; Webb, G., A mathematical model of cell-to-cell HIV-1 that include a time delay, Journal of mathematical biology, 46, 425-444, (2003) · Zbl 1023.92011 |

[10] | Herz, A.V.M.; Bonhoeffer, S.; Anderson, R.M.; May, R.M.; Nowak, M.A., Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proceedings of national Academy of sciences USA, 93, 7247-7251, (1996) |

[11] | Nelson, P.W.; Perelson, A.S., Mathematical analysis of a delay differential equation models of HIV-1 infection, Mathematical biosciences, 179, 73-94, (2002) · Zbl 0992.92035 |

[12] | Hale, J.K., Theory of functional differential equations, (1997), Springer-Verlag New York |

[13] | Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press San Diego · Zbl 0777.34002 |

[14] | Freedman, H.I.; Sree Hari Rao, V., The trade-off between mutual interference and time lags in predatorprey systems, Bulletin of mathematical biology, 45, 991-1003, (1983) · Zbl 0535.92024 |

[15] | Marsden, J.E.; McCracken, M., The Hopf bifurcation and its applications, (1976), Springer-Verlag · Zbl 0346.58007 |

[16] | Liu, Wenxiang; Freedman, H.I., A mathematical model of vascular tumor treatment by chemotherapy, Mathematical and computer modelling, 42, 9-10, 1089-1112, (2005) · Zbl 1080.92045 |

[17] | Li, Xiuling; Wei, Junjie, On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos, solitions & fractals, 26, 2, 519-526, (2005) · Zbl 1098.37070 |

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