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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 1: 185-208. doi: 10.3934/mbe.2015.12.185
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Virus dynamics model with intracellular delays and immune response

  • 1.
    Department of Mathematics, Harbin Institute of Technology, Harbin, 150001
  • 2.
    Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080
  • Received: 01 October 2013 Accepted: 29 June 2018 Published: 01 December 2014

  • MSC : 34C23, 34D23, 92D30.

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    In this paper, we incorporate an extra logistic growth term for uninfected CD4$^+$ T-cells into an HIV-1 infection model with bothintracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basicreproduction number $R_0<1 then="" the="" infection-free="" steady="" state="" is="" globally="" asymptotically="" stable="" second="" when="" r_0="">1$, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the virus is completely determined by $R_0 $. Furthermore, given both the two delays are zero, then the infected steady state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small. When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcationas well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al[15]. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcationdiagram of viral load due to the logistic term of target cells and the two time delays.

    Citation: Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response[J]. Mathematical Biosciences and Engineering, 2015, 12(1): 185-208. doi: 10.3934/mbe.2015.12.185

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  • In this paper, we incorporate an extra logistic growth term for uninfected CD4$^+$ T-cells into an HIV-1 infection model with bothintracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basicreproduction number $R_0<1 then="" the="" infection-free="" steady="" state="" is="" globally="" asymptotically="" stable="" second="" when="" r_0="">1$, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the virus is completely determined by $R_0 $. Furthermore, given both the two delays are zero, then the infected steady state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small. When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcationas well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al[15]. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcationdiagram of viral load due to the logistic term of target cells and the two time delays.


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