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Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment

  • * Corresponding author: Zhijun Liu

    * Corresponding author: Zhijun Liu 

The work is supported by National Natural Science Foundation of China (No.11871201) and Youth Talents Project of Science and Technology Research Plan of Hubei Provincial Education Department (No.Q20171904)

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  • It is well known that CTL (cytotoxic T lymphocyte) immune response could be broadly classified into lytic and nonlytic components, nonlinear functions can better reproduce saturated effects in the interaction processes between cell and viral populations, and distributed intracellular delay can realistically reflect the stochastic element in the delay effects. For these reasons, we develop an HTLV-I (Human T-cell leukemia virus type I) infection model with nonlinear lytic and nonlytic CTL immune responses, nonlinear incidence rate, distributed intracellular delay and immune impairment. Through conducting complete analysis, it is revealed that all these factors influence the concentration level of infected T-cells at the chronic-infection equilibrium, whereas intracellular distributed delay and nonlinear incidence rate may change the expression of the basic reproduction number $ \mathfrak{R}_0 $ in the context where the model proposed still preserves the threshold dynamics. Our analysis results obtained may improve several existing works by comparison. We also perform global sensitivity analysis for $ \mathfrak{R}_0 $ in order to explore the effective strategies of lowering the concentration level of infected T-cells.

    Mathematics Subject Classification: Primary: 92D30; Secondary: 34K20.

    Citation:

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  • Figure 1.  The long-term and short-term behaviors for infected T-cells $ y(t) $ of model (3) with the weak or strong kernel and the same initial condition are respectively shown as follows: (a) the global dynamics of $ y(t) $; (b) the short-term dynamics of $ y(t) $

    Figure 2.  PRCC values for $ \mathfrak{R}_0 $ in the cases where input parameters respectively follow (a) uniform and (b) normal distributions

    Figure 3.  Simulations for the concentrations of infected T-cells of model (22) when (a) $ \sigma $ varies by 0.75 and 0.5, (b) $ a $ varies by 1.5 and 2 times, (c) $ \alpha_1 $ varies by 1.5 and 2 times and (d) $ \beta $ varies by 0.75 and 0.5 of their baseline values in Table 1, respectively

    Table 1.  Biological description of parameters in models (1) and (3)

    Para.(Unit) Description Value Range Ref.
    $ \lambda $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) The recruitment rate of healthy T-cells $ 105 $ [10,200] [20,21,22,34,37]
    $ d $ ($ \mbox{d}^{-1} $) The death rate of healthy T-cells $ 0.11 $ [0.01, 0.2] [20,21,22,34,37]
    $ \beta $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) Viral infectivity rate $ 0.026 $ [0.001, 0.05] [20,21,22,34,37]
    $ \rho $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) The death rate of infected not productive cells $ 0.11 $ [0.01, 0.2] [36]
    $ \tau $ ($ \mbox{d} $) The intracellular latent delay $ 5 $ [0, 10] [1]
    $ a $ ($ \mbox{d}^{-1} $) The sum of the released rate of viral particles and the death rate of infected T-cells $ 1.005 $ [0.01, 0.2] [20,21,22,34,37]
    $ q_0 $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) The efficacy of NL-CTL response $ 0.5 $ [0, 1] [34,37]
    $ p_0 $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) The strength of L-CTL response $ 0.1 $ [0, 1] [20,21,22,34,37]
    $ c $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) The proliferation rate of CTLs $ 0.2 $ [0, 1] [20,21,22,34,37]
    $ b $ ($ \mbox{d}^{-1} $) The decay rate of CTLs $ 0.4 $ [0, 1] [20,21,22,34,37]
    $ \sigma $ (-) A fraction of cells newly infected by contacts that survive the antibody immune response $ 0.5 $ [0, 1] [9,22]
    $ \alpha_1 $ ($ \mu \mbox{l} $) The inhibitory rate from healthy T-cells $ 0.003 $ [0, 10] Estimated
    $ \alpha_2 $ ($ \mu \mbox{l} $) The inhibitory rate from infected T-cells $ 0.005 $ [0, 10] Estimated
    $ \omega $ ($ \mu \mbox{l} $) The inhibitory rate from NL-CTL response $ 10 $ [0, 10] [32]
    $ m $ ($ \mu \mbox{l}^{-1}\mbox{d}^{-1} $) Immune impairment rate of virus $ 0.01 $ [0, 1] [1]
     | Show Table
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    Table 2.  PRCC values for $ \mathfrak{R}_0 $

    Para. Distribution1 PRCC1 p-value1 Distribution2 PRCC2 p-value2 Rank
    $ \lambda $ U(10, 200) $ -0.0193 $ $ 0.2548 $ N(105, 30) $ 0.0230 $ $ 0.1732 $ 7
    $ d $ U(0.01, 0.2) $ -0.0180 $ $ 0.2880 $ N(0.11, 0.03) $ 0.0149 $ $ 0.3779 $ 8
    $ \beta $ U(0.001, 0.05) $ 0.6968 $ $ 0 $ N(0.026, 0.005) $ 0.5697 $ $ 0 $ 4
    $ \alpha_1 $ U(0, 10) $ -0.7167 $ $ 0 $ N(5, 1.667) $ -0.7789 $ $ 0 $ 3
    $ \sigma $ U(0, 1) $ 0.7411 $ $ 0 $ N(0.5, 0.167) $ 0.7885 $ $ 0 $ 1
    $ \rho $ U(0.01, 0.2) $ -0.3574 $ $ 0 $ N(0.11, 0.03) $ -0.4692 $ $ 0 $ 6
    $ \tau $ U(0, 10) $ -0.3753 $ $ 0 $ N(5, 1.667) $ -0.5536 $ $ 0 $ 5
    $ a $ U(0.01, 0.2) $ -0.7211 $ $ 0 $ N(1.005, 0.332) $ -0.7811 $ $ 0 $ 2
     | Show Table
    DownLoad: CSV
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