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Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion

  • * Corresponding author: Wei Wang

    * Corresponding author: Wei Wang 

This work is supported by the NNSF of China (11901360) to W. Wang and supported by the National Key R-D Program of China (No. 2017YFF0207401) and the NNSF of China (No. 11971055) W. Ma

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  • In this paper, we study the global dynamics of a viral infection model with spatial heterogeneity and nonlinear diffusion. For the spatially heterogeneous case, we first derive some properties of the basic reproduction number $ R_0 $. Then for the auxiliary system with quasilinear diffusion, we establish the comparison principle under some appropriate conditions. Some sufficient conditions are derived to ensure the global stability of the virus-free steady state. We also show the existence of the positive non-constant steady state and the persistence of virus. For the spatially homogeneous case, we show that $ R_0 $ is the only determinant of the global dynamics when the derivative of the function $ g $ with respect to $ V $ (the rate of change of infected cells for the repulsion effect) is small enough. Our simulation results reveal that pyroptosis and Beddington-DeAngelis functional response function play a crucial role in the controlling of the spreading speed of virus, which are some new phenomena not presented in the existing literature.

    Mathematics Subject Classification: Primary: 34D20, 35C07; Secondary: 35Q92, 92D3.


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  • Figure 1.  a. Initial distribution $ \omega_0(x) $. b. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. c. The contour of b

    Figure 2.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a

    Figure 3.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a

    Figure 4.  a. Evolution of $ \omega(t, \, x) $ from the initial distribution, where $ dashed $ $ line\ (red) $: $ g_0 = 0 $, $ solid\ line\ (black) $: $ g_0 = 3.8\times10^{-8} $. b. The contour of a

    Table 1.  Summary of model parameters

    Parameters Descriptions
    $ \xi(x) $ Generation of uninfected cells
    $ \beta(x) $ Infection rate
    $ q(x) $ Pyroptosis effect of inflammatory cytokines on uninfected cells
    $ \alpha_1(x) $ Death rate due to pyroptosis
    $ \alpha_2(x) $ Production rate of inflammatory cytokines
    $ k(x) $ Production rate of virus
    $ d_U(x) $ Death rate of uninfected cells
    $ d_V(x) $ Death rate of infected cells
    $ d_M(x) $ Death rate of inflammatory cytokines
    $ d_{\omega}(x) $ Death rate of virus
    $ D_0 $ Diffusion rate of cells (uninfected cells and infected cells)
    $ D_1 $ Diffusion rate of inflammatory cytokines
    $ D_2 $ Diffusion rate of virus
    $ a $ Rate of the inhibitory effect on virus
    $ b $ Rate of the inhibitory effect on inflammatory cytokines
     | Show Table
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