Article Contents
Article Contents

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

• * 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

• 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.

 Citation:

• 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
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