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

Wei Wang; Wanbiao Ma; Xiulan Lai

doi:10.3934/dcdsb.2020271

Article Contents

2021, Volume 26, Issue 7: 3989-4011. Doi: 10.3934/dcdsb.2020271

This issue Previous Article On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies Next Article

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

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
3.
Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

^* Corresponding author: Wei Wang
^* Corresponding author: Wei Wang

Received: December 31, 2019

Revised: June 30, 2020

Early access: September 2020

Published: July 2021

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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Abstract

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.

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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

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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

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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

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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

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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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