# American Institute of Mathematical Sciences

September  2019, 24(9): 4783-4797. doi: 10.3934/dcdsb.2019030

## Global dynamics of a virus infection model with repulsive effect

 Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China

Received  October 2017 Revised  April 2018 Published  February 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11671359, No. 11271342), the provincial Natural Science Foundation of Zhejiang (No. LY19A010027, No. LY18A010013) and the Science Foundation of Zhejiang Sci-Tech University under Grant No. 15062173-Y

This paper is devoted to investigate a virus infection model with a spatially heterogeneous structure and nonlinear diffusion. First we establish the properties of the basic reproduction number $R_0$ for infected cells and free virus particles. Then we prove that the comparison principle can be applied to an auxiliary system with quasilinear diffusion under appropriate conditions. Then the sufficient conditions for the globally asymptotical stability of infection-free steady state are obtained, which indicates that $R_0<1$ is necessary for infected cells and free virus particles to be extinct. Next we prove the existence of positive non-constant steady states and the persistence of infected cells and free virion where $R_0>1$ is required. Finally, it is shown that, for the spatially homogeneous case when the infected cells rate of change of the repulsive effect is small enough, $R_0$ is the only determinant of the global dynamics of the underlying virus infection system. The obtained results give an insight into the optimal control of the virion.

Citation: Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030
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