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## Dynamical analysis of a diffusive SIRS model with general incidence rate

 1 School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China 2 Department of Mathematics, Sichuan University, Chengdu 610064, China 3 Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia 4 Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

* Corresponding author: Lan Zou, Email: lanzou@163.com

Received  May 2019 Revised  August 2019 Published  December 2019

Fund Project: Partially supported by National Natural Science Foundation of China (No. 11671114, 11831012 and 11771168), and Natural Science Foundation of Zhejiang Province (SY20A010005).

In this paper, we propose a diffusive SIRS model with general incidence rate and spatial heterogeneity. The formula of the basic reproduction number $\mathcal R_0$ is given. Then the threshold dynamics, including globally attractive of the disease-free equilibrium and uniform persistence, are established in terms of $\mathcal{R}_0$. Special cases and numerical simulations are presented to support our main results.

Citation: Yu Yang, Lan Zou, Tonghua Zhang, Yancong Xu. Dynamical analysis of a diffusive SIRS model with general incidence rate. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020017
##### References:

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##### References:
Variation of populations of system (15) with $D = 1$, $\alpha_2 = 0$ and other parameters in (16)
Variation of populations of system (15) with $D = 1$, $\alpha_2 = 0.1$ and other parameters in (16)
The relationship of $\mathcal R_0$ and $\gamma_1$
The relationship of $\mathcal R_0$ and $\alpha_2$
When $\alpha_2 = 0$ and $D = 10^{-5}$, the evolution of the infective individuals $I(x, t)$ with parameters in (16) and $\mathcal R_0\approx 1.9375$
The relation between $\mathcal R_0$ and $c$ in $\beta(x)$
When $\alpha_2 = 0$ and $D = 10^{6}$, the evolution of the infective individuals $I(x, t)$ with parameters in (16) and $\mathcal R_0\approx 1.0373$
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