### Mathematical Biosciences and Engineering

2020, Issue 1: 418-441. doi: 10.3934/mbe.2020023
Research article Special Issues

# Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment

• Received: 23 June 2019 Accepted: 06 September 2019 Published: 14 October 2019
• In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number $\mathcal{R}_{0}>1$. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c = 0, some new dynamical behaviors appear in the model studied here; in particular, $\mathcal{R}_{0}$ is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.

Citation: Jinzhe Suo, Bo Li. Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 418-441. doi: 10.3934/mbe.2020023

### Related Papers:

• In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number $\mathcal{R}_{0}>1$. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c = 0, some new dynamical behaviors appear in the model studied here; in particular, $\mathcal{R}_{0}$ is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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