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

Jinzhe Suo; Bo Li; Jinzhe Suo; Bo Li

doi:10.3934/mbe.2020023

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 418-441. doi: 10.3934/mbe.2020023

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

Jinzhe Suo ,
Bo Li ^,

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China

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

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Abstract

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.

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