Global stability of the steady states of an epidemic model incorporating intervention strategies

Yongli Cai; Yun Kang; Weiming Wang

doi:10.3934/mbe.2017056

Article Contents

2017, Volume 14, Issue 5&6: 1071-1089. Doi: 10.3934/mbe.2017056

This issue Previous Article Preface Next Article Effect of seasonal changing temperature on the growth of phytoplankton

Global stability of the steady states of an epidemic model incorporating intervention strategies

1.
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
2.
Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

* Corresponding author: Weiming Wang
* Corresponding author: Weiming Wang

Received: June 30, 2016

Accepted: September 30, 2016

Published: November 2017

The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to improvement of our original manuscript. This research was supported by the National Science Foundation of China (11601179, 61373005 & 61672013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (16KJB110003). The research of YK is partially supported by NSF-DMS (Award Number 1313312) and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472).

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Abstract

In this paper, we investigate the global stability of the steady states of a general reaction-diffusion epidemiological model with infection force under intervention strategies in a spatially heterogeneous environment. We prove that the reproduction number $\mathcal{R}_0$ can be played an essential role in determining whether the disease will extinct or persist: if $\mathcal{R}_0<1$ , there is a unique disease-free equilibrium which is globally asymptotically stable; and if $\mathcal{R}_0>1$ , there exists a unique endemic equilibrium which is globally asymptotically stable. Furthermore, we study the relation between $\mathcal{R}_0$ with the diffusion and spatial heterogeneity and find that, it seems very necessary to create a low-risk habitat for the population to effectively control the spread of the epidemic disease. This may provide some potential applications in disease control.

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Mathematics Subject Classification: Primary: 35B36, 45M10; Secondary: 92C15.

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Figure 1. In the low-risk domain of model (4), (a) the influence of the diffusion coefficient $d$ on $\mathcal{R}_0$; (b) the influence of the spatial heterogeneity of environment on $\mathcal{R}_0$. The parameters are taken as (33)

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Figure 2. In the high-risk domain of model (4), (a) the influence of the diffusion coefficient $d$ on $\mathcal{R}_0$; (b) the influence of the spatial heterogeneity of environment on $\mathcal{R}_0$. The parameters are taken as (33)

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