Global behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate
type: Item
instance of: scholarly article
title: Global behaviors of a class of discrete SIRS epidemic models with nonlinear incidence rate
zbMATH Open document ID: 1406.92629
DOI: 10.1155/2014/249623
author: Lei Wang
author: Long Zhang
author: Zhidong Teng
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: We study a class of discrete SIRS epidemic models with nonlinear incidence rate \(F(S) G(I)\) and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number \(\mathcal{R}_0 < 1\), then the disease-free equilibrium is globally asymptotically stable, and if \(\mathcal{R}_0 > 1\), then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when \(\mathcal{R}_0 > 1\), the disease in the model is permanent. Some special cases of \(F(S) G(I)\) are discussed. Particularly, when \(F(S) G(I) = \beta S I /(1 + \lambda I)\), it is obtained that the endemic equilibrium is globally asymptotically stable if and only if \(\mathcal{R}_0 > 1\). Furthermore, the numerical simulations show that for general incidence rate \(F(S) G(I)\) the endemic equilibrium may be globally asymptotically stable only as \(\mathcal{R}_0 > 1\).
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 34D23
zbMATH DE Number: 7021999
zbMATH Keywords: discrete SIRS epidemic models
zbMATH Keywords: nonlinear incidence rate
zbMATH Keywords: disease-induced mortality
zbMATH Keywords: global asymptotic stability
Wikidata QID: Q59036875
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/249623
OpenAlex ID: W2035691413
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description: scientific article; zbMATH DE number 7021999