Global dynamics for a class of discrete SEIRS epidemic models with general nonlinear incidence.

IF 4.1 3区数学 Q1 Mathematics

Advances in Difference Equations Pub Date : 2016-01-01 Epub Date: 2016-05-06 DOI:10.1186/s13662-016-0846-y

Xiaolin Fan, Lei Wang, Zhidong Teng

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Abstract

In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number $R_{0} \leq 1$ , then disease-free equilibrium is globally attractive, and if $R_{0} > 1$ , then the disease is permanent. When the model degenerates into SEIR model, it is proved that if $R_{0} > 1$ , then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when $R_{0} > 1$ , the endemic equilibrium of general SEIRS models is also globally attractive.

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具有一般非线性发病率的一类离散 SEIRS 流行病模型的全局动力学。

本文研究了一类具有一般非线性发病率的离散 SEIRS 流行病模型。特别是，还考虑了具有标准入射率的离散 SEIRS 流行模型。得到了具有正初始条件的解的实在性和有界性。结果表明，如果基本繁殖数 R 0 ≤ 1，则无疾病平衡是全局有吸引力的；如果 R 0 > 1，则疾病是永久性的。当模型退化为 SEIR 模型时，证明了如果 R 0 > 1，则模型有一个唯一的流行均衡，该均衡具有全局吸引力。此外，数值示例还验证了一个重要的未决问题，即当 R 0 > 1 时，一般 SEIRS 模型的流行均衡也具有全局吸引力。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Difference Equations 数学-数学

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0.00%

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审稿时长

4-8 weeks

期刊介绍： The theory of difference equations, the methods used, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last 15 years, the proliferation of the subject has been witnessed by hundreds of research articles, several monographs, many international conferences, and numerous special sessions. The theory of differential and difference equations forms two extreme representations of real world problems. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. The actual behavior of the population is somewhere in between. The aim of Advances in Difference Equations is to report mainly the new developments in the field of difference equations, and their applications in all fields. We will also consider research articles emphasizing the qualitative behavior of solutions of ordinary, partial, delay, fractional, abstract, stochastic, fuzzy, and set-valued differential equations. Advances in Difference Equations will accept high-quality articles containing original research results and survey articles of exceptional merit.