An analytical approach to applying the Lyapunov direct method to an epidemic model with age and stage structures

IF 1.8 3区数学 Q1 MATHEMATICS, APPLIED

Nonlinear Analysis-Real World Applications Pub Date : 2025-01-07 DOI:10.1016/j.nonrwa.2024.104312

Jianquan Li , Yuming Chen , Xiaojian Xi , Nini Xue

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引用次数: 0

Abstract

Usually, it is very challenging to construct appropriate Lyapunov functionals for proving the global stability of age-structured models. In this paper, we propose an analytical approach to applying the Lyapunov direct method for such models. The novelty of this approach lies in successfully handling the two challenges when applying the method. On the one hand, according to the integral terms involved in the model, we propose an easy-to-follow way to determine the kernel functions in the Lyapunov functional candidate. On the other hand, we establish a new integral inequality, which is conducive to arranging the derivative of the functional so that it is easy to see whether the derivative is negative definite or negative semi-definite. As an application, we investigate the global stability of the endemic steady state of an age-structured epidemic model with two infectious stages. Moreover, the Lyapunov functional obtained for the endemic steady state is also helpful for proving the global stability of the disease-free steady state and the persistence of the disease.

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李雅普诺夫直接法在具有年龄和阶段结构的流行病模型中的应用分析方法

通常，构造合适的Lyapunov泛函来证明年龄结构模型的全局稳定性是非常具有挑战性的。在本文中，我们提出了一种分析方法来应用李雅普诺夫直接方法对这类模型。该方法的新颖之处在于在应用该方法时成功地处理了这两个挑战。一方面，根据模型中涉及的积分项，我们提出了一种易于理解的方法来确定Lyapunov候选函数中的核函数。另一方面，我们建立了一个新的积分不等式，这有利于对泛函的导数进行排列，以便于判别导数是负定还是负半定。作为应用，我们研究了具有两个传染阶段的年龄结构流行病模型的地方性稳态的全局稳定性。此外，获得的地方病稳态的Lyapunov泛函也有助于证明无病稳态的全局稳定性和疾病的持续性。

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

Nonlinear Analysis-Real World Applications 数学-应用数学

CiteScore

3.80

自引率

5.00%

发文量

176

审稿时长

59 days

期刊介绍： Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems. The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.