On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof

Rehman, Attiq ul; Singh, Ram; Agarwal, Praveen

doi:10.3390/foundations2010013

Cited by 5 publications

(3 citation statements)

References 17 publications

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“…In terms of stability, they discovered that Caputo outperformed the other two operators. In [ 23 ], utilizes the Mittag-Leffler stability idea and presents fractional Lyapunov functions for epidemic models. The global stability of the epidemic model in an equilibrium state is established.…”

Section: Introductionmentioning

confidence: 99%

Modeling and analysis of Hepatitis B dynamics with vaccination and treatment with novel fractional derivative

Zehra,

Jamil,

Farman

et al. 2024

PLoS ONE

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These days, fractional calculus is essential for studying the dynamic transmission of illnesses, developing control systems, and solving several other real-world issues. In this study, we develop a Hepatitis B (HBV) model to observe the dynamics of vaccination and treatment effects to control the disease by using novel fractional operator. Modified Atangana-Baleanu-Caputo (MABC) is a new definition of the used derivative that is based on a modification of the Atangana and Baleanu derivatives. By employing the MABC fractional derivative, which incorporates the concepts of non-locality and memory effects our model captures the complex dynamics of HBV transmission more accurately than traditional models. An objective of this study is to analyze the effect of immunization and treatment techniques on the course of the hepatitis B virus, with a particular focus on the changing order of differentiation. Thereby, our paper deals with the stability analysis, positiveness, existence and uniqueness of the solution and simulations. Analysis of reproductive number R0 with the impact of different parameters is also treated. The proposed model’s existence and uniqueness findings are examined through the use of Banach’s fixed point and Leray-Schauder nonlinear alternative theorems. The equilibria for the models are determined to be globally stable using Lyapunov functions. The simulations for certain parameters are achieved by applying the Lagrange interpolation for the numerical computations and also the results are compared with the ABC operator results. The model is validated using numerical simulations, which are also used to assess how well different intervention techniques work to lower the impact of HBV infection and prevent its spread throughout the community. The results of this research assist in developing public health policies intended to lower the incidence of HBV infection worldwide and offer insightful information about how well treatment and vaccination strategies work to prevent HBV disease.

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Section: Introductionmentioning

confidence: 99%

Modeling and analysis of Hepatitis B dynamics with vaccination and treatment with novel fractional derivative

Zehra,

Jamil,

Farman

et al. 2024

PLoS ONE

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“…In addition, in this paper, to illustrate an application of our inequalities for Lyapunov functions, we define generalized Mittag-Leffler stability in time for the above mentioned Riemann-Liouvile-type fractional derivatives and apply our inequalities to obtain sufficient conditions for this type of stability. The Mittag-Leffler stability is studied by Lyapunov functions and applied to some models, such as some epidemiological models with Caputo fractional derivative in [24], and tempered fractional neural networks in [25].…”

Section: Introductionmentioning

confidence: 99%

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Agarwal,

Hristova,

O’Regan

2023

Mathematics

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In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

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“…In [17], fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established.…”

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confidence: 99%

Editorial for the Special Issue of Foundations “Recent Advances in Fractional Differential Equations and Inclusions”

Ntouyas

2023

Foundations

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On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof

Abstract: In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established.

Cited by 5 publications

References 17 publications

Modeling and analysis of Hepatitis B dynamics with vaccination and treatment with novel fractional derivative

Modeling and analysis of Hepatitis B dynamics with vaccination and treatment with novel fractional derivative

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

Editorial for the Special Issue of Foundations “Recent Advances in Fractional Differential Equations and Inclusions”

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