The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis

Naim, Mouhcine; Sabbar, Yassine; Zahri, Mostafa; Ghanbari, Behzad; Zeb, Anwar; Gul, Nadia; Djilai, Salih; Lahmidi, Fouad

doi:10.1088/1402-4896/ac9e7a

Cited by 6 publications

(3 citation statements)

References 68 publications

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“…The subsequent manuscript proceeds from the fact that fractional systems can characterize the virological properties better compared to the ordinary formulation and treat a Caputo fractional order viral formulation under some thought-provoking assumptions [19]. Overall, the manuscript can provide the enrichment and generalization of the work of Dhar et al through the consideration of a general hypothetical setting.…”

Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

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Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

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Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

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“…Singular systems of differential equations, see [2,3,5,7,12,17,20], and difference equations, see [8,21] have attracted the interest of several researchers in the last few decades. Some interesting results have also been obtained for singular systems of equations evolving fractional operators, see [1,6,9,11,13,15,29,34]. This type of systems appear in control theory, see [3,10,33], and in several applications in electrical engineering such as the modeling of electrical circuits, see [20], electricity markets, see [14], and power system dynamics, see [28,32].…”

Section: Introductionmentioning

confidence: 99%

On the relations between a singular system of differential equations and a system with delays

Dassios

2022

mmnsa

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In this article, we consider a class of systems of differential equations with multiple delays. We define a transform that reformulates the system with delays into a singular linear system of differential equations whose coefficients are non-square constant matrices, and the number of their columns is greater than the number of their rows. By studying only the singular system, we provide a form of solutions for both systems.

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“…Oscillatory and chaotic dynamics of the HIV-1 model has been studied through FC in the literature [7]. FC calculus has been used in virology [8,9]. Zeb et al used a piecewise fractional order model of COVID-19 [10].…”

Section: Introductionmentioning

confidence: 99%

Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

Ahmad

Rahman

et al. 2022

Mathematics

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Recently, several research articles have investigated the existence of solutions for dynamical systems with fractional order and their controllability. Nevertheless, very little attention has been given to the observability of such dynamical systems. In the present work, we explore the outcomes of controllability and observability regarding a differential system of fractional order with input delay. Laplace and inverse Laplace transforms, along with the Mittage–Leffler matrix function, are applied to the proposed dynamical system in Caputo’s sense, and a general solution is obtained in the form of an integral equation. Then, we set out conditions for the controllability of the underlying model, regarding the linear case. We then expound controllability conditions for the nonlinear case by utilizing the fixed point result of Schaefer and the Arzola–Ascoli theorem. Using the fixed point concept, we prove the observability of the linear case using the observability Grammian matrix. The necessary and sufficient conditions for the nonlinear case are investigated with the help of the Banach contraction mapping theorem. Finally, we add some examples to elaborate on our work.

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The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis

Cited by 6 publications

References 68 publications

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

On the relations between a singular system of differential equations and a system with delays

Controllability and Observability Results of an Implicit Type Fractional Order Delay Dynamical System

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