Analysis of a TB and HIV co-infection model under Mittag-Leffler fractal-fractional derivative

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.

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

“…Many diseases, such as HIV/AIDS and COVID-19, have become endemic, causing thousands of deaths each year. Some works on communicable diseases have been published in the literature [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Insights into COVID-19 stochastic modelling with effects of various transmission rates: simulations with real statistical data from UK, Australia, Spain, and India

Xu,

Pang,

Liu

et al. 2024

In the literature \cite{saeed}, the COVID-19 model has been constructed using deterministic approach. The present manuscript examines a stochastic model designed to capture the interplay between COVID-19 and varying infection rates on disease dynamics. We present the necessary criteria for a global solution to the considered model to exist and be unique. To illustrate several outcomes pertaining to the ergodic properties of the given system, the we utilize nonlinear analysis. Furthermore, the model undergoes simulation and is compared with deterministic dynamics. To verify the efficacy of the considered model and demonstrate its utility, we compare the dynamics of the infected population to real statistical data from multiple countries, such as the United Kingdom, Australia, Spain, and India. The proposed model has proven to be a reliable and effective tool for understanding the intricate nature of COVID-19 dynamics. Moreover, we provide a visually striking depiction of the impact of different infection rates on the propagation of the model under investigation. This visualization provides valuable insight into the multifaceted nature of the pandemic and significantly contributes to the comprehension of COVID-19 dynamics.

“…In addition to using fractal dimensions for statistical self-similarity, one may also use mathematical models that might approximate the production of actual fractal objects based on direct observation [15]. The FF operators have several applications in applied sciences such as mathematical physics [16,17], chaotic systems [18], mathematical biology [19] and many more [20,21].…”

Section: Introductionmentioning

confidence: 99%

Investigation of fractal fractional nonlinear Korteweg-de-Vries-Schrödinger system with power law kernel

Khan

Ahmad

2023

Self Cite

In this research article, we apply the Yang transform homotopy perturbation method (YTHPM) to solve the Schrödinger-KdV equation with Caputo FF operator. We analyze and prove the existence and uniqueness of the solution, as well as provide graphical representations of the results. Using the YTHPM, we derive an approximate solution to the Schrödinger-KdV equation. The Banach fixed point theorem is used to prove the solution's existence and uniqueness, and graphical representations are provided to showcase its behavior and accuracy. Our findings demonstrate that the YTHPM and the Caputo fractal-fractional operator are effective in solving the Schrödinger-KdV equation.