Fractal Fractional Operators Research Articles

ANALYSIS AND DYNAMICS OF CHOLERA EPIDEMIC SYSTEM IN SOCIETY VIA FRACTAL-FRACTIONAL OPERATOR

To comprehend the dynamics of disease propagation within a society, mathematical formulation is a crucial tool to understand the complex dynamics. In order to transform the mathematical model with the objective of bolstering the immune system into a fractional-order model, we use the definition of Fractal-Fractional with Mittag-Leffler kernel. For an assessment of the stable position of a recently modified system, qualitative as well as quantitative assessments are carried out. We validate the property positivity and reliability of the developed system by evaluating its boundedness and uniqueness, which are important features of an epidemic model. The positive solutions with linear growth have been verified by the global derivative, and the level of effects of different parameters in each sub-section is determined through employing Lipschitz criteria. By employing Lyapunov’s first and second derivatives of the function, the framework is examined on a global scale to evaluate the overall effect with symptomatic and asymptomatic measures. Bifurcation analysis was performed to check the behavior of each sub-compartment under different parameters effects. The Mittag-Leffler kernel is used to obtain a robust solution via Fractal-Fractional operator for continuous monitoring of spread and control of cholera disease under different dimensions. Simulations are carried out to observe both the symptomatic and asymptomatic consequences of cholera globally, also to observe the actual behavior of cholera disease for control measures, and it has been confirmed that those with strong immune systems individuals recover early due to early detection measures. The actual state of cholera disease can be controlled by taking the following measures: early detection of disease for both individuals receiving medication and those who do not require medication because of their robust immune systems. This kind of research will be beneficial in determining how diseases spread and in developing effective control plans based on our validated findings.

SENSITIVITY ANALYSIS AND NUMERICAL MODELING OF INFLUENZA PROPAGATION AND INTERVENTION STRATEGIES UNDER THE FRACTAL-FRACTIONAL OPERATOR

This paper introduces a novel nonlinear fractal fractional model to comprehensively analyze influenza epidemics. By integrating fractional calculus and considering nonlinear interactions among individuals, the model utilizes the Fractal-Fractional (FF) operator. This operator, combining fractal and fractional calculus, establishes a unique framework for investigating influenza virus propagation dynamics and potential vaccination strategies. Amid growing concerns over influenza outbreaks, fractional derivatives are employed to address intricate challenges. The proposed model sheds light on virus spread dynamics and countermeasures. The integration of the FF operator enriches analysis, while the application of the fixed point theory of Schauder and Banach demonstrates solution existence and uniqueness. Model validation employs numerical simulations with MATLAB12̂ and the Adams–Bashforth method, confirming its ability to capture influenza propagation dynamics accurately. Ulam–Hyers stability techniques ensure the model’s reliability. Beyond its scientific contributions, the model underscores the significance of studying influenza epidemics via mathematical modeling in understanding disease dynamics and guiding effective intervention strategies. Through a synergy of mathematical innovation and epidemiological insights, this study establishes a robust foundation for more effective strategies against influenza epidemics.

Assessing the global dynamics of Nipah infection under vaccination and treatment: A novel computational modeling approach.

In biology and life sciences, fractal theory and fractional calculus have significant applications in simulating and understanding complex problems. In this paper, a compartmental model employing Caputo-type fractional and fractal-fractional operators is presented to analyze Nipah virus (NiV) dynamics and transmission. Initially, the model includes nine nonlinear ordinary differential equations that consider viral concentration, flying fox, and human populations simultaneously. The model is reconstructed using fractional calculus and fractal theory to better understand NiV transmission dynamics. We analyze the model's existence and uniqueness in both contexts and instigate the equilibrium points. The clinical epidemiology of Bangladesh is used to estimate model parameters. The fractional model's stability is examined using Ulam-Hyers and Ulam-Hyers-Rassias stabilities. Moreover, interpolation methods are used to construct computational techniques to simulate the NiV model in fractional and fractal-fractional cases. Simulations are performed to validate the stable behavior of the model for different fractal and fractional orders. The present findings will be beneficial in employing advanced computational approaches in modeling and control of NiV outbreaks.

Quantum–Fractal–Fractional Operator in a Complex Domain

In this effort, we extend the fractal–fractional operators into the complex plane together with the quantum calculus derivative to obtain a quantum–fractal–fractional operators (QFFOs). Using this newly created operator, we create an entirely novel subclass of analytical functions in the unit disk. Motivated by the concept of differential subordination, we explore the most important geometric properties of this novel operator. This leads to a study on a set of differential inequalities in the open unit disk. We focus on the conditions to obtain a bounded turning function of QFFOs. Some examples are considered, involving special functions like Bessel and generalized hypergeometric functions.

A new grey model with generalized fractal-fractional derivative for prediction of tourism development

A new fractional order grey prediction model is proposed for accurate forecasting of tourism development in China. The model combines generalized fractal-fractional derivative operators with difference and accumulation generation operators. Experimental comparisons with existing models show significant improvements in accuracy and efficiency. The model is applied to forecast tourism development in China and results are compared with actual data to verify effectiveness. The proposed model combines fractal-fractional operators to improve prediction accuracy and efficiency, accounting for various factors affecting tourism development. Comparisons with existing models show superiority in accuracy and efficiency. The model accurately predicts tourism development in China, resulting in improved forecasting compared to existing methods. Comparison with actual data further validates the model by displaying agreement between predicted and actual values. Overall, the proposed model effectively captures tourism development dynamics in China for accurate forecasting.

Dynamic analysis of fractal-fractional cancer model under chemotherapy drug with generalized Mittag-Leffler kernel.

Cancer's complex and multifaceted nature makes it challenging to identify unique molecular and pathophysiological signatures, thereby hindering the development of effective therapies. This paper presents a novel fractal-fractional cancer model to study the complex interplay among stem cells, effectors cells, and tumor cells in the presence and absence of chemotherapy. The cancer model with effective treatment through chemotherapy drugs is considered and discussed in detail. The numerical method for the fractal-fractional cancer model with a generalized Mittag-Leffler kernel is presented. The Routh-Hurwitz stability criteria are applied to confirm the local asymptotically stability of an endemic equilibrium point of the cancer model without treatment and with effective treatment under some conditions. The existence and uniqueness criteria of the fractal-fractional cancer model are derived. Furthermore, the stability analysis of the fractal-fractional cancer model is performed. The temporal concentration pattern of stem cells, effectors cells, tumor cells, and chemotherapy drugs are procured. The usage of chemotherapy drugs kills the tumor cells or decreases their density over time and as a consequence takes a longer time to reach to equilibrium point. The decay rate of stem cells and tumor cells plays a crucial role in cancer dynamics. The notable role of fractal dimensions along with fractional order is observed in capturing the cancer cell concentration. A dynamic analysis of the fractal-fractional cancer model is demonstrated to examine the impact of chemotherapy drugs with a generalized Mittag-Leffler kernel. The model possesses three equilibrium points among them two correspond to the cancer model without treatment namely the tumor-free equilibrium point and endemic equilibrium point. One additional endemic equilibrium point exists in the case of effective treatment through chemotherapy drugs. The Routh-Hurwitz stability criteria are applied to confirm the local asymptotically stability of an endemic equilibrium point of the cancer model with and without treatment under some conditions. The chemotherapy drug plays a crucial role in controlling the growth of tumor cells. The fractal-fractional operator provided robustness to study cancer dynamics by the inclusion of memory and heterogeneity.

Mathematical analysis with control of liver cirrhosis causing from HBV by taking early detection measures and chemotherapy treatment

Reformulating the physical processes associated with the evolution of different ailments in accordance with globally shared objectives is crucial for deeper comprehension. This study aims to investigate the mechanism by which the HB virus induces harmful inflammation of the liver, with a focus on early detection and therapy using corticosteroids or chemotherapy. Based on the developed hypothesis, a new mathematical model has been created for this purpose. The recently developed system for HBV is SEI1I2R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SEI_{1} I_{2}R$$\\end{document}, which is examined both quantitatively and qualitatively to determine its actual effect on stability. Reliable conclusions are ensured by examining the system’s boundedness, positivity, existence, uniqueness, and conducting local and global stability analysis-all crucial components of epidemic models. Global stability is tested using Lyapunov first derivative functions to assess the overall impact of asymptomatic persons and chemotherapy treatment. Additionally, the Lipschitz condition is used to confirm the unique solutions for the newly built HBV model using methods from fixed point theory, thus meeting the requirements for uniqueness and existence. Since the population must maintain this property, positivity is confirmed using global derivatives and Lipschitz criteria to calculate the rate of change in each sub-compartment. Applying the Mittag-Leffler kernel with a fractal-fractional operator to continuously monitor the HBV virus for liver cirrhosis infection yields dependable results. Furthermore, the current situation regarding the HBV outbreak pertaining to liver cirrhosis infection, along with the control measures implemented following early diagnosis through asymptomatic measures and chemotherapy treatment under constant observation, are established to prevent chronic stage infections. Simulations have been used to study the true behavior and impact of HBV in asymptomatic persons receiving chemotherapy for liver cirrhosis infection in the community. This research is essential for understanding the spread of viruses and developing control strategies based on our validated findings to mitigate the risk factors associated with liver cirrhosis.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.

Computation of Thermal Radiative Flux from a Porous Medium by Using Fractal–Fractional Operator

Computation of Thermal Radiative Flux from a Porous Medium by Using Fractal–Fractional Operator

Analysis and dynamical transmission of tuberculosis model with treatment effect by using fractional operator

ABSTRACT Each year, millions of people die from the airborne infectious illness tuberculosis (TB). Several drug-susceptible (DS) and drug-resistant (DR) forms of the causative agent, Mycobacterium tuberculosis (MTB), are currently common in the majority of affluent and developing nations, particularly in Bangladesh, and completely drug-resistant strains are beginning to arise. The main purpose of this research is to develop and examine a non-integer-order mathematical model for the dynamics of tuberculosis transmission using the fractal fractional operator. By demonstrating characteristics such as the boundedness of solutions, positivity, and reliance of the solution on the original data, the biological well-posedness of the mathematical model formulation was investigated for TB cases from 2002 to 2017 in KPK Pakistan. Ulam-Hyres stability is also used to assess both local and global aspects of TB bacterial infection. Sensitivity analysis of the TB model with therapy was also examined. The advanced numerical technique is used to find the solution of the fractional-order system to check the impact of fractional parameters. Simulation highlights that all classes have converging qualities and retain established positions with time, which shows the actual behavior of bacterial infection with TB.

Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ−fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators.

Mathematical analysis of Ebola considering transmission at treatment centres and survivor relapse using fractal-fractional Caputo derivatives in Uganda

In this article, we seek to formulate a robust mathematical model to study the Ebola disease through fractal-fractional operators. The study thus incorporates the transmission rate in the treatment centers and the relapse rate, since the Ebola virus persists or mostly hides in the immunologically protected sites of survivors. The Ebola virus disease (EVD) is one of the infectious diseases that has recorded a high death rate in countries where it is endemic, and Uganda is not an exception. The world at large has suffered from this deadly disease since 1976 when it was declared epidemic by the World Health Organization. The study employed fractal-fractional operators to identify the epidemiological patterns of EVD, especially in treatment centers and relapse. Memory loss and relapse are mostly observed in EVD survivors and this justifies the use of fractional operators to capture the true dynamics of the disease. Through dynamical analysis, the model is proven to be positive and bounded in the region. The model is further explicitly shown to have a solution that is unique and stable. The reproduction number was duly computed by using the next-generation matrix approach. By taking EVD epidemic cases in Uganda, the study fitted all parameters to real data. It has been shown through sensitivity index analysis that the transmission rate outside treatment centers and relapse have a significant effect on the endemic state of the disease, as they lead to an increase in the basic reproduction ratio.

Mathematical modeling of allelopathic stimulatory phytoplankton species using fractal–fractional derivatives

In the current study, we employ the novel fractal–fractional operator in the Atangana–Baleanu sense to investigate the dynamics of an interacting phytoplankton species model. Initially, we utilize the Picard-Lindelöf theorem to validate the uniqueness and existence of solutions for the model. We then explore equilibrium points within the phytoplankton model and conduct Hyers–Ulam stability analysis. Additionally, we present a numerical scheme utilizing the Newton polynomial to validate our analytical findings. Numerical simulations illustrate the dynamical behavior of the model across various fractal and fractional parameter values, visualized through graphical representations. Our simulations reveal that the stability of equilibrium points is not significantly impacted with the long-term memory effect, which is characterized by fractal–fractional order values. However, an increase in fractal–fractional parameters accelerates the convergence of solutions to their intended equilibrium states.

A chaotic study of love dynamics with competition using fractal-fractional operator

PurposeThe objective of this work is to analyze the necessary conditions for chaotic behavior with fractional order and fractal dimension values of the fractal-fractional operator.Design/methodology/approachThe numerical technique based on the fractal-fractional derivative is implemented over the fractional model and analyzes the condition at the distinct values of fractional order and fractal dimension.FindingsThe obtained numerical solution from the numerical technique is analyzed at distinct fractional order and fractal dimension values, and it has been figured out that the behavior of the solution either chaotic or non-chaotic agrees with the condition.Originality/valueThe necessary condition is associated with the fractional order only. So, our work not only studies the condition with fractional order but also examines the model by simultaneously adjusting fractal dimension values. It is found that the model still has chaotic or non-chaotic behavior at certain fractal dimension values and fractional order values corresponding to the condition.

Dynamics of an epidemic model for COVID-19 with Hattaf fractal-fractional operator and study of existence of solutions by means of fixed point theory

Dynamics of an epidemic model for COVID-19 with Hattaf fractal-fractional operator and study of existence of solutions by means of fixed point theory

Numerical study and dynamics analysis of diabetes mellitus with co-infection of COVID-19 virus by using fractal fractional operator

COVID-19 is linked to diabetes, increasing the likelihood and severity of outcomes due to hyperglycemia, immune system impairment, vascular problems, and comorbidities like hypertension, obesity, and cardiovascular disease, which can lead to catastrophic outcomes. The study presents a novel COVID-19 management approach for diabetic patients using a fractal fractional operator and Mittag-Leffler kernel. It uses the Lipschitz criterion and linear growth to identify the solution singularity and analyzes the global derivative impact, confirming unique solutions and demonstrating the bounded nature of the proposed system. The study examines the impact of COVID-19 on individuals with diabetes, using global stability analysis and quantitative examination of equilibrium states. Sensitivity analysis is conducted using reproductive numbers to determine the disease’s status in society and the impact of control strategies, highlighting the importance of understanding epidemic problems and their properties. This study uses two-step Lagrange polynomial to analyze the impact of the fractional operator on a proposed model. Numerical simulations using MATLAB validate the effects of COVID-19 on diabetic patients and allow predictions based on the established theoretical framework, supporting the theoretical findings. This study will help to observe and understand how COVID-19 affects people with diabetes. This will help with control plans in the future to lessen the effects of COVID-19.

Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines (IL2&IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ IL_{2} \\& IL_{12}$$\\end{document}) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCDIL2IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$IL_{2} IL_{12}$$\\end{document}Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

Studies in fractal–fractional operators with examples

By using the generalization of the gamma function (p-gamma function: Γp(.)), we introduce a generalization of the fractal–fractional calculus which is called p-fractal fractional calculus. We extend the proposed operators into the symmetric complex domain, specifically the open unit disk. Normalization for each operator is formulated. This allows us to explore the most important geometric properties. Examples are illustrated including the basic power functions.

Numerical investigation of pine wilt disease using fractal–fractional operator

Numerical investigation of pine wilt disease using fractal–fractional operator

A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances

Background:Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal–fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population. Methods:Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model’s equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel. Results:The stability analysis of the fractal–fractional model has been confirmed for both Ulam–Hyers and generalized Ulam–Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal–fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values. Conclusion:According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control.