Fractal Fractional Model Research Articles

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.

Fractal-fractional model for the receptor in a 3D printing system

A critical hurdle in three-dimensional printing is the low printing accuracy. In particular, the deformation of the printed object on the receptor makes accurate printing impossible due to insufficient solvent evaporation. We address this challenge by establishing a Murray-like receptor to control the morphology of the printed object, which can be used for accurate printing through efficient residual energy absorption and active vibration attenuation. A fractal oscillator is established and He’s frequency formulation is used to reveal the fractal response of the Murray-like receptor, and an experiment was carried out to show that the receptor system can keep the printed object in good shape without any deformation.

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.

Fractal fractional model for tuberculosis: existence and numerical solutions

This paper deals with the mathematical analysis of Tuberculosis by using fractal fractional operator. Mycobacterium TB is the bacteria that causes tuberculosis. This airborne illness mostly impacts the lungs but may extend to other body organs. When the infected individual coughs, sneezes or speaks, the bacterium gets released into the air and travels from one person to another. Five classes have been formulated to study the dynamics of this disease: susceptible class, infected of DS, infected of MDR, isolated class, and recovered class. To study the suggested fractal fractional model’s wellposedness associated with existence results, and boundedness of solutions. Further, the invariant region of the considered model, positive solutions, equilibrium point, and reproduction number. One would typically employ a fractional calculus approach to obtain numerical solutions for the fractional order Tuberculosis model using the Adams-Bashforth-Moulton method. The fractional order derivatives in the model can be approximated using appropriate numerical schemes designed for fractional order differential equations.

FRACTAL FRACTIONAL MODEL OF RADIATIVE HEAT AND MASS TRANSFER CHARACTERISTICS OF TIME DEPENDENT FLOW WITH VARIABLE VISCOSITY, INDUCED MAGNETIC FIELD, DUFOUR AND SORET EFFECTS: AN ENTROPY GENERATION

: In this article, the Caputo Fabrizio fractal fractional order derivatives operator with an exponential kernel was employed to examine the radiative heat and mass transfer characteristics of time dependent flow with variable fluid viscosity, induced magnetic field, Soret and Dufour effects. The local mathematical model for the flow problem is formulated by take into account the impacts of thermal radiation, heat source, and viscous dissipation. The governing equations in-terms of fractal fractional model with an exponential kernel were solved numerically using finite difference method. The influence of flow variables such as induced magnetic field, concentration field, entropy rate, thermal field, and velocity field profiles against the pertinent parameters are discussed through graphs. Increase the values of magnetic Prandtl number results to rises of induced magnetic field. Higher Dufour number significantly grows the thermal field. The fractal fractional parameters enhance the velocity field, thermal field, Bejan number, entropy rate, concentration field and induced magnetic field profiles. The velocity field profiles recede with higher values of fluid variable viscosity parameter whereas the thermal field and induced magnetic field has an opposite effect. Larger Soret number amplifies the concentration field. Increase of Brinkman number, thermal radiation parameter, and magnetic Prandtl number intensifies the entropy generation rate. Increases of Brinkman number, magnetic Prandtl number, Soret number and Dufour number leads to a decrease of Bejan number whereas Bejan number rises with an increase of thermal radiation parameter and heat source

Investigating a Fractal–Fractional Mathematical Model of the Third Wave of COVID-19 with Vaccination in Saudi Arabia

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.

Modeling Rift Valley fever transmission: insights from fractal-fractional dynamics with the Caputo derivative

<abstract><p>The infection caused by Rift Valley fever (RVF) virus is a dangerous vector-borne disease found in humans, domestic, and wild animals. It is transferred through insect vectors to ruminant host and then spread through direct contact of infected animals with their body fluid or organs. In this paper, a fractal-fractional model for the transmission of RVF in the Caputo's sense was presented. We analyzed the model and determined the basic reproduction number through the next-generation matrix technique, indicated by $ \mathcal{R}_0 $. The global sensitivity technique is used for the sensitivity test of $ \mathcal{R}_0 $ to find out the most sensitive input-factors to the reproduction parameter $ \mathcal{R}_0 $. The existence and uniqueness results of the proposed fractal-fractional model were established. Then, we presented the fractal-fractional dynamics of the proposed RVF model through a novel numerical scheme under the fractal-fractional Caputo operator. In the end, the recommended model of RVF was highlighted numerically with the variation of different input parameters of the system. The key factors of the system were highlighted to the policymakers for the control and prevention of the infection.</p></abstract>

On the fractal-fractional Mittag-Leffler model of a COVID-19 and Zika Co-infection

The World Health Organization declared COVID-19 a global pandemic in March 2020, which had a significant impact on global health and economies. There have been several Zika outbreaks in different regions such as Africa, Southeast Asia, and the Americas. Therefore, it is essential to study the dynamics of these two diseases, taking into account their memory and recurrence effects. A new fractal-fractional hybrid Mittag-Leffler model of COVID-19 and Zika co-dynamics is designed and studied to evaluate the effects of COVID-19 on Zika and vice-versa. The stability analysis of the local asymptotic type at disease-free equilibrium is conducted for the hybrid model. The existence of unique solutions to the model is established via some fixed point results. The fractal-fractional model is proved to be Hyers–Ulam stable. With the help of Newton polynomials, we obtain some numerical algorithms to approximate the solutions of the fractal-fractional hybrid Mittag-Leffler model graphically. The impact of fractional and fractal orders on the dynamics of each of the epidemiological classes is also assessed. In addition, empirical evidence from numerical simulations suggests that implementing measures to contain the transmission of the SARS-CoV-2 virus can significantly contribute to the reduction of co-infections involving the Zika virus. Therefore, it is imperative for healthcare systems to maintain a state of constant vigilance in order to detect any atypical patterns or probable occurrences of co-infections, particularly in areas where both diseases are widespread. Additionally, it is vital to consult the most recent directives provided by health authorities, as our comprehension of diseases may undergo advancements over the course of time.

A fractal-fractional sex structured syphilis model with three stages of infection and loss of immunity with analysis and modeling

Treponema pallidum, a spiral-shaped bacterium, is responsible for the sexually transmitted disease syphilis. Millions of people in less developed countries are getting the disease despite the accessibility of effective preventative methods like condom use and effective and affordable treatment choices. The disease can be fatal if the patient does not have access to adequate treatment. Prevalence has hovered between endemic levels in industrialized countries for decades and is currently rising. Using the Mittag-Leffler kernel, we develop a fractal-fractional model for the syphilis disease. Qualitative as well as quantitative analysis of the fractional order system are performed. Also, fixed point theory and the Lipschitz condition are used to fulfill the criteria for the existence and uniqueness of the exact solution. We illustrate the system’s Ulam–Hyers stability for disease-free and endemic equilibrium. The analytical solution is supported by numerical simulations that show how the dynamics of the spread of syphilis within the population are influenced by fractional-order derivatives. The outcomes show that the suggested methods are effective in delivering better results. Overall, this research helps to develop more precise and comprehensive approaches to understanding and regulating syphilis disease transmission and progression.

Theoretical and numerical treatment for the fractal-fractional model of pollution for a system of lakes using an efficient numerical technique

This article proposes an efficient simulation to investigate the fractal-fractional (FF) pollution model's solution behavior for a network of three lakes connected by channels. With the aid of an efficient numerical technique for integration, the system of fractional differential equations (FDES) is numerically integrated. Special attention is given to studying the uniqueness and existence of the suggested model's solution by applying the Picard-Lindelöf's theorem and Banach's fixed point theorem. In this work, we deal with three cases (periodic, exponentially decaying, and linear) of the input models. The results are compared with that of the fourth-order Runge-Kutta technique (RK4) as well as the q-homotopy analysis transform method. The given figures for studying and simulating the proposed system through different values of FF-operators and comparisons show that the solutions for the system demonstrated the dynamic and natural behavior of the model. Our findings show that the used technique provides a straightforward and efficient tool to solve such problems. The key benefit of the suggested method is that it only requires a few easy steps, doesn't produce secular terms, and doesn't rely on a perturbation parameter.

Fractional order age dependent Covid-19 model: An equilibria and quantitative analysis with modeling

The presence of different age groups in the populations being studied requires us to develop models that account for varying susceptibilities based on age. This complexity adds a layer of difficulty to predicting outcomes accurately. Essentially, there are three main age categories: 0−19 years, 20−64 years, and >64 years. However, in this article, we only focus on two age groups (20−64 years and >64 years) because the age category 0−19 years is generally perceived as having a lower susceptibility to the virus due to its consistently low infection rate during the pandemic period of this research, particularly in the countries being examined. In this paper, we presented an age-dependent epidemic model for the COVID-19 Outbreak in Kuwait, France, and Cameroon in the fractal-fractional (FF) sense of derivative with the Mittag–Leffler kernel. The study includes positivity, stability, existence results, uniqueness, stability, and numerical simulations. Globally, the age-dependent COVID-19 fractal fractional model is examined using the first and second derivatives of Lyapunov. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model. The numerical scheme of this paper is based on the Newton polynomial and is tested for a particular case with numerical values from Kuwait, France, and Cameroon. In our analysis, we explore the significance of these distinct parameters incorporated into the model, focusing particularly on the impact of vaccination and fractional order on the progression of the epidemic. The results are getting closer to the classical case for the orders reaching 1 while all other solutions are different with the same behavior. Consequently, the fractal fractional order model provides more substantial insights into the epidemic disease. We open a novel viewpoint on enhancing an age-dependent model and applying it to real-world data and parameters. Such a study will help determine the behavior of the virus and disease control methods for a population.

Mathematical study of fractal-fractional leptospirosis disease in human and rodent populations dynamical transmission

In both industrialized and developing nations, leptospirosis is one of the most underdiagnosed and under-reported diseases. It is known that people are more likely to contract a disease depending on their employment habits and the environment they live in, which varies from community to community. The absence of global data for morbidity and mortality has contributed to leptospirosis' neglected disease status even though it is a life-threatening illness and is widely acknowledged as a significant cause of pulmonary hemorrhage syndrome. This study aims to examine the impact of rodent-borne leptospirosis on the human population by constructing and evaluating a compartmental mathematical model using fractional-order differential equations. The model considers both the presence of disease-causing agents in the environment and the rate of human infection resulting from interactions with infected rodents and the environment. Through this approach, the research investigates the dynamics and implications of leptospirosis transmission in the context of human-rodent interactions and environmental factors. We create a fractal-fractional model using the mittag-leffler kernel. The positivity and boundedness of solutions are first discussed. The model equilibria and fundamental reproduction number are then presented. With the use of the Lyapunov function method, the solutions are subjected to global stability analysis. The fixed-point theory is used to derive the fractional-order model's existence and uniqueness. Solutions are produced using a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel to explore the effect of the fractional operator with numerical simulations, which shows the influence of the sickness due to the effect of different parameters involved. Such a study will aid in the development of control strategies to combat the disease in the community and an understanding of the behavior of the Leptospira virus.

A fractal–fractional model of Ebola with reinfection

In this article, we have studied the dynamics of the Ebola virus disease by means of fractional differentiation combined with a fractal dimension. It has been shown explicitly that the Ebola model is positively invariant. The fixed point theorem procedures check the solution’s existence and uniqueness to the model using the Mittag-Leffler kernel. The stability of the Ebola fractal–fractional model is studied using the Hyers–Ulam stable analysis. The numerical simulation for the trajectories of the proposed model is obtained using Lagrangian interpolation. Also, the numerical sensitivity analysis of the model is presented. For instance, a reduction in the rate of human-to-human contact results in a decline in the rate of infection, implying that the disease at a point will die out. Again, an increase in the loss of immunity rate to unity leads to a massive spread of the Ebola disease in the population. The findings in our work depict that we can achieve an Ebola-free state should the health sector give more education on maintaining a strong immune system and reducing human-to-human contact, especially during outbreaks of the Ebola disease, using measures like isolation and quarantine. Finally, using the fractal–fractional operator, we observed that any amount of memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease.

Different strategies for diabetes by mathematical modeling: Modified Minimal Model

A fractal-fractional-order of a modified minimal model of glucose-insulin interaction model is proposed in sense of Atangana-Baleanu operator based on the intravenous glucose tolerance test (IVGT). Based on the fractal-fractional sense, we examine the existence of the solution, Hyers-Ulam stability, uniqueness, non-negativity, positivity, and boundedness. The model is analyzed qualitatively using its equivalent integral using an iterative convergence sequence and fixed point approach. A numerical scheme for the fractal-fractional model is then applied to a case study. Modified minimal models are reasonable and acceptable. We are interested to see that both analytical and simulation results agree.

Strange Fractal Attractors and Optimal Chaos of Memristor–Memcapacitor via Non-local Differentials

The multi-dimensional electronic devices are so called memory circuit elements (memristor or memcapacitor); such memory circuit elements usually rely on previous applied voltage, current, flux or charge based on memory capability with their resistance, capacitance or inductance. In view of above fact, this manuscript investigates the non-integer modeling of memristor–memcapacitor in discrete-time domain through non-singular kernels of fractal fractional differentials and integrals operators. The governing equations of memristor–memcapacitor have been developed for the sake of the dynamical characteristics of simple chaotic circuit. The fractal fractional differentials and integrals operators have been invoked for non-integer modeling of memristor–memcapacitor that can exhibit a combination of dynamical chaotic phenomena. The numerical schemes, numerical simulations, stability analysis and equilibrium points have been highlighted in detail. The comparative chaotic graphs have been discussed in three ways (i) by keeping fractal component fixed and varying fractional component distinctly, (ii) by keeping fractional component fixed and varying fractal component distinctly and (iii) by varying both fractal component and fractional component distinctly. Our results suggest that fractal-fractional model of memristor–memcapacitor retains the memory characteristics.

A fractal–fractional order model for exploring the dynamics of Monkeypox disease

This study explores the biological behaviour of the Monkeypox disease using a fractal–fractional operator. We discuss the existence and uniqueness of the solution of the model using the fixed-point concept. We further show that the Monkeypox fractal–fractional model is stable through the Hyers–Ulam and Hyers–Ulam Rassias stability criteria. The epidemiological threshold of the model is obtained. The numerical simulation for the proposed model is obtained using the Newton polynomial. For instance, the disease dies out at lower fractional values. We investigated the effects of some key parameters on the dynamics of the disease. The variation of the parameters shows that quarantine and isolation are effective approaches to managing, controlling, or eradicating the Monkeypox disease.

Investigating the impact of memory effects on computer virus population dynamics: A fractal–fractional approach with numerical analysis

In this study, we investigate the impact of memory effects through the application of the fractal–fractional derivative operator on population dynamics in computer viruses. By utilizing numerical analysis techniques, we explore the influence of varying fractal–fractional operator orders on virus propagation and conduct a parameter study to examine the effects of fractal dimension and fractional order on computer virus spread. Furthermore, in order to comprehensively compare and analyze the outcomes from various perspectives, we introduce a novel application of the operational matrix technique to the proposed dynamical system, transforming the fractal–fractional model into the Caputo derivative form. Two different numerical techniques, Adams–Bashforth and Taylor operational matrix, have been employed to solve the two distinct forms of the model. The obtained results yielded highly consistent outcomes for the two numerical methods employed, despite their utilization of distinct derivative operators. The reported results highlight the memory effect associated with fractal–fractional derivatives, where past states and behaviors continue to influence the system. The ability to capture these memory effects through fractal–fractional derivatives can be considered a benefit, enabling a more comprehensive understanding, analysis and modeling of computer virus dynamics. Additionally, we investigate four variants to ensure numerical stability, employing both the Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) criteria.

Dynamics of Age-Structure Smoking Models with Government Intervention Coverage under Fractal-Fractional Order Derivatives

The rising tide of smoking-related diseases has irreparably damaged the health of both young and old people, according to the World Health Organization. This study explores the dynamics of the age-structure smoking model under fractal-fractional (F-F) derivatives with government intervention coverage. We present a new fractal-fractional model for two-age structure smokers in the Caputo–Fabrizio framework to emphasize the potential of this operator. For the existence-uniqueness criterion of the given model, successive iterative sequences are defined with limit points that are the solutions of our proposed age-structure smoking model. We also use the functional technique to demonstrate the proposed model stability under the Ulam–Hyers condition. The two age-structure smoking models are numerically characterized using the Newton polynomial. We observe that in Groups 1 and 2, a change in the fractal-fractional orders has a direct effect on the dynamics of the smoking epidemic. Moreover, testing the inherent effectiveness of government interventions shows a considerable impact on potential, occasional, and temporary smokers when the fractal-fractional order is 0.95. It is the view that this study will contribute to the applicability of the schemes, the rich dynamics of the fractal, and the fractional perspective of future predictions.

ON FRACTAL FRACTIONAL HEPATITIS B EPIDEMIC MODEL WITH MODIFIED VACCINATION EFFECTS

Fractional calculus is strongly tied to the dynamics of complicated real-world phenomena. The goal of this study is to employ a fractal fractional mathematical model with vaccine effects to solve HBV infection. We provide a novel numerical technique for fractal–fractional model with many graphical results. Moreover, we present by comparing these two operators for many values of the fractal and fractional order parameters [Formula: see text] and [Formula: see text]. The effects of vaccination have been examined more thoroughly because vaccination results are not cent percent in some epidemic circumstances, e.g. vaccinated people can also be deemed as susceptible people. The dynamical behavior of the fractal–fractional HBV model with modified vaccination effects is examined utilizing the Atangana–Baleanu approach.

A fractal-fractional model on impact stress of crusher drum

In this paper, a fractal-fractional model of the impact stress on the crusher drum is established by using He?s fractal derivative and the fluid-solid coupling vibration equation. The two-scale transform is used to obtain its solution, which can be used to improve the safety performance of beating machines.