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Öğe A fractal-fractional sex structured syphilis model with three stages of infection and loss of immunity with analysis and modeling(Elsevier, 2023) Farman, Muhammad; Shehzad, Aamir; Akgul, Ali; Hincal, Evren; Baleanu, Dumitru; El Din, Sayed M.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.Öğe A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances(Elsevier Ltd, 2024) Farman, Muhammad; Shehzad, Aamir; Nisar, Kottakkaran Sooppy; Hincal, Evren; Akgul, AliBackground: 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. © 2024Öğe Analysis and controllability of diabetes model for experimental data by using fractional operator(Elsevier, 2024) Farman, Muhammad; Ahmad, Aqeel; Zehra, Anum; Nisar, Kottakkaran Sooppy; Hincal, Evren; Akgul, AliDiabetes is a silent illness that is endangering public health in society. Diabetes is a chronic disease affecting millions of people worldwide, and understanding the underlying mechanisms of glucose homeostasis is crucial for managing this condition. Diabetes is a significant public health issue due to the early morbidity, mortality, shortened life expectancy, and financial and other expenses to the patient, their careers, and the health care system. In this study, we propose a mathematical model consisting of fl-cells, insulin, glucose, and growth hormone that incorporates the fractional operator. Using the Lyapunov function, we treated a global stability analysis and investigated the impact of a new wave of dynamical transmission on the equilibrium points of the second derivative. With the Lipschitz criteria and linear growth, the exact singular solution for the proposed model has been determined. Furthermore, we present a detailed analysis of infections, and numerical simulations are conducted using the Mittag-Leffler Kernel mathematical framework to illustrate the theoretical conclusions for various orders of the fractional derivative. Controllability and observability of the linear system are treated for close loop design to check the relation between the glucose and insulin systems. Overall, our results provide a comprehensive understanding of glucose homeostasis and its underlying mechanisms, contributing to the development of effective diabetes management strategies. The proposed model and mathematical framework offer a valuable tool for investigating complex systems and phenomena, with applications beyond the field of diabetes research and helpful to designing the closed loop for the glucose-insulin system.Öğe Flip bifurcation analysis and mathematical modeling of cholera disease by taking control measures(Nature Portfolio, 2024) Ahmad, Aqeel; Abbas, Fakher; Farman, Muhammad; Hincal, Evren; Ghaffar, Abdul; Akgul, Ali; Hassani, Murad KhanTo study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases which are spread in the world wide. The objective of the research study is to assess the early diagnosis and treatment of cholera virus by implementing remedial methods with and without the use of drugs. A mathematical model is built with the hypothesis of strengthening the immune system, and a ABC operator is employed to turn the model into a fractional-order model. A newly developed system SEIBR, which is examined both qualitatively and quantitatively to determine its stable position as well as the verification of flip bifurcation has been made for developed system. The local stability of this model has been explored concerning limited observations, a fundamental aspect of epidemic models. We have derived the reproductive number using next generation method, denoted as R 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}$$\end{document} , to analyze its impact rate across various sub-compartments, which serves as a critical determinant of its community-wide transmission rate. The sensitivity analysis has been verified according to its each parameters to identify that how much rate of change of parameters are sensitive. Atangana-Toufik scheme is employed to find the solution for the developed system using different fractional values which is advanced tool for reliable bounded solution. Also the error analysis has been made for developed scheme. Simulations have been made to see the real behavior and effects of cholera disease with early detection and treatment by implementing remedial methods without the use of drugs in the community. Also identify the real situation the spread of cholera disease after implementing remedial methods with and without the use of drugs. Such type of investigation will be useful to investigate the spread of virus as well as helpful in developing control strategies from our justified outcomes.Öğe Fractional order age dependent Covid-19 model: An equilibria and quantitative analysis with modeling(Elsevier, 2023) Jamil, Saba; Farman, Muhammad; Akgul, Ali; Saleem, Muhammad Umer; Hincal, Evren; El Din, Sayed M.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.Öğe Generalized Ulam-Hyers-Rassias stability and novel sustainable techniques for dynamical analysis of global warming impact on ecosystem(Nature Portfolio, 2023) Farman, Muhammad; Shehzad, Aamir; Nisar, Kottakkaran Sooppy; Hincal, Evren; Akgul, Ali; Hassan, Ahmed MuhammadMarine structure changes as a result of climate change, with potential biological implications for human societies and marine ecosystems. These changes include changes in temperatures, flow, discrimination, nutritional inputs, oxygen availability, and acidification of the ocean. In this study, a fractional-order model is constructed using the Caputo fractional operator, which singular and nol-local kernel. A model examines the effects of accelerating global warming on aquatic ecosystems while taking into account variables that change over time, such as the environment and organisms. The positively invariant area also demonstrates positive, bounded solutions of the model treated. The equilibrium states for the occurrence and extinction of fish populations are derived for a feasible solution of the system. We also used fixed-point theorems to analyze the existence and uniqueness of the model. The generalized Ulam-Hyers-Rassias function is used to analyze the stability of the system. To study the impact of the fractional operator through computational simulations, results are generated employing a two-step Lagrange polynomial in the generalized version for the power law kernel and also compared the results with an exponential law and Mittag Leffler kernel. We also produce graphs of the model at various fractional derivative orders to illustrate the important influence that the fractional order has on the different classes of the model with the memory effects of the fractional operator. To help with the oversight of fisheries, this research builds mathematical connections between the natural world and aquatic ecosystems.Öğe Mathematical analysis and dynamical transmission of monkeypox virus model with fractional operator(Wiley, 2023) Farman, Muhammad; Akgul, Ali; Garg, Harish; Baleanu, Dumitru; Hincal, Evren; Shahzeen, SundasMonkeypox virus is one of the major causes of both smallpox and cowpox infection in our society. It is typically located next to tropical rain forests in remote villages in Central and West Africa. The disease is brought on by the monkeypox virus, a member of the Orthopoxvirus genus and the Poxviridae family. For analysis and the dynamical behaviour of the monkeypox virus infection, we developed a fractional order model with the Mittag-Leffler kernel. The uniqueness, positivity, and boundedness of the model are treated with fixed point theory results. A Lyapunov function is used to construct both local and global asymptotic stability of the system for both endemic and disease-free equilibrium points. Finally, numerical simulations are carried out using the effective numerical scheme with an extended Mittag-Leffler function to demonstrate the accuracy of the suggested approaches.
