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Öğe ANALYSIS AND DYNAMICS OF CHOLERA EPIDEMIC SYSTEM IN SOCIETY VIA FRACTAL-FRACTIONAL OPERATOR(2025-01-01) Abbas, Fakhar; Ghaffar, Abdul; Akgül, Ali; Ahmad, Aqeel; Mustafa, Ghulam; Hendy A.S.; Abdallah, Suhad Ali Osman; El-Gawaad, N.S. AbdTo 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.Öğe ANALYSIS OF FRACTIONAL ORDER DIARRHEA MODEL USING FRACTAL FRACTIONAL OPERATOR(World Scientific Publ Co Pte Ltd, 2022) Yao, Shao-Wen; Ahniad, Aqui; Inc, Mustafa; Farman, Muhammad; Ghaffar, Abdul; Akgul, AliIn this paper, we construct a scheme of fractional-order mathematical model for the population infected by diarrhea disease by using the four compartments S, I, T and R. The fractal-fractional derivative operator (FFO) with generalized Mittag-Leffler kernel is employed to obtain the solution of the proposed system. The system is analyzed qualitatively as well as verify non-negative unique solution. The existence and uniqueness results of fractional-order model under Atangana-Baleanu fractal-fractional operator have been proved by fixed point theory. Also error analysis has been made for the proposed fractional-order model. Simulation has been carried out for derived fractional-order scheme to check the effectiveness of the results which will help, how to prevent and control such type of epidemic in society.Öğe Effect of slow release nitrogenous fertilizers and biochar on growth, physiology, yield, and nitrogen use efficiency of sunflower under arid climate(Springer Heidelberg, 2022) Waqar, Muhammad; Habib-ur-Rahman, Muhammad; Hasnain, Muhammad Usama; Iqbal, Shahid; Ghaffar, Abdul; Iqbal, Rashid; Hussain, Muhammad IftikharSunflower plants need nitrogen consistently and in higher amount for optimum growth and development. However, nitrogen use efficiency (NUE) of sunflower crop is low due to various nitrogen (N) losses. Therefore, it is necessary to evaluate the advanced strategies to minimize N losses and also improve sunflower productivity under arid climatic conditions. A field trial was conducted with four slow release nitrogenous fertilizers [SRNF (bacterial, neem, and sulfur-coated urea and N loaded biochar)] and three N levels (100% = 148 kg N-ha(-1), 80% = 118 kg N-ha(-1), and 60% = 89 kg N-ha(-1)) of recommended application (100%) for sunflower crop under arid climatic conditions. Results showed that neem-coated urea at 148 kg N-ha(-1) significantly enhanced crop growth rate (CGR) (19.16 g-m(-2)-d(-1)) at 60-75 days after sowing (DAS); leaf area index (2.12, 3.62, 5.97, and 3.00) at 45, 60, 75, and 90 DAS; and total dry matter (14.27, 26.29, 122.67, 410, and 604.33 g m(-2)) at 30, 45, 60, 75, and 90 DAS. Furthermore, higher values of net leaf photosynthetic rate (25.2 mu mol m(-2) -s(-1)), transpiration rate (3.66 mmol-s(-1)), and leaf stomatal conductance (0.39 mol-m(-2)-s(-1)) were recorded for the same treatment. Similarly, neem-coated urea produced maximum achene yield (2322 kg ha(-1)), biological yield (9000 kg-ha(-1)), and harvest index (25.8%) of the sunflower crop. Among various N fertilizers, neem-coated urea showed maximum NUE (20.20 kg achene yield kg(-1) N applied) in comparison to other slow release N fertilizers. Similarly, nitrogen increment N-60 showed maximum NUE (22.40 kg grain yield-kg(-1) N applied) in comparison to N-80 and N-100. In conclusion, neem-coated urea with 100% and 80% of recommended N would be recommended for farmers to get better sunflower productivity with sustainable production and to reduce the environmental nitrogen losses.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 Generation of fractal curves using new binary 8-point interpolatory subdivision scheme(Taylor & Francis Ltd, 2024) Ghaffar, Abdul; Javed, Sadia; Mustafa, Ghulam; Akgul, Ali; Hassani, Murad KhanThe subdivision scheme is a valuable tool for designing shapes and representing geometry in computer-aided geometric design. It has excellent geometric properties, such as fractals and adjustable shape. In this research paper, we explore the generation of fractal curves using a novel binary 8-point interpolatory subdivision scheme with two parameters. We analyse different properties of the proposed scheme, including convergence, special cases, and fractals. Additionally, we demonstrate through various examples the relationship between the shape parameters and the fractal behaviour of the resulting curve. Our research also identifies a specific range of shape parameters that can effectively produce fractal curves. The findings of this study provide a fast and efficient method for generating fractals, as demonstrated by numerous examples. Modelling examples show that the 8-point interpolatory scheme can enhance the efficiency of computer design for complex models.
