Journal of Mathematical Modeling and Fractional Calculus
The Journal of Mathematical Modeling and Fractional Calculus is dedicated to providing a platform for researchers to publish high-quality research articles that apply mathematical modeling principles to solve real-world problems across various scientific disciplines. The journal focuses on new mathematical challenges and their applications in applied science, contributing to the advancement of scientific knowledge within the community.
Investigation of COVID-19 epidemic mathematical model incorporating media coverage impact and control
In order to estimate and control the recent corona virus disease 2019 outbreak, under me dia impact. Most countries impose stringent intervention measures to halt the spread of COVID-19. Globally, the COVID-19 epidemic has raised a lot of concerns. A non-smooth SIR system is proposed to investigate the impact of three control strategies (vaccination, therapy, and media coverage) on the spread of an infectious disease. Social media platforms like Face book, Twitter, and others have been crucial in spreading information during the COVID-19 outbreak, both factual and misleading, which has led to widespread confusion. When the susceptible population surpasses the threshold value, the traditional epidemic model comprises media coverage, linear functions that describe vaccine injection, and treatment tactics. COVID-19 model under media coverage investigated both qualitatively and quantitatively. models key proportion are also verified like boundedness, uniqueness and existence. Stability Analysis his also know development on local and global scale in order to properly manage funds and human resources and create successful disease preventive marketing efforts, organizations, institutions, enterprizes, policymakers, and educators who are in charge of infection and disease control must keep a close eye on infection rates.
Copyright (c) 2025 Journal of Mathematical Modeling and Fractional Calculus
Analysis and treatment impact to control the transmission dynamics of skin sores disease with novel hybrid fractional operators
Skin sores (impetigo) are a frequent and contagious skin infection in young children, caus ing blisters and ulcers. It is usually not hazardous and resolves within a week of treatment or a few weeks if no treatment is used. Treatment is commonly recommended since it can shorten the length of the condition and reduce the risk of the virus spreading to others. We developed and tested a non linear hybrid fractional order model to investigate skin sores infection and the transmission dynamics. The study examines the mathematical properties of the suggested model, such as the feasible region, equilibrium points, basic reproduction number, and existence of the system’s unique solutions using Banach fixed-point theory. It also uses the appropriate lyapunov function to examine the stability of equilibrium states. This work gives a thorough examination of several hybrid fractional operators and numerically simulates the suggested skin sores system using the Laplace-Adomian decomposition approach, demonstrating its efficacy in simulating theoretical scenarios. This research advances our knowledge of the mechanisms underlying disease transmission by taking into account fractional-order dynamics and a variety of routes, offering suggestions for better disease management and control.
Dynamics of a Food Chain Model Using the Caputo Fractional Derivative
This work investigates a fractional-order food chain model based on the Caputo operator, taking into account resource availability, competition, and predation. The dynamics of the food chain are bet ter represented using fractional calculus, which takes into account long-range interactions and previous dependency. Analytical and numerical simulations reveal information about the resilience, persistence, and stability of biological communities under fractional order dynamics. The model includes a three species food chain, as well as environmental contamination. We begin by confirming the model’s uniqueness, nonnegativity, and boundedness. We also examine numerous conditions for the presence of equilibrium and local stability. Second, a controller is proposed, and the global stability of the positive equilibrium point is investigated using the Lyapunov method. The proposed model solution was estimated using the fractional iterative technique, and numerical simulations were carried out to validate the theoretical results.
Finite Volume Simulation of Nanofluid Heat Transfer with Free Slip
This study numerically investigates the viscous heating mechanism in a nanofluid-filled cavity subjected to external forces applied to the top lid, combined with laminar mixed convection of the nanofluid. The vertical walls of the cavity are assumed to be insulated, non-conductive, and impermeable to mass transfer. The horizontal walls are differentially heated, with the lower wall maintained at a higher temperature while the upper wall remains cooler. The primary objective of this research is to introduce a novel method that accurately incorporates height in the solution of heat transfer equations, using transient analysis for numerical iterations. Although the fluid flow reaches a steady state, the square cavity’s walls are fully insulated, and a constant heat flux is generated by the motion of the top lid. This work aims to analyze the effects of viscous heating in a fully insulated lid-driven cavity with Neumann boundary conditions under both no-slip and free-slip conditions, while varying Rayleigh and Prandtl numbers as independent parameters. The simulations are performed for a specific case where the Prandtl number Pr = 6.2 is fixed, while Rayleigh numbers and the volume fraction of the nanofluid (water mixed with copper nanoparticles) range from 0% to 5%. The time dependent vorticity-stream function and thermal energy equations are discretized and solved using a custom finite volume method combined with the artificial compressibility approach, implemented in MATLAB. Despite viscous heating having a limited effect, the Neumann boundary conditions with no-slip and free-slip assumptions influence heat retention within the insulated cavity. The free-slip condition acts as a lubricant, leading to reduced temperature distribution, particularly under lower Rayleigh numbers and higher Prandtl numbers, compared to the no-slip condition. This is attributed to the free-slip effects under these conditions, which enhance heat dissipation and increase the fluid velocity, further stabilizing the system.
On Double Delay Terms Robbin Boundary Value Problem with Fractional Derivative
In this manuscript, the existence theory of a mixed delay Robbin boundary value problem (RBVP) together with impulsive conditions is investigated. For the required results, we use topological degree of non-compactness. The concerned tools have the ability to relax the strong compact criteria to some weaker one. As compared to fixed point theory, the aforesaid proposed tools are more powerful and applicable in dealing various nonlinear problems of fractional order differential equations (FODEs) with different kinds of boundary conditions. Proper example is given to illustrate the results.
Fractional Integral Operator Associated with the I-Functions
In this paper we have evaluated certain fractional integral operators associated with Saxenas I-function using the operators of fractional integration. The results obtained are of general character and are reduced to the known results on specializing the parameters. By leveraging the properties of fractional calculus, we investigate new integral transforms involving in this function. We present novel results that extend the existing theory and provide a framework for solving complex integral operators in broadening the scope in this function. These results have potential applications in various fields such as astrophysics, engineering, and applied mathematics, where the I-function and fractional calculus play a crucial role
Fractional Integral operators and I-function
Integrating Novel Deep Neural Networks on the Analysis of Fractional Epidemiological Models in both Populations with Disease Infection
Deep Neural Networks (DNNs) have become one of the most significant tools utilized in analyzing complicated systems, surpassing the ability to comprehend complexities. The study provides the advantage of DNNs' interest to improve the analysis of intricate patterns and produce insightful findings. It is crucial to comprehend the dynamics of infectious disease spread in ecological systems, especially when it comes from predator to prey, because of its significance for a variety of real-world situations. In complicated regulation environments, the size range of prey and predator populations is influenced by the complicated interactions between infections and predator-prey relationships. The analysis of fractional epidemiological models in both populations with disease infection (FEM-BPDI) is performed through the novel application of artificial intelligence, particularly Deep Neural Networks (DNNs). Datasets for Deep Neural Networks are generated using the fde12 solver. Training, testing, and validation phases are applied to the DNNs models to acquire solutions for the FEM-BPDI under different epidemiological scenarios. Several statistical metrics, such as mean-squared error analyses, auto correlation of error (ACE), correlation input and error (CIE), error histogram visualizations, and expected regression measurements, are used to show the efficiency that DNNs are at solving the FEM-BPDI. The model's stability and resilience in forecasting disease dynamics are highlighted by a low Mean Squared Error (MSE) that is obtained. Also, the small or negligible Absolute error provides more evidence of the suggested technique's efficacy. This study highlights the critical role that AI-powered Deep Neural Networks play in improving the comprehension and forecasting of fractional epidemiological models in the context of dual population dynamics, providing invaluable insight into the dynamics of disease transmission and ecosystem control.
An analysis of the Caputo Fabrizio fractional modeling for infectious diseases caused by chlamydia
These days, infectious illness mathematical modeling is a major global trend. With the use of current data, mathematical models enable us to predict the occurrence of disease outbreaks in the future. In this work, we use a fractal fractional operator with two fractal and fractional orders to solve a system of fractional differential equations using a Caputo Fabrizio type kernel. A six chambered 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. Initially, we calculate the system’s equilibrium points and fundamental reproduction number R0. We then look at the system’s stability at the equilibrium point. Through the application of the Picard Lindelof methodology, we establish the existence of a unique solution for the given fractional CF-system of the hearing loss model and use fixed point theory to examine the stability of the iterative process. By taking into account the therapy as a control technique to lower the number of infected individuals, the system’s optimal control is established. Calculating the estimated solution of the system involves applying the Euler technique for the fractional order Caputo Fabrizio derivative. In two scenarios, R0 < 1 and R0 > 1, we provide a numerical simulation of the disease’s spread with regard to the basic reproduction number and the transmission rate. We compute the results for various fractional order derivatives and compare the findings in order to examine the impact of the fractional order derivative on the behavior and value of each variable in the model. Additionally, we examine the sensitivity of R0 with regard to each model parameter and ascertain the influence of each parameter, taking into account the significance of reproduction number in the persistence of disease transmission. Finally, it can be said that once more, fractional operator mathematical models can help in making better decisions on how to manage financially turbulent situations.
Dynamical transmission of Varicella Virus in Jordan with SVIR Model through analysis and numerical simulations
Chickenpox, another name for varicella virus, is a highly contagious illness. In this work, we use mathematical modelling and simulation to examine the dynamics of transmission and propose an SVIR model for the varicella virus in Jordan. The dynamics of the population in various classes are examined using a suggested model, which offers a more thorough comprehension of the course of the epidemic and the efficacy of control efforts. Lipschitz condition and fixed-point theory were also used to establish the solution's existence and uniqueness. Additionally, using figures and mathematics, it investigates the effects of different factors on the reproductive number and sensitivity analysis of the suggested model. The Routh-Hurwitz stability criterion is used to determine local stability at equilibrium sites that are both disease-free and endemic. Lyapunov functions with the first derivative test used to establish the global stability of model equilibria analysis for both >1 and <1. the suggested model is numerically simulated using Matlab software to ascertain how parameters affect the scenarios. The findings of this study confirm the theoretical and biological phenomena of the model and aid in elucidating the mechanisms of varicella virus transmission and directing public safety policy decisions. In the conclusion, Since the varicella virus puts a lot of demand on the country's healthcare system, the Jordanian Ministry of Health should launch vaccination campaigns in order to eliminate the disease because a universal varicella vaccination is necessary. Long-lasting protection from varicella immunization lowers the risk of epidemics and promotes community health, and lowering the overall burden of varicella virus infection frees up funds for other health concerns.
Mathematical Modeling of Covid-19 Model dynamical transmission with different way of infections
The COVID-19 pandemic is causing a lot of pain on a global scale. This work aims to develop new mathematical models for the outbreak by utilising fractional derivatives. Through the control of certain diseases, the adoption of changed methodologies and fundamental explanations can have a substantial impact on the fitness of society. Examining the dynamics and numerical approxima tions for the suggested arbitrary-order coronavirus illness system is the primary goal. For the Covid-19 model, the study introduces fractional derivatives using sophisticated methods such as the Atangana Baleanu, Sumudu transform, and Atangana-Toufik scheme. These methods yield accurate findings while examining the outbreak. Solutions involving various fractional operators are constructed using the Generalized Mittag-Leffler law. Covid-19 effects at various fractional values are explained and theoretical results are validated using numerical simulations. This aids in comprehending the outbreak and its management tactics.
Analysis of Energy Sector Model with Laplace Transform under modified fractional operator: Mathematical Analysis
Population growth is driving up energy demand, and burning fossil fuels accounts for a sizable amount of global energy production. Since this adds to the rise in greenhouse gas emissions, lowering energy sector emissions is essential to achieving climate change goals. In this research, three hybrid fractional operators are used to investigate carbon emissions from the power sector. A hy brid fractional operator is used to modify differential equations throughout time and space in order to create a mathematical model that describes the human population, energy consumption, and at mospheric carbon dioxide concentration. Compartmental models offer a mathematical framework for comprehending systems and forecasting outcomes, which makes them essential fgor comprehending real-world events. The effectiveness of these novel operators is supported by a number of analyses, and the Lipschitz condition ensures unique results. The order of the fractional derivative has a major impact on the dynamical process that is used to construct the non-integer order model.
Modeling and Stability Analysis of the Unemployment Model with Caputo
The problem of unemployment is acute, joblessness affects millions of people worldwide, which is why governments are constantly looking for efficient solutions to combat this social and economic ill. In response, we construct a model that encompasses the nonlinear nature of the unemployment rate. A fractional-order system of differential equations is employed in the structure of the considered model to produce a more accurate analysis of the unemployment dynamics of equilibrium positions and their stability or instability. In the model, there are three main dynamical parameters whose time evolution is described by fractional-order differential equations involving Caputo derivatives. The existence and uniqueness of the solutions are established by using the fixed point indices and the stability of the model is determined by applying the Hyers-Ulam stability test. A Newton polynomial approach is used for numerical simulation and investigates the results at fractional orders ω = 0.85 and ω = 1. The results presented in this paper suggest that there is one and only one stable positive equilibrium that is locally and globally asymptotically stable under some conditions. Computations demonstrate that for different values of ω, obtained with Caputo derivatives coinciding with the classical sense at ω = 1. In this study, by determining the employment rate and job creation rate that produce an unemployment rate of 7% the model complies with the governments policy objectives of low unemployment. These results provide important implications for the dynamic employment policies suggested and point to the use of fractional-order methodologies in other economic-social systems.
Analysis and Numerical Treatment of Chain Smokers using Advanced Fractional
In this paper, a fractal fractional derivative is used to examine an environmentally friendly approach of understanding the workings of smoking in humans. We proposed a fractional differential equation system to represent a time-fractional order smoking model with illness effects. Studies are conducted using methodologies. Lipschitz circumstances and linear growth are utilized to demonstrate the presence and distinction of the suggested model in relation to the impact of the global offset. It is confirmed that the fractional order model’s solutions are bounded and positive. When conducting the initial and subsequent derivative assessments, the Lyapunov function is employed to verify analysis of global stability. In order to examine the influence of smoking on humans, the fractional operator is studied. To do this, solutions are constructed applying the extended version of the Mittag-Leffler kernel using a two-step Lagrange polynomial method. Numerical simulation is performed to see how the fractional order smoking model behaves. Such research will aid in understanding the behavior of smoking and in developing human defenses.
Integral Transform and Generalized M-Series Fractional Integral Operators
This paper presents new theorems that build upon existing research by extending the ap plication of the Mellin transform within the framework of fractional integral operators. Traditionally, the Mellin transform has been a powerful tool for analyzing asymptotic behavior, scaling properties, and integral representations of special functions. However, by incorporating fractional integral oper ators, its analytical flexibility is significantly enhanced, allowing for a more in-depth study of special function properties, particularly in fractional calculus. Furthermore, the inclusion of the generalized I-function and M-series broadens this mathematical framework by generalizing established results and encompassing a wider class of special functions. Through their evaluation alongside fractional inte gral operators, this study introduces new integral representations, special cases, and applications that have not been previously explored. This extension greatly increases the applicability of these mathe matical tools in diverse fields such as mathematical physics, engineering, and applied analysis, where fractional calculus and integral transforms play a crucial role in solving complex differential equa tions and boundary value problems. Ultimately, these theorems not only refine existing mathematical structures but also create new opportunities for future research on the interplay between integral trans forms, special functions, and fractional calculus, contributing to both theoretical progress and practical advancements.
Mellin transform and Fractional Integral
Control of Marburg Virus Disease Spread in Humans under Hypersensitive Response through Fractal-Fractional
The primary objective of this research is to examine the model of Marburg virus disease with therapy to prevent the infections from spreading throughout the community due to saliva, urine, and feces. A mathematical model is established using the created hypothesis for a healthy environment in order to examine the different rates of Marburg virus disease after taking control measures with treatment for different protection. The model is then converted into a fractional ordered model for continuous monitoring, including theoretical solutions, by using the Fractal Fractional operator. The model’s global stability is studied through Lyapunov derivative by equilibrium and endemic point of model. Both qualitative and quantitative analysis is done on a proposed model with the power law kernel, with particular focus on unique solutions, positivity, and boundedness. By using of fixed point theory and the Lipschitz condition, there is only one exact solution. By confirming the Lyapunov function globally, equilibrium points that are both endemic and disease-free are checked for stability. Numerical simulations are utilized to investigate the effects of the fractal fractional operator on the generalized form of the power law kernel using a two-step Lagrange polynomial approach for ongoing monitoring of the Marburg virus disease and treatment strategies. The simulations show how different parameters affect the illness. In order to simulate the actual behavior and management of Marburg virus disease produced by saliva, urine, and feces and to implement control measures for a healthy environment with a hypersensitive reaction (HR), simulations have been constructed. This type of research will be helpful in figuring out how diseases spread and in developing management strategies based on our verified results for human in community.
Analyzing the Dynamics of Tuberculosis through a Fractional-Order Model
Tuberculosis (TB) remains a significant public health challenge worldwide, exacerbated by the complex and persistent nature of Mycobacterium tuberculosis infection. Traditional integer order models provide insight into TB transmission and control but may fall short in capturing the nuances of disease progression and memory effects inherent in biological systems. This study presents a fractional-order mathematical model of TB dynamics to better reflect the complexities of TB infec tion, latency, and progression. Utilizing fractional calculus, the model introduces a novel approach that accounts for memory and hereditary properties, offering a more accurate representation of dis ease transmission and latency. Analytical methods are applied to explore the behavior of the disease dynamics. Numerical simulations validate the model’s effectiveness in describing TB dynamics. The fractional-order TB model has the potential to inform more effective intervention strategies and im prove predictions of long-term outcomes.
Epidemic drinking model: a dynamical analysis employing the Mittag-Leffler kernel
One of the most helpful operators for using fractional differential equations to explain non local behaviors is the fractal fractional with Mittage-Leffler kernel fractional derivative. We proposed the fractal fractional Mittage-Leffler kernel for the drinking epidemic which is the world wide largest issue now a days. Qualitative and quantitative analysis for model and scheme are analyzed for drinking system in community. Additionally, using several well-known theorems from fixed point theory, we confirm the existence, uniqueness, analysis, and Ulam-Hyres stability of solutions to a specific class of fractional operators. Connect the drinking fractional system to a recently proposed bivariate Mittag Leffler function to solve it. Some important properties were also verified for Mittage-Leffler fractional derivative on the fractional drinking system. Comparison has been made to the integer order derivative to verify the efficiency of results and effect of drinkers in community.
Investigation of COVID-19 outbreak as a case study in Italy using different fractional operators
This work analyzes the COVID-19 fractional order SEIQRD compartmental model using the six primary categories of the Caputo approach. Numerous conclusions have been drawn on the the solution’s boundedness and non-negativity, as well as the existence and uniqueness of the new mod el. Our findings reveal that the When R0 < 1, the system is locally asymptotically stable at infection-free equilibrium. We also observed that In the absence of illness, the system is globally asymptotically stable (R of Covid 19 <1). This study’s primary goal is to look into the dynamics of COVID-19 transmission in Italy, the nation where the virus was initially discovered in January 2020. To account for the uncertainty arising from the limited knowledge about the Corona virus (COVID-19), The order of fractions A fractional order was used to apply the SEIQRD compartmental model framework. The dynamics of the equilibrium are examined using the La Salle invariant principle and the Routh-Hurwitz consistency criterion. Additionally, the suggested model’s approximate solution is calculated using the fractional-order Taylor’s method. Through the comparison of simulation results with empirical data, the model’s validity is shown. The study’s findings regarding the impacts of using face masks showed that wearing face masks frequently can prevent the COVID-19 virus from spreading.
Hybrid Fractional Operator for Epidemic Model analysis and application
The Constant proportional Caputo (CPC) fractional derivative has been one of the most use ful operators for modeling non-local behaviors by fractional differential equations. We proposed the CPC operator for the drinking epidemic which is the world wide largest issue nowadays. In compar ison to the integer-order models, the fractional order differential equations models appear to be more compatible with this illness. Qualitative and quantitative analysis for model and scheme are treated. We also present the uniqueness and Ulam-Hyres stability of solutions to a particular class of fractional starting value issues involving the Hilfer proportional fractional derivative using certain well-known theorems from the fixed point theory. The bivariate Mittag-Leffler function that was recently pub lished will be used to solve the drinking pandemic model, we first construct the inverse operator and Laplace transform of the new formulation also eigenfunctions for the proposed scheme. Some impor tant properties were also verified for the Constant proportional Caputo (CPC) fractional derivative on the epidermic model.
Uniqueness, stability analysis and approximate solution of a fractional order COVID-19 Model
In this paper, we investigate a fractional model for the COVID-19 epidemic that contains an antiretroviral treatment compartment. We implement novel methods to acquire effective results. We discuss equilibrium point, reproductive number, and sensitivity analysis. We utilize the Sumudu transform technique to obtain approximate solutions for the model, and we explore chaos control to assess stability around equilibrium points. We demonstrate the numerical simulations to prove the accuracy of the proposed techniques. The graphs illustrate how varying fractional orders impact the dynamics of each epidemiological group, revealing the memory and time-dependent effects on disease spread and control strategies.