Fractional-order Mathematical Model Research Articles

Fractional mathematical model for the transmission dynamics and control of Lassa fever

This paper aims to investigate various epidemiological aspects of Lassa fever viral infection using a fractional-order mathematical model so as to assess the impacts of treatment and vaccination on the spread of Lassa fever transmission dynamics. Initially, the model employs integer-order nonlinear differential equations, incorporating imperfect vaccination and treatment as control measures for the human population. Subsequently, the model is redefined using a fractional order derivative with power law to enhance understanding of disease dynamics. The paper establishes conditions ensuring the existence and uniqueness of the model’s solution in the fractional case, alongside presenting stability analysis of the endemic equilibrium using the Lyapunov function approach. Numerical simulations are performed using the fractional Adams–Bashforth–Moulton method, shedding light on the influence of model parameters and fractional order values on Lassa fever dynamics and control. Additional numerical simulations conducted utilizing surface and contour plots revealed that elevating the contact rates and diminishing the efficacy of vaccines would escalate the prevalence of Lassa fever among the human populace. It was also observed that optimizing treatment and enhancing vaccination strategies would ultimately mitigate the prevalence of Lassa fever within the human population.

Fractional-order pro-tumor and anti-tumor macrophages model: Dynamical analysis and optimal control

The development of new techniques for anti-cancer therapy has become a very active area of research in recent years. One of the promising directions of anti-cancer therapy aims at using the genetically engineered macrophages as killers of tumor cells. The present work introduces a new fractional-order mathematical model to simulate the nonlinear interaction between genetically engineered macrophages and tumor cells. The proposed model incorporates the pro-tumor impacts and anti-tumor impacts which are triggered by the macrophages. The existence and uniqueness of positive solutions of the model are proved. Then, the stability conditions for the equilibrium points are obtained. The bifurcation diagrams explore the impacts of crucial parameters on the dynamical behaviors of the model. Finally, the optimal control scheme is employed to efficiently eliminate tumor cells and inhibit their prevalence in the body. Using Adms-Bashforth-Moulton method, numerical experiments are carried out to validate theoretical analysis.

Analysis and dynamical structure of glucose insulin glucagon system with Mittage-Leffler kernel for type I diabetes mellitus.

In this paper, we propose a fractional-order mathematical model to explain the role of glucagon in maintaining the glucose level in the human body by using a generalised form of a fractal fractional operator. The existence, boundedness, and positivity of the results are constructed by fixed point theory and the Lipschitz condition for the biological feasibility of the system. Also, global stability analysis with Lyapunov's first derivative functions is treated. Numerical simulations for fractional-order systems are derived with the help of Lagrange interpolation under the Mittage-Leffler kernel. Results are derived for normal and type 1 diabetes at different initial conditions, which support the theoretical observations. These results play an important role in the glucose-insulin-glucagon system in the sense of a closed-loop design, which is helpful for the development of artificial pancreas to control diabetes in society.

Non-integer Time Fractional-Order Mathematical Model of the COVID-19 Pandemic Impacts on the Societal and Economic Aspects of Nigeria

Non-integer Time Fractional-Order Mathematical Model of the COVID-19 Pandemic Impacts on the Societal and Economic Aspects of Nigeria

A Fractional-order mathematical model to analyze the stability and develop a sterilization strategy for the habitat of stray dogs

Today, the socio-cultural lack of some countries with increased urbanization has led to the unconscious breeding of stray dogs. The failure to care for the offspring of possessive dogs or ignoring the responsibility to find a suitable family for the offspring increased the dog population on the streets and in the shelters. In this study, our main target is to analyze the habitat of stray dogs and the strategy of how to control the population without damaging the ecosystem of the species. For this aim, we establish a fractional-order differential equation system to investigate the fractal dimension with long-term memory that invovles two compartments; the non-sterilized dog population (x(t)) and the sterilized one (y(t)). Firstly, we analyze the stability of the equilibrium points using the Routh-Hurwitz criteria to discuss cases that should not affect the ecosystem of the dog population, but control the stray dog population in the habitat. Since the intervention to the stray dog population occurs at discrete time impulses, we use the Euler method's discretization process to analyse the local and global stability around the equilibrium points. Besides this, we show that the solutions of the system represent semi-cycle behaviors. At the end of the study, we use accurate data to demonstrate the sterilization rate of stray dogs in their habitat.

The analysis of a novel COVID-19 model with the fractional-order incorporating the impact of the vaccination campaign in Nigeria via the Laplace-Adomian Decomposition Method

This study underscores the crucial role of COVID-19 vaccinations in managing the pandemic, with a specific focus on Nigeria. Employing a fractional-order mathematical modeling approach, the research assesses vaccination efficacy, minimum effectiveness, and duration. The model’s numerical solution is derived through the Laplace Adomian Decomposition Method (LADM), utilizing rapidly converging infinite series. Simulation results illustrate the impact of COVID-19 transmission and vaccination rates. The study concludes that implementing a vaccination strategy in an integer order proves to be the most effective approach to controlling the spread of COVID-19. These findings have significant implications for researchers, policymakers, and healthcare workers. They emphasize the central role of fractional calculus in facilitating vaccine implementation in the ongoing battle against COVID-19. The study calls for global efforts to maximize vaccination implementation for the overall benefit of public health.

Optimal Control Applied to Piecewise-Fractional Ebola Model

A recently proposed fractional-order mathematical model with Caputo derivatives was developed for Ebola disease. Here, we extend and generalize this model, beginning with its correction. A fractional optimal control (FOC) problem is then formulated and numerically solved with the rate of vaccination as the control measure. The research presented in this work addresses the problem of fitting real data from Guinea, Liberia, and Sierra Leone, available at the World Health Organization (WHO). A cost-effectiveness analysis is performed to assess the cost and effectiveness of the control measure during the intervention. We come to the conclusion that the fractional control is more efficient than the classical one only for a part of the time interval. Hence, we suggest a system where the derivative order changes over time, becoming fractional or classical when it makes more sense. This type of variable-order fractional model, known as piecewise derivative with fractional Caputo derivatives, is the most successful in managing the illness.

An Efficient Numerical Scheme for Fractional Order Mathematical Model of Cytosolic Calcium Ion in Astrocytes

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided.

Approximate Solution of the Fractional Order Mathematical Model on the Transmission Dynamics on The Co-Infection of COVID-19 and Monkeypox Using the Laplace-Adomian Decomposition Method

A fractional order compartmental model on the transmission dynamics of the co-infection of COVID-19 and Monkeypox is presented. The approximate solutions of the fractional order model are obtained using the Laplace-Adomian Decomposition method in the form of an infinite series which was shown to converge to the exact value. Using the MATLAB fmincon algorithm, we carried out a data fitting analysis using real life COVID-19 and Monkeypox data so as to obtain estimates for some of the key parameters used in the formulation of model. The results of our analysis showed that an increase in the effective treatment capacity in the human population will significantly reduce the burden of these diseases in the human population.

State estimation-based parameter identification for a class of nonlinear fractional-order systems

Parametric identification is an important part of system theory since knowledge of the parameters allows the analysis and control of the system. The aim of this paper is to propose a novel robust (against measurement noise) parameter identification method for a class of nonlinear fractional-order systems. In order to solve the parametric identification we carry out this problem to a state estimation problem, we introduce a Fractional Algebraic Identifiability (FAI) property which allows to represent the system parameters as a function of the inputs and outputs of the system, this parameter identification method provides an on-line identification process (while the system is operating), we also propose a fractional-order differentiator which allows to reduce the effect of measurement noise as well as to provide the estimation of a fractional-order derivative of the system output. Moreover, we use the Mittag–Leffler boundedness to demonstrate the convergence of this method, a different approach for this stability analysis method is given in this paper. Finally, we illustrate the accuracy and robustness of our proposed method by means of the parametric identification of two nonlinear fractional-order systems: a time-varying nonlinear fractional-order system and a nonlinear fractional-order mathematical model of a simple pendulum.

Simulation of Fractional Order 2D-Mathematical Model Using α-Fractional Differential Transform Method

In this paper, we will introduce a well-known transformation technique, the modified α-fractional differential transform, to the differential equation of fractional order. We derive some new results with proof using new techniques that never existed before. By using this new technique, we are attempting to solve the nonlinear fractional-order mathematical epidemic model. Furthermore, the fractional epidemic model’s solution obtained by using this new technique is correlated with the solution of the same model calculated for a different fractional order by the modified α-fractional differential transform method. Moreover, using the Python software, we can numerically and graphically represent the solution of fractional differential equations.

Dynamical and computational analysis of fractional order mathematical model for oscillatory chemical reaction in closed vessels

One of the most fascinating chemical reactions is an oscillating one. The reactant and the autocatalyst are the two chemical species that are considered in this system. Firstly, we convert the oscillatory chemical reaction model into a fractional order oscillatory chemical reaction model for derivatives of arbitrary order provided in the sense of Caputo. The recommended methodology is centered on the shifted Jacobi collocation technique (JCT) and the shifted Jacobi operational matrix. In this work, we offer computational simulations of fractional order oscillatory chemical reaction models by using collocation technique and Newton polynomial interpolation (NPI) technique. The primary benefit of the collocation method is to study a general estimation for temporal and spatial discretizations. The numerical strategy also simplifies fractional differentiation equations (FDEs) by simplifying them into a simple issue that can be resolved by finding solutions to a few algebraic equations. We also present a comparison between the collocation and NPI techniques through the figures. The mathematical outcomes and data demonstrate that the offered strategy is an efficient procedure with outstanding reliability for resolving differential equations of arbitrary order. Some theorems related to the analysis of the collocation technique are also presented and explained here.

Designing a novel fractional order mathematical model for COVID-19 incorporating lockdown measures

This research focuses on the design of a novel fractional model for simulating the ongoing spread of the coronavirus (COVID-19). The model is composed of multiple categories named susceptible S(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(t)$$\\end{document}, infected I(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I(t)$$\\end{document}, treated T(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(t)$$\\end{document}, and recovered R(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(t)$$\\end{document} with the susceptible category further divided into two subcategories S1(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_{1} (t)$$\\end{document} and S2(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${S}_{2} (t)$$\\end{document}. In light of the need for restrictive measures such as mandatory masks and social distancing to control the virus, the study of the dynamics and spread of the virus is an important topic. In addition, we investigate the positivity of the solution and its boundedness to ensure positive results. Furthermore, equilibrium points for the system are determined, and a stability analysis is conducted. Additionally, this study employs the analytical technique of the Laplace Adomian decomposition method (LADM) to simulate the different compartments of the model, taking into account various scenarios. The Laplace transform is used to convert the nonlinear resulting equations into an equivalent linear form, and the Adomian polynomials are utilized to treat the nonlinear terms. Solving this set of equations yields the solution for the state variables. To further assess the dynamics of the model, numerical simulations are conducted and compared with the results from LADM. Additionally, a comparison with real data from Italy is demonstrated, which shows a perfect agreement between the obtained data using the numerical and Laplace Adomian techniques. The graphical simulation is employed to investigate the effect of fractional-order terms, and an analysis of parameters is done to observe how quickly stabilization can be achieved with or without confinement rules. It is demonstrated that if no confinement rules are applied, it will take longer for stabilization after more people have been affected; however, if strict measures and a low contact rate are implemented, stabilization can be reached sooner.

A Fractional-Order Mathematical Model of Banana Xanthomonas Wilt Disease Using Caputo Derivatives

This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.

Real-world validation of fractional-order model for COVID-19 vaccination impact

<abstract><p>In this manuscript, we develop a fractional-order mathematical model to characterize the propagation dynamics of COVID-19 outbreaks and assess the influence of vaccination interventions. The model comprises a set of eight nonlinear fractional-order differential equations in the Caputo sense. To establish the existence and uniqueness of solutions, we employ the fixed-point technique. Furthermore, we employ the effective fractional Adams-Bashforth numerical scheme to explore both the approximate solutions and the dynamic behavior inherent to the examined model. All of the results are numerically visualized through the consideration of various fractional orders. Furthermore, the real data from three different countries are compared with the simulated results, and good agreements are obtained, revealing the effectiveness of this work.</p></abstract>

Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Fractional Order Mathematical Model to Investigate Topical Drug Diffusion in Human Skin

Impact of social media on academics: a fractional order mathematical model

ABSTRACT Due to the COVID pandemic and lockdown, usage of social platforms increased for academic and non-academic purposes. As a result, students are at significant risk of developing social media addiction, so techniques to control social media addiction throughout society are required. There are several positive and negative ways in which social media affects the academic performance of a student. Most of the mathematical models exclude the past of an individual, which is critical for controlling social media consumption. Hence, this study offers a fractional-order mathematical model to analyze the impact of social media on academics. There are two equilibrium points for the proposed model: social web-free and endemic equilibrium. Based on an evaluation of the threshold value, the social web-free equilibrium point is globally asymptotically stable whenever the threshold value is less than one. Endemic equilibrium points exist when the threshold value is greater than 1. Additionally, numerical simulations have been performed to examine changes in population dynamics and validate analytical outcomes. In summary, the findings of this research reveal that social media addiction decreases as the order of the derivative decreases, demonstrating the high efficiency of a fractional-order model over an integer-order model.

An incommensurate fractional order model for complex dynamics of viral infection with immunity

This paper deals with an incommensurate fractional order mathematical model for dynamic analysis of viral infection with immunity. The primary focus of the work is to explore stability analysis of this version of incommensurate fractional order model with harmonic mean type incidence function and fractional derivative in Caputo sense. First, well-posed ness of the model has been established by analyzing existence and uniqueness of the solution. In the next section, stability analysis of the equilibrium points has been caried out based on the basic reproduction number. Sensitivity analysis of the threshold parameter have been performed in the following sections. Finally, rigorous numerical simulations have been performed to support the theoretical findings as well as to observe the effect of various fractional orders and incidence function.

Fractional order mathematical model of Ebola virus under Atangana–Baleanu–Caputo operator

The aim of this paper is to analyze a fractional model of the Ebola virus. This study is important because it contributes to our understanding of the Ebola virus transmission dynamics using the notion of non-local differential operators. We aim to apply the recently implemented Atangana–Baleanu–Caputo (ABC) fractional derivative with the Mittag-Leffler kernel to study the Ebola virus model closely. The Picard–Lindelof approach is used to do a comprehensive study of the existence and uniqueness of the model’s solutions. The approximate solutions of the fractional order Ebola virus model were obtained using a numerical technique with the ABC operator, a combination of the fundamental theorem of fractional calculus and the two-step Lagrange polynomial interpolation. This innovative approach may offer new insights into the Ebola virus model that were not previously explored. Finally, the numerical simulations illustrate how the control parameters impact specific compartments within the model. The geometrical representation gives significant information about the model’s complexity and reliable information about the model. We simulate each model compartment at various fractional orders and compare them with integer-order simulations, highlighting the effectiveness of modern derivatives. The fractional analysis underscores the enhanced accuracy of non-integer order derivatives in capturing the Ebola virus model’s dynamics.

Fractional model of HIV transmission on workplace productivity using real data from Indonesia

A mathematical model approach to control the spread of HIV and AIDS is needed to predict the future effect of HIV and AIDS on work productivity. In this paper, we consider the analysis of fractional-order mathematical models of the spread of HIV with productivity in the workplace. First, we estimate the epidemiological parameters of the HIV/AIDS model using the annual data of AIDS reported in Indonesia from 2006 to 2018. Based on the model analysis, two equilibria are determined, namely the HIV disease-free and endemic equilibrium’s. The disease-free equilibrium of HIV is locally asymptotically stable if the basic reproduction number is less than one, while the endemic equilibrium is globally asymptotically stable if the reproduction number is greater than one. The sensitivity analysis and numerical simulations are then carried out with variations in fractional order values to determine the dynamics of HIV spread with on-site productivity. Based on numerical simulation results, it was found that the transition rate of HIV-productive workers to AIDS sufferers could reduce the labor population of people living with AIDS and increase the workforce population vulnerable to HIV infection.