Fractional Euler Method Research Articles

Utilizing fractional derivatives and sensitivity analysis in a random framework: a model-based approach to the investigation of random dynamics of malware spread

In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number (R0\\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 underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of R0\\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} with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well.

Numerical Comparisons Between Some Recent Modifications of Fractional Euler Methods

For dealing with systems of Fractional Differential Equations (FDEs), the primary goal of this work is to offer several graphical comparisons performed with the use of specific recent adjustments of the so-called Fractional Euler Method (FEM). These numerical adjustments are the Modified FEM (or MFEM) and Improved Modified FEM (or IMFEM) coupled with the FEM itself. This would reveal the best approximate solution to the fractional-order system consisting of FDEs in comparison with its exact solution.

A powerful tool for dealing with high-dimensional fractional-order systems with applications to fractional Emden–Fowler systems

In this study, an approximation solution for a high-dimensional system in terms of the Caputo fractional derivative operator is obtained using the improved modified fractional Euler method, or IMFEM for short. To accomplish this aim, a result that can transform such a system into a double-equation, one-dimensional fractional-order system, is provided theoretically. Some physical applications, including fractional-order systems of equations of Emden–Fowler type, are discussed, and their graphs are plotted using MATLAB to demonstrate the IMFEM schema’s validity.

Modeling and analysis of visceral leishmaniasis dynamics using fractional‐order operators: A comparative study

An in‐depth understanding of the mechanism concerning the parasitic disease called visceral leishmaniasis (VL) remains challenging. Thus, we modeled the dynamics of this illness using two fractional‐order operators, including Caputo–Fabrizio and Atangana–Baleanu. In the proposed dynamical model, the endemic and disease‐free equilibrium points were considered the symmetrical components. The fractional Euler method was applied to simulate the developed model, thus determining the equilibrium points' stability. The numerical simulation results were compared with the measured data to validate the model. The results obtained from the optimum fractional operator disclosed the minimum absolute and relative errors. The primary outcome of our study is the successful application of fractional‐order operators, specifically the Atangana–Baleanu operators with = 0.98, in modeling the dynamics of VL. Notably, the numerical simulation results, validated against real data from Sudan, demonstrated that the Atangana–Baleanu operators with = 0.98 yielded the best performance, with minimum absolute and relative errors. This underscores the precision of our fractional calculus‐based dynamical model in predicting VL dynamics compared to the classical framework, particularly for fractional‐order values of = 0.99 and = 0.98.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

A Mathematical Study on a Fractional-Order SEIR Mpox Model: Analysis and Vaccination Influence

This paper establishes a novel fractional-order version of a recently expanded form of the Susceptible-Exposed-Infectious-Recovery (SEIR) Mpox model. This model is investigated by means of demonstrating some significant findings connected with the stability analysis and the vaccination impact, as well. In particular, we analyze the fractional-order Mpox model in terms of its invariant region, boundedness of solution, equilibria, basic reproductive number, and its elasticity. In accordance with an effective vaccine, we study the progression and dynamics of the Mpox disease in compliance with various scenarios of the vaccination ratio through the proposed fractional-order Mpox model. Accordingly, several numerical findings of the proposed model are depicted with the use of two numerical methods; the Fractional Euler Method (FEM) and Modified Fractional Euler Method (MFEM). Such findings demonstrate the influence of the fractional-order values coupled with the vaccination rate on the dynamics of the established disease model.

Numerical Solution for Fractional-Order Mathematical Model of Immune-Chemotherapeutic Treatment for Breast Cancer Using Modified Fractional Formula

Cancer is a complex and diverse group of diseases characterized by the uncontrolled growth and spread of abnormal cells in the body. Tumors, which are commonly associated with cancer, refer to abnormal masses of tissue that can develop in various organs or tissues. Cancer can arise from almost any cell type in the body and can affect different organs and systems. The disease occurs when the normal processes of cell division and growth go awry, leading to the formation of malignant tumors. These tumors have the potential to invade nearby tissues and spread to distant parts of the body through a process known as metastasis. In this paper, we aim to present a numerical solution for a recent fractional-order model related to Immune-Chemotherapeutic Treatment for Breast Cancer (ICT) using a novel numerical scheme called the Modified Fractional Euler Method (MFEM). We will also compare our proposed scheme with the traditional numerical scheme, Fractional Euler Method (FEM), through numerical simulations.

Stability and optimal control analysis for studying the transmission dynamics of a fractional-order MSV epidemic model

In this research paper, we study the transmission dynamics of the maize streak virus disease in agriculture using fractional calculus. Our fractional-order model (FOM) consists of fractional differential equations within Caputo fractional derivative (CFD). Further, we investigate the mathematical analysis of this FOM such as the positivity and boundedness of the projected solutions, compute the control reproduction number (Rc) by the next-generation matrix method, examine the local stability by using Routh–Hurwitz conditions and global stability by using fractional LaSalle’s invariance principle for all possible equilibrium points. A sensitivity analysis is conducted on the parameters integrated into Rc to evaluate how accurately the parameter estimation fits. Additionally, we present further numerical results on the FOM to demonstrate the obtained analytical results. Then, we formulate the fractional optimal control problem (FOCP) depending on the suggested FOM with three control efforts u1, u2 and u3 called respectively, prevention, quarantine and insecticide chemical. Also, we study this FOCP analytically to derive the fractional necessary optimality conditions (NOCs) based on a kind of Pontryagin’s maximum principle. After that, we solve these fractional NOCs numerically where the state and co-state variables are subject to the left CFD. We present various strategies with different combinations of suggested controls to examine the best strategy for decreasing the transmission of MSV in maize plants, where each one of these strategies is able to reduce the maize streak disease (MSD) at the period of control time. We perform simulations of outcomes using MATLAB by developing the forward–backward sweep method utilizing the fractional Euler method. Moreover, some figures are presented to show the impact of both control interventions, change in weight factors and fractional order within the suggested strategies. We can observe that the fractional order plays a significant role in limiting the spread of MSD where the obtained results illustrated that the MSD can be reduced through the suggested FOM, which has a notable influence on plant epidemiology.

Analysis, modeling and simulation of a fractional-order influenza model

The primary goal of this study is to provide a novel mathematical model for Influenza using the Atangana–Baleanu Caputo fractional-order derivative operator (ABC-Operator) in place of the standard operator. There will be an examination of how the influenza-positive solutions reacts to real-world data. The fractional Euler Method will be utilized to reveal the dynamics of the influenza mathematical model. Both the stability of the disease-free equilibrium and the endemic equilibrium points, two symmetrical extrema of the proposed dynamical model, are examined. It will be shown, using numerical comparisons, that the findings obtained by employing the fractional-order model are considerably more similar to certain actual data than the integer-order model's results. These should shed light on the significance of fractional calculus when confronting epidemic risks.

New Algorithms for Dealing with Fractional Initial Value Problems

This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) and Improved Modified Fractional Euler Method 2 (IMFEM 2), endeavor to further enhance FEM and MFEM in terms of attaining more accuracy. By utilizing certain theoretical results, the resultant error bounds of the proposed methods are analyzed and estimated. Several numerical comparisons are carried out to validate the efficiency of our proposed methods.

Application of Fractional SIQRV Model for SARS-CoV-2 and Stability Analysis

Throughout history, infectious diseases have been the cause of outbreaks and the deaths of people. It is crucial for endemic disease management to be able to forecast the number of infections at a given moment and the frequency of new infections so that the appropriate precautions can be taken. The COVID-19 pandemic has highlighted the value of mathematical modeling of pandemics. The susceptible–infected–quarantined–recovered–vaccinated (SIQRV) epidemic model was used in this work. Symmetrical aspects of the proposed dynamic model, disease-free equilibrium, and stability were analyzed. The symmetry of the population size over time allows the model to find stable equilibrium points for any parameter value and initial conditions. The assumption of the strong symmetry of the initial conditions and parameter values plays a key role in the analysis of the fractional SIQRV model. In order to combat the pandemic nature of the disease, control the disease in the population, and increase the possibility of eradicating the disease, effective control measures include quarantine and immunization. Fractional derivatives are used in the Caputo sense. In the model, vaccination and quarantine are two important applications for managing the spread of the pandemic. Although some of the individuals who were vaccinated with the same type and equal dose of vaccine gained strong immunity thanks to the vaccine, the vaccine could not give sufficient immunity to the other part of the population. This is thought to be related the structural characteristics of individuals. Thus, although some of the individuals vaccinated with the same strategy are protected against the virus for a long time, others may become infected soon after vaccination. Appropriate parameters were used in the model to reflect this situation. In order to validate the model, the model was run by taking the COVID-19 data of Türkiye about a year ago, and the official data on the date of this study were successfully obtained. In addition to the stability analysis of the model, numerical solutions were obtained using the fractional Euler method.

Farklı İnsidans Oranlarının Etkisi Altında Bir SEIR Epidemiyolojik Modelinin Optimal Kontrolü

In this study, optimal control problem for a fractional SEIR epidemiological model under the effect of bilinear and saturate incidence rate functions is investigated. These rates play an important role in the realistic modeling of an epidemic by describing the interaction between susceptible and infected individuals of a population. This interaction is highly decisive in whether the disease will turn into a pandemic or not. Therefore, these functions can be defined in different forms depending on the course of the epidemic. The model discussed in this study is defined in terms of Caputo. Dimensional compatibility is guaranteed before posing the optimal control problem. The main objective of the proposed optimal control problem is to minimize the number of infected individuals and the cost of education given to susceptible individuals as a preventive measure. Euler-Lagrange equations corresponding to the optimality conditions of the considered model are first determined by Hamiltonian’s formalism. Afterward, the optimal system with right and left fractional Caputo derivatives are solved numerically by the forward-backward sweep method combined with the fractional Euler method. Optimal solutions are interpreted graphically for varying values of the incidence rate coefficients and the fractional parameter. According to the simulation results, it is seen that the education given to susceptible individuals is significantly effective in slowing down the epidemic.

Fractional-order modeling and optimal control of a new online game addiction model based on real data

The spread of online games has become more and more serious in recent years, which not only endangers the healthy growth of young people, but also has a certain potential harm to society. Therefore, how to effectively control the spread of games has become an urgent problem to be solved. To solve this problem, a new fractional model of online game addiction is developed in this paper. In the aspect of qualitative analysis, the non-negativity and boundedness of the model solutions are demonstrated. All equilibria and basic reproduction number are solved. The global asymptotic stability of all equilibria are proved by constructing ingenious Lyapunov functions. In terms of quantitative analysis, different from previous fractional order literature, we use a new method to fit the real data of game users in China from 2010 to 2021, and obtain the optimal fractional order α=0.98942 and model parameters. The estimated results show that the value of the basic reproduction number during this period is R0=7.1045. By setting different values of the parameters, the effects of all parameters on the model are shown. With the Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) techniques, the influence of all parameters in the basic reproduction number R0 are quantified and the means to control the spread of the game were generalized: isolation and treatment. Finally, the Fractional Euler Method (FEM) and the fractional Forward Backward Sweep Method (FBSM) are used together to solve the fractional optimal control problem and the optimal control strategy is determined. The results of this paper provide a new research idea for parameter estimation of fractional equation, and a more reliable result for controlling the spread of online games.

A Numerical Confirmation of a Fractional-Order COVID-19 Model’s Efficiency

In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards.

On the existence of solution for a [formula omitted]-dimensional system of conformable fractional derivative

In this paper, we discuss the existence of solution for a k-dimensional system of conformable fractional derivative. Using the fractional Euler method, we get an approximate solution to the system. Moreover we prove the existence of Laplace transform for our equation. Finally to illustrate the nature of our system, we provide numerical examples and its plot. The results show that the present procedure is very effective and convenient.

Optimal control strategies for a computer network under virus threat

This study proposes different optimal control strategies to eliminate the damage of virus propagation in a computer network with a minimum cost of installing anti-virus software. The model discussed to be developed is a fractional-order SEIR epidemiological model. Unlike the existing studies on the model, it has been considered that the recovered computers may malfunction due to any mechanical reason, and the model has been developed accordingly. In addition, unit consistency for the model is provided. Although there are a limited number of studies in the literature, the control variable affects the equilibrium points and thus the stability of the controlled system. Therefore, the reproduction number is recalculated for the controlled system. Before the optimal control problem is formulated, the existence of optimal control is proved. Then, the optimal system is obtained using Hamiltonian formalism. The numerical solutions of the optimal system are achieved by the fractional Euler method combined with the forward–backward sweep algorithm. The graphs drawn with MATLAB software show the efficiency of the fractional parameter for different control scenarios. Comparing the three proposed control strategies, it is clear that Strategy 1 is the most effective anti-virus installation strategy. In other words, if an anti-virus software is installed on both infected computers and susceptible computers connected to exposed and infected computers, virus propagation in the network can be prevented quickly as intended. Moreover, this is achieved by minimizing the cost of installing an antivirus program, thanks to the optimal control strategy.

New fractional modelling and control analysis of the circumscribed self-excited spherical strange attractor

The purpose of this study is to present and examine a novel non-integer model of the circumscribed self-excited spherical strange attractor, which has not been worked yet. To design the fractional-order model, we use the Caputo-Fabrizio derivative. In order to ensure the existence of the solution Picard-Lindel and fixed-point theories are provided. Moreover, the stability of the considered fractional-order model is shown using the Picard iteration and fixed point theory approach. To get the approximate solutions of the proposed fractional model an efficient numerical scheme called the fractional Euler method(FEM) is used. To see the performance of the used method, the behavior of the numerical solutions of the model is examined under various initial conditions(ICs) and fractional orders. Considerable chaotic behaviors of the solutions are obtained which prove the accuracy and reliability of FEM.

A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions

The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible (textsc {S}), exposed (textsc {E}), asymptomatic infected (textsc {I}_1), symptomatic infected (textsc {I}_2), and recovered (textsc {R}) classes named mathrm {SEI_{1}I_{2}R} model, using the Caputo fractional derivative. Here, mathrm {SEI_{1}I_{2}R} model describes the effect of asymptomatic and symptomatic transmissions on coronavirus disease outbreak. The existence and uniqueness of the solution are studied with the help of Schaefer- and Banach-type fixed point theorems. Sensitivity analysis of the model in terms of the variance of each parameter is examined, and the basic reproduction number (R_{0}) to discuss the local stability at two equilibrium points is proposed. Using the Routh–Hurwitz criterion of stability, it is found that the disease-free equilibrium will be stable for R_{0} < 1 whereas the endemic equilibrium becomes stable for R_{0} > 1 and unstable otherwise. Moreover, the numerical simulations for various values of fractional-order are carried out with the help of the fractional Euler method. The numerical results show that asymptomatic transmission has a lower impact on the disease outbreak rather than symptomatic transmission. Finally, the simulated graph of total infected population by proposed model here is compared with the real data of second-wave infected population of COVID-19 outbreak in India.

Fractional stochastic modelling of COVID-19 under wide spread of vaccinations: Egyptian case study

This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.