System Of Fractional Order Differential Equations Research Articles

Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm

Colon cancer is a complex disease with genetically unstable cell lines. In order to better understand the complexity of colon cancer cells and their metastatic mechanisms, we develop a mathematical model in this study. The model is based on a system of fractional-order differential equations and Fractional-Cancer-Informed Neural Networks (FCINN). The model captures a dynamic network of interactions between dendritic cells (DCs), cytotoxic T-cells (CD8+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8^+$$\\end{document}), helper T-cells (CD4+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4^+$$\\end{document}), and colon cancer cells through fractional differential equations. By varying the fractional order between 0 and 1, we can classify patients into different groups based on their immune patterns. The goal of this paper is to identify different immune patterns and cancer cell behaviors, as well as the parameters that play an important role in metastasis, control, or elimination of cancer cells in the model. However, several parameters in the model are difficult to estimate in a patient-specific manner. To address this challenge, we use FCINN as an effective deep-learning tool for parameter estimation and numerical simulation of the model. Our findings suggest that the most effective factors in controlling the progression and preventing metastasis of colon cancer are the initial number of cancer cells, the inhibiting rates of tumor cells by DCs, the source of DCs, and the activation of helper T-cells by DCs. These findings suggest that DCs can be used as an immunotherapy tool for the control and treatment of colon cancer.

Optimal treatment and stochastic stability on a fractional-order epidemic model incorporating media awareness

Infectious diseases have become a severe concern in our society in recent times. In this respect, conducting a qualitative investigation of infectious disease systems is very crucial. Nowadays, developing a mathematical model and its analysis for disease control has been very popular. Our present study has formulated a SIR-type epidemic model by considering a saturated incidence function with media impact. To account for the influence of memory, we have modified the model by employing a system of fractional-order differential equations of Caputo’s notion. We have determined all the system equilibrium points and discussed their stability criteria based on the basic reproduction number (R0) as the threshold parameter. It is observed that for the threshold R0<1, disease-free equilibrium becomes both locally and globally asymptotically stable. A transcritical bifurcation occurs at the threshold R0=1. Further, as the threshold R0 exceeds unity, the endemic equilibrium becomes asymptotically stable. A fractional-order optimal control problem with treatment as the control parameter is developed to reduce the impact of disease. Using Pontryagin’s Maximum Principle yields a theoretical and numerical solution to the fractional order optimal control problem. It is observed that both the media and memory effect dramatically reduces infection rates. Considering environmental variations, the stochastic stability of the endemic equilibrium point has also been studied. Sensitivity analysis is carried out to comprehend how the parameters impact the threshold value (R0). Finally, extensive numerical simulations are performed to demonstrate our theoretical examination.

On the Solvability of Iterative Systems of Fractional-Order Differential Equations with Parameterized Integral Boundary Conditions

The aim of this paper is to determine the eigenvalue intervals of $\mu_{\mathtt{k}},~1\le \mathtt{k}\le \mathtt{n}$ for which an iterative system of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example of their feasibility.

Fractional mathematical model of Listeria infection caused by pre-cooked package food

This article introduces a novel mathematical approach for modeling and analyzing Listeria infection dynamics by considering a system of fractional-order differential equations to understand the complexities. The homotopy perturbation general transform method (HPGTM) is applied to derive approximate solutions for the fractional-order Listeria infection model to obtain valuable insights. The obtained numerical and graphical solutions are rigorously analyzed to explore the effect of fractional orders on the dynamics of Listeria infections. Furthermore, we explore the existence, uniqueness, and stability properties of the solutions, unveiling the critical parameters influencing infection spread and persistence.

Investigating the dynamics of a novel fractional-order monkeypox epidemic model with optimal control

In this paper, a novel fractional-order monkeypox epidemic model is introduced, wherefractional-order derivativesin the sense of Caputo are applied to achieve more realistic results for the proposednonlinear model. The newly developed model, which models the transmission and spread of monkeypox across the interacting populations of humans and rodents, is controlled by a 14-dimensional system of fractional-order differential equations. To comply with empirical and reported observations, the state variables of the proposed model are categorized into three main groups of state variables: the population who are at high risk of being infected, people with low infection probability, and finally, rodents who can carry and transmit the virus. The high-risk group represents individuals who might be more vulnerable to the virus due to their habits, workplace, or hygienic behaviors. The existence, uniqueness, non-negativity, and boundedness of the solution to the proposed model are proved. The next-generation matrix approach is used to determine the control monkeypox reproduction number, R0, and the equilibrium points for the proposed model are obtained. The effect of the main parameters in the model is thoroughly investigated to provide new insight into the new dynamics of the model. The region of stability of the disease-free points (DFE) is obtained in the space of parameters, and the effect of the parameters is examined. In addition, theoptimal control strategyis applied to the model to provide insight into some prevention control to stop the disease from spreading and to provide new control strategies during the monkeypox outbreak. Numerical simulations are performed to validate the theoretical results of the different optimal control strategies.

ON SYSTEMS OF FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS FOR ORDER 1 &lt; 𝜗 ≤ 2

This work is devoted to establish a numerical scheme for system of fractional-order differential equations (FODEs) with order [Formula: see text]. The scheme is established by using Bernstein polynomials (BPs). Based on the said materials, some operational matrices are formed. With the help of obtained operational matrices, the considered system is reduced to some algebraic system of equations. On using MATLAB-16, the system is then solved to get the required numerical solution for the proposed system. Several examples are treated with the help of the proposed method for numerical solutions. Further, error analysis is also recorded for different fractional orders and various scale levels. The mentioned results are displayed graphically. Comparison with exact solution at traditional order derivative is also given. It should be kept in mind that the proposed method does not require any kind of discretization or collocation. Also, there is no external parameter which controls the method. Due to these features, the proposed method is powerful and efficient for different classes of FODEs to compute their numerical solutions. The efficiency of the proposed method can be enhanced by increasing the scale level.

Mathematical Modelling of Dengue Transmission with Intervention Strategies Using Fractional Derivatives.

This paper deals with a deterministic mathematical model of dengue based on a system of fractional-order differential equations (FODEs). In this study, we consider dengue control strategies that are relevant to the current situation in Malaysia. They are the use of adulticides, larvicides, destruction of the breeding sites, and individual protection. The global stability of the disease-free equilibrium and the endemic equilibrium is constructed using the Lyapunov function theory. The relations between the order of the operator and control parameters are briefly analysed. Numerical simulations are performed to verify theoretical results and examine the significance of each intervention strategy in controlling the spread of dengue in the community. The model shows that vector control tools are the most efficient method to combat the spread of the dengue virus, and when combined with individual protection, make it more effective. In fact, the massive use of personal protection alone can significantly reduce the number of dengue cases. Inversely, mechanical control alone cannot suppress the excessive number of infections in the population, although it can reduce the Aedes mosquito population. The result of the real-data fitting revealed that the FODE model slightly outperformed the integer-order model. Thus, we suggest that the FODE approach is worth to be considered in modelling an infectious disease like dengue.

Formulation, Solution’s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations

In the recent past, multi-term fractional equations have been studied using symmetry methods. In some cases, many practical test problems with some symmetries are provided to demonstrate the authenticity and utility of the used techniques. Fractional-order differential equations can be formulated by using two types of differential operators: single-term and multi-term differential operators. Boundary value problems with single- as well as multi-term differential operators have been extensively studied, but several multi-term fractional differential equations still need to be formulated, and examination should be done with symmetry or any other feasible techniques. Therefore, the purpose of the present research work is the formulation and study of a new couple system of multi-term fractional differential equations with delay, as well as supplementation with nonlocal boundary conditions. After model formulation, the existence of a solution and the uniqueness conditions will be developed, utilizing fixed point theory and functional analysis. Moreover, results related to Ulam’s and other types of functional stability will be explored, and an example is carried out to illustrate the findings of the work.

On the fractional-order mathematical model of COVID-19 with the effects of multiple non-pharmaceutical interventions

&lt;abstract&gt;&lt;p&gt;In this article, the Caputo fractional derivative operator of different orders $ 0 &amp;lt; \alpha\leq1 $ is applied to formulate the fractional-order model of the COVID-19 pandemic. The existence and boundedness of the solutions of the model are investigated by using the Gronwall-Bellman inequality. Further, the uniqueness of the model solutions is established by using the fixed-point theory. The Laplace Adomian decomposition method is used to obtain an approximate solution of the nonlinear system of fractional-order differential equations of the model with a different fractional-order $ \alpha $ for every compartment in the model. Finally, graphical presentations are presented to show the effects of other fractional parameters $ \alpha $ on the obtained approximate solutions.&lt;/p&gt;&lt;/abstract&gt;

Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

Exact Stability and Instability Regions for Two-Dimensional Linear Autonomous Multi-Order Systems of Fractional-Order Differential Equations

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

一类具有Caputo导数的非线性分数阶微分方程耦合系统的正解

本文研究了一类非线性分数阶微分方程耦合系统的正解存在性，此耦合系统具有Caputo导数和边界条件。通过运用一个新的研究具有矢量的算子的不动点方法，Krasnoselskii锥不动点定理，得到系统的正解存在性。进一步拓展定理得到正解的局限性和多重性。 In this paper, we study the existence of positive solutions for a class of nonlinear coupled system of fractional-order differential equations with Caputo derivatives and boundary conditions. By using a new method to study the fixed points of operators with vectors, Krasnoselskii fixed point theorem of cone, the existence of positive solutions of the system is obtained. We also investigate the localization and multiplicity of the positive solutions by further extending the theorem.

Mathematical modeling of the immune-chemotherapeutic treatment of breast cancer under some control parameters

We consider a system of fractional-order differential equations to analyze breast cancer growth in the immune-chemotherapeutic treatment process under some control parameters: ketogenic diet, immune booster, and anti-cancer drugs. The established model assumes the growth of the tumor density under chemotherapy treatment and the immune response during the interaction between the normal cells and tumor cells. For the local stability of the critical points (tumor-free critical point, dead critical point, and co-existing critical point), we used the Routh-Hurwitz criteria to show the necessary effect of the immune booster; moreover, we addressed the ketogenic rate in the treatment process. Our theoretical and numerical studies pointed out that on early detection of the tumor density (with weak Allee effect) the treatment should be supported by ketogenic nutrition. Several examples are shown to present our theoretical findings.

Modeling and Estimation of Nuclear Reactor Performance Using Fractional Neutron Point Kinetics with Temperature Effect and Xenon Poisoning

This paper discusses modeling of the performance of a VVER-type nuclear power plant using the Fractional Neutron Point Kinetics (FNPK) model. The modeling intent is to achieve a nonlinear system of fractional-order differential equations that are solved using SIMULINK by developing a scheme with the FOMCON Toolbox in the Matlab® environment. The model is shown to be identifiable, and the goodness of fit between the measurements and the model’s output is improved as a result of employing the optimized fractional differential equation. Benchmarking the results of the FNPK model against those from the optimized Classical Neutron Point Kinetics model shows that the FNPK model is superior.

New explicit and accelerated techniques for solving fractional order differential equations

This paper proposes new direct and acceleration numerical methods for solving Fractional order Differential Equations (FDEs). For the Caputo differential operator with fractional order 0 < ν < 1, we rewrite the FDE as an equivalent integral form circumventing the derivative of the solution by the integral by parts. We obtain a discrete formulation directly from the integral form using the Lagrange interpolate polynomials. We provide truncation errors depending on the fractional order ν for linear and quadratic interpolations, which are O(h2−ν) and O(h3−ν) respectively. In order to overcome a strong singular issue as ν ≈ 0 we propose the explicit predictor-corrector scheme with perturbation technique. In case of ν ≈ 1, the truncation errors are reduced by O(h) and O(h2). In order to accelerate the convergence rate, we decompose ν=ν/2+ν/2, convert the FDE into the system of FDEs, and apply the predictor-corrector scheme. Numerical tests for linear, nonlinear, variable order and time-fractional sub-diffusion problems demonstrate that the proposed methods give a prominent performance. We also compare the numerical results with other high-order explicit and implicit schema.

Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method

This manuscript deals a numerical technique based on Haar wavelet collocation which is developed for the approximate solution of some systems of linear and nonlinear fractional order differential equations (FODEs). Based on these techniques, we find the numerical solution to various systems of FODEs. We compare the obtain solution with the exact solution of the considered problems at integer orders. Also, we compute the maximum absolute error to demonstrate the efficiency and accuracy of the proposed method. For the illustration of our results we provide four test examples. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is defined in the Caputo sense.

Mittag-Leffler stability for a new coupled system of fractional-order differential equations with impulses

This paper is devoted to investigation of the Mittag-Leffler stability problem for a new coupled system of fractional-order differential equations with impulses on networks. By using the direct graph theory, a new coupled model with two fractional-order impulsive equations on each vertex is constructed, and the related Lyapunov function is presented. By the Lyapunov direct method, sufficient conditions are derived to ensure the equilibrium point of the coupled fractional-order impulsive model is globally Mittag-Leffler stable. Our new results show a relation between the stability criteria and some topology property of the system. Finally, a numerical example is provided to illustrate the effectiveness of our results.

Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect

This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies.

Optimal Leader\u2013Follower Control for the Fractional Opinion Formation Model

This paper deals with an opinion formation model, that obeys a nonlinear system of fractional-order differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader–follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented.

Spectral tau algorithm for solving a class of fractional optimal control problems via Jacobi polynomials

This paper is dedicated to analyzing and presenting an efficient numerical algorithm for solving a class of fractional optimal control problems (FOCPs). The basic idea behind the suggested algorithm is based on transforming the FOCP under investigation into a coupled system of fractional-order differential equations whose solutions can be expanded in terms of the Jacobi basis. With the aid of the spectral-tau method, the problem can be reduced into a system of algebraic equations which can be solved via any suitable solver. Some illustrative examples and comparisons are presented aiming to demonstrate the accuracy, applicability, and efficiency of the proposed algorithm.