ABSTRACT
In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The hybrid variable-order derivative is defined as a linear combination of the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. A symmetry parameter σ is presented in order to be consistent with the physical model problem. The existence, uniqueness, boundedness and positivity of the proposed model are given. Moreover, the stability of the proposed model is discussed. The theta finite difference method with the discretization of the hybrid variable-order operator is developed for solving numerically the model problem. This method can be explicit or fully implicit with a large stability region depending on values of the factor Θ. The convergence and stability analysis of the proposed method are proved. Moreover, the fourth order generalized Runge–Kutta method is also used to study the proposed model. Comparative studies and numerical examples are presented. We found that the proposed model is also more general than the model in the previous study;the results obtained by the proposed method are more stable than previous research in this area.
ABSTRACT
The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
ABSTRACT
With the development of Medical Internet of Things (MIoT) technology and the global COVID-19 pandemic, hospitals gain access to patients' health data from remote wearable medical equipment. Federated learning (FL) addresses the difficulty of sharing data in remote medical systems. However, some key issues and challenges persist, such as heterogeneous health data stored in hospitals, which leads to high communication cost and low model accuracy. There are many approaches of federated distillation (FD) methods used to solve these problems, but FD is very vulnerable to poisoning attacks and requires a centralised server for aggregation, which is prone to single-node failure. To tackle this issue, we combine FD and blockchain to solve data sharing in remote medical system called FedRMD. FedRMD use reputation incentive to defend against poisoning attacks and store reputation values and soft labels of FD in Hyperledger Fabric. Experimenting on COVID-19 radiography and COVID-Chestxray datasets shows our method can reduce communication cost, and the performance is higher than FedAvg, FedDF, and FedGen. In addition, the reputation incentive can reduce the impact of poisoning attacks. © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
ABSTRACT
In the present paper, a reaction-diffusion epidemic mathematical model is proposed for the analysis of the transmission mechanism of the novel coronavirus disease 2019 (COVID-19). The mathematical model contains six-time and space-dependent classes, namely; Susceptible, Exposed, Asymptomatically infected, Symptomatic infected, Quarantine, and Recovered or Removed (SEQIaIsR). The threshold number R0 is calculated by utilizing the next-generation matrix approach. Values of the parameters are estimated with the help of the least square curve fitting tools. In addition to the simple explicit procedure, the mathematical epidemiological model with diffusion is simulated through the operator splitting approach based on finite difference and meshless methods. Stability analysis of the disease free and endemic equilibrium points of the model is investigated. Simulation results of the model with and without diffusion are presented in detail. A comparison of the obtained numerical results of both the models is performed in the absence of an exact solution. The correctness of the solution is verified through mutual comparison and partly, via theoretical analysis as well.
ABSTRACT
A compartmental mathematical model of spreading COVID-19 disease in Wuhan, China is applied to investigate the pandemic behaviour in Iran. This model is a system of seven ordinary differential equations including individual behavioural reactions, governmental actions, holiday extensions, travel restrictions, hospitalizations, and quarantine. We fit the Chinese model to the Covid-19 outbreak in Iran and estimate the values of parameters by trial-error approach. We use the Adams-Bashforth predictor-corrector method based on Lagrange polynomials to solve the system of ordinary differential equations. To prove the existence and uniqueness of solutions of the model we use Banach fixed point theorem and Picard iterative method. Also, we evaluate the equilibrium points and the stability of the system. With estimating the basic reproduction number R 0 , we assess the trend of new infected cases in Iran. In addition, the sensitivity analysis of the model is assessed by allocating different parameters to the system. Numerical simulations are depicted by adopting initial conditions and various values of some parameters of the system.
ABSTRACT
We have been developing an interactive and multimodal platform to facilitate learning fluid dynamics with the rationale of using an immersive environment as a visualisation medium. Before the pan-demic, we used our in-house virtual reality app to teach fluid dynamics (FD), significantly enhancing student engagement. Since the COVID-19 pandemic struck, we have explored AR and MR applications for scaling our remote online and hybrid teaching efforts. The work presented in this paper has two objectives. (i) Provide an AR learning medium for remotely located students. (ii) Provide a student-paced instructional learning medium using MR for the hy-brid or onsite students. To achieve this, we describe a methodology in four parts. (i) A computational fluid dynamics data processing and distribution pipeline for generating 3D models for AR and MR. (ii) A platform-independent FD learning platform that uses WebXR for rendering models in AR. (iii) Hololens-based instructional medium in MR for learning FD.(iv) A pedagogy design. We discuss the results of a feasibility study on 18 hybrid learning students to assess the effectiveness of the pedagogy design using MR. We conclude that by using our platform, students can interactively visualise our in-house fluid dynamics models aligned with the course work and acquire knowledge naturally and intuitively. © 2022 IEEE.
ABSTRACT
In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used;however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.
ABSTRACT
In this work, we generalize the multi-vaccination for COVID-19 mathematical model to the variable-order fractional derivative case by replacing the first-order derivative with the variable-order fractional Atangana-Baleanu-Caputo operator. Moreover, the parameters of the proposed model are dependent on the variable-order fractional derivative. We use an amended numerical technique to build an asymptotically stable difference scheme. The proposed scheme accurately simulates the behavior of the solution of the presented model. Some properties, boundedness, and positivity of the studied system are proved analytically. The introduced scheme preserves these properties of the analytic solutions. Numerical treatment is run out for testing the reliability, applicability, and simplicity of this scheme. Comparitive study bettween the obtained results using the discretization of Atangana-Baleanu-Caputo derivative and nonstandard finite differeance method with the results which obtained using the discretization of the operator Yang-Abdel-Cattani is given.
ABSTRACT
This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data. © 2022 THE AUTHORS
ABSTRACT
Fuzziness or uncertainties arise due to insufficient knowledge, experimental errors, operating conditions and parameters that provide inaccurate information. The concepts of susceptible, infectious and recovered are uncertain due to the different degrees in susceptibility, infectivity and recovery among the individuals of the population. The differences can arise, when the population groups under the consideration having distinct habits, customs and different age groups have different degrees of resistance, etc. More realistic models are needed which consider these different degrees of susceptibility infectivity and recovery of the individuals. In this paper, a Susceptible, Infected and Recovered (SIR) epidemic model with fuzzy parameters is discussed. The infection, recovery and death rates due to the disease are considered as fuzzy numbers. Fuzzy basic reproduction number and fuzzy equilibrium points have been derived for the studied model. The model is then solved numerically with three different techniques, forward Euler, Runge-Kutta fourth order method RK-4) and the nonstandard finite difference (NSFD) methods respectively. The NSFD technique becomes more efficient and reliable among the others and preserves all the essential features of a continuous dynamical system.
ABSTRACT
In this paper, we propose a modified fractional diffusive SEAIR epidemic model with a nonlinear incidence rate. A constructed model of fractional partial differential equations (PDEs) is more general than the corresponding model of fractional ordinary differential equations (ODEs). The Caputo fractional derivative is considered. Linear stability analysis of the disease-free equilibrium state of the epidemic model (ODEs) is presented by employing Routh–Hurwitz stability criteria. In order to solve this model, a fractional numerical scheme is proposed. The proposed scheme can be used to find conditions for obtaining positive solutions for diffusive epidemic models. The stability of the scheme is given, and convergence conditions are found for the system of the linearized diffusive fractional epidemic model. In addition to this, the deficiencies of accuracy and consistency in the nonstandard finite difference method are also underlined by comparing the results with the standard fractional scheme and the MATLAB built-in solver pdepe. The proposed scheme shows an advantage over the fractional nonstandard finite difference method in terms of accuracy. In addition, numerical results are supplied to evaluate the proposed scheme’s performance.
ABSTRACT
A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt.
ABSTRACT
In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with ξ as order of the fractional derivative. Moreover, a valued structure preserving the numerical design, coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model. Taking into account the importance of the positivity and boundedness of the state variables, some productive results have been proved to ensure these essential features. Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values. The Routh–Hurwitz criterion is applied for the local stability analysis. An appropriate example with fitted and estimated set of parametric values is presented for the simulations. Graphical solutions are displayed for the chosen values of ξ (fractional order of the derivatives). The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases. In the end, outcomes of the study are presented. © 2022 Tech Science Press. All rights reserved.
ABSTRACT
In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald-Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.
ABSTRACT
Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists.