Several collected data representing the spread of some infectious diseases have demonstrated that the spread does not really exhibit homogeneous spread. Clear examples can include the spread of Spanish flu and Covid-19. Collected data depicting numbers of daily new infections in the case of Covid-19 from countries like Turkey, Spain show three waves with different spread patterns, a clear indication of crossover behaviors. While modelers have suggested many mathematical models to depicting these behaviors, it becomes clear that their mathematical models cannot really capture the crossover behaviors, especially passage from deterministic resetting to stochastics. Very recently Atangana and Seda have suggested a concept of piecewise modeling consisting in defining a differential operator piece-wisely. The idea was first applied in chaos and outstanding patterns were captured. In this paper, we extend this concept to the field of epidemiology with the aim to depict waves with different patterns. Due to the novelty of this concept, a different approach to insure the existence and uniqueness of system solutions are presented. A piecewise numerical approach is presented to derive numerical solutions of such models. An illustrative example is presented and compared with collected data from 3 different countries including Turkey, Spain and Czechia. The obtained results let no doubt for us to conclude that this concept is a new window that will help mankind to better understand nature.

Modeling third waves of Covid-19 spread with piecewise differential and integral operators: Turkey, Spain and Czechia

In the present paper, we formulate novel fractional tuberculosis (TB) model with a generalized Atangana–Baleanu (GAB) fractional derivative. The parameters in this model are fitted from the real statistical data of cases of TB infections in Yemen from 2000 to 2019. Firstly, pulmonary TB cases of infected people in Yemen during the period (2000–2019) were studied and analyzed. The future trends for possible cases of pulmonary tuberculosis in Yemen for the years (2020–2021) were predicted using the Expert Modeler model and apply statistical analysis programs (SPSS, version23 & Eviews, 9). In addition, we have investigated the spread of TB in Yemen and compared it with some countries around the world. Further, the fundamental properties of this novel fractional tuberculosis (TB) model with a generalized Atangana–Baleanu (GAB) fractional derivative are given. The fixed point theory is used to obtained existence and uniqueness results for this model. Finally, the solutions are obtained using iterative techniques of Adams–Bashforth and are presented using graphical simulations. Here we get numerical values of parameters used in this novel model to display the significance of our novel model, which provides useful information about the dynamics of spread and control of TB.

A fractional dynamics of tuberculosis (TB) model in the frame of generalized Atangana–Baleanu derivative

In December 2019, a new outbreak in Wuhan, China has attracted world-wide attention, the virus then spread rapidly in most countries of the world, the objective of this paper is to investigate the mathematical modelling and dynamics of a novel coronavirus (COVID-19) with Caputo-Fabrizio fractional derivative in the presence of quarantine and isolation strategies. The existence and uniqueness of the solutions for the fractional model is proved using fixed point iterations, the fractional model are shown to have disease-free and an endemic equilibrium point.We construct a fractional version of the four-steps Adams-Bashforth method as well as the error estimate of this method. We have used this method to determine the numerical scheme of this model and Matlab program to illustrate the evolution of the virus in some countries (Morocco, Qatar, Brazil and Mexico) as well as to support theoretical results. The Least squares fitting is a way to find the best fit curve or line for a set of points, so we apply this method in this paper to construct an algorithm to estimate the parameters of fractional model as well as the fractional order, this model gives an estimate better than that of classical model.

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Fractional Model and Numerical Algorithms for Predicting COVID-19 with Isolation and Quarantine Strategies

This paper introduces a Fractional Order Field Oriented Control (FO-FOC) with sensorless Model Reference Adaptive System (MRAS) for the Permanent Magnet Synchronous Motor (PMSM). In this work, the conventional integer controllers of the vector control method are replaced with the Fractional Order PI (FOPIα) controllers. Particle Swarm Optimization technique (PSO) is employed to tune the gain, integration parameter and the fractional order parameter for each of the speed, current and MRAS controllers to get the optimum values. The presented methodology is tested using MATLAB Simulink simulation and the results show improved performance of the field oriented control technique by the employment of the fractional order controllers instead of the conventional integer at different operating points. Furthermore, the additional degree of freedom in the fractional controllers helps the PSO technique to get the optimum cost at a very low number of iterations compared with that of the integer one.

PSO technique applied to sensorless field-oriented control PMSM drive with discretized RL-fractional integral

In this analysis, we use the high order cubic B-spline method to create approximating polynomial solutions for fractional Painlevé and Bagley-Torvik equations in the Captuo, Caputo-Fabrizio, and conformable fractional sense concerning boundary set conditions. Using a piecewise spline of a 3rd-degree polynomial; the discretization of the utilized fractional model problems is gained. Taking advantage of the Taylor series expansion; the error order behavior spline theorem is proved. We demonstrate applications of our spline method to several certain kinds including the 1st(2nd) Painlevé and Bagley-Torvik fractional models. For more detail, using Mathematica 11 several drawings and many tables were calculated and their explanations were mentioned. The computational results indicate that the suggested spline approach is most acceptable in terms of cost efficiency and precision of calculations. Highlight, conclusion, and future notes are provided to extract the ability of the discussed approach and the tendency of the utilized fractional models to extrapolate new application areas in the meshless numerical training. 

The novel cubic B-spline method for fractional Painlevé and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense

The objective of this paper is to investigate the mathematical modeling and dynamics of a novel coronavirus (2019-nCoV) with Atangana-Baleanu and Caputo-Fabrizio fractional derivatives in the presence of isolation strategy. The existence and uniqueness of the solutions for multi-order fractional model is proved using fixed point iterations, the fractional model are shown to have disease-free and an endemic equilibrium point. We have used a numerical scheme based on the newton polynomial to determine the numerical solutions of this model and Matlab program to illustrate the evolution of the virus in Morocco, as well as to support theoretical results. We have estimated the value of the fractional order for each model and used the least squares method for estimating model parameters. Further, the sensitivity analysis of the parameter is performed as a result, our approach gives a good approximation to real data COVID-19 of Morocco. © 2021, Azerbaijan National Academy of Sciences. All rights reserved.

A fractional multi-order model to predict the COVID-19 outbreak in Morocco

Modeling and numerical investigation of a conformable co-infection model for describing Hantavirus of the European moles

In December 2019, a new virus belonging to the coronavirus strain has been discovered in Wuhan, China, this virus has attracted world-wide attention and it spread rapidly in the world, reaching nearly 216 countries in the world in November 2020. In this chapter, we study the fractional incommensurate SIQR (susceptible, infections,quarantined and removed) COVID-19 model with nonlinear saturated incidence rate using Atangana-Baleanu fractional derivatives. The existence and uniqueness of the solutions for the fractional model is proved using fixed point theorem, the model are shown to have two equilibrium point (disease-free and an endemic equilibrium). Some numerical simulations using Euler method are also carried out to support our theoretical results. We estimated the value of the fractional orders and the parameters of the proposed model using the least squares method.. Further, the sensitivity analysis of the parameter is performed as a result, our incommensurate model gives a good approximation to real data of COVID-19.

https://medrxiv.org/content/10.1101/2021.07.18.21260711v1.full.pdf

On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate

In this paper, the existence and uniqueness of the weak solution for a linear parabolic equation with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear differential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the monotone sequence converge to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and Aubin–Lions lemmas are also proved.

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Abdelilah Lamrani Alaoui

Papers

Fractional Model and Numerical Algorithms for Predicting COVID-19 with Isolation and Quarantine Strategies

A fractional multi-order model to predict the COVID-19 outbreak in Morocco

Modeling and numerical investigation of a conformable co-infection model for describing Hantavirus of the European moles

On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate

Monotone Iterative Technique for Nonlinear Periodic Time Fractional Parabolic Problems