Advancements in Bullen-type inequalities via fractional integral operators and their applications.

In this paper, we investigate Bullen-type inequalities applicable to functions that are twice-differentiable. To explore these advanced inequalities, we utilize generalized convexity and Riemann-type fractional integrals. A comparative analysis is provided to highlight the more refined inequalities from among the explored results. By exploring the limiting cases, a relation with existing literature is established. Several examples are also presented to illustrate the outcomes and their accuracy is validated through graphical analysis. Additionally, applications in generalized means are also discussed.

Refinements of Pólya-SzegO and Chebyshev type inequalities via different fractional integral operators.

Various differential and integral operators have been introduced and applied for the generalization of several integral inequalities. The purpose of this article is to create a more generalized fractional integral operator of Saigo type. This operator will be used alongwith the existing Saigo type and Q-Saigo type fractional integral operators to establish extended and generalized versions of several inequalities, including Pólya-Szego and Chebyshev type inequalities.

Predictive dynamical modeling and stability of the equilibria in a discrete fractional difference COVID-19 epidemic model.

The SARSCoV-2 virus, also known as the coronavirus-2, is the consequence of COVID-19, a severe acute respiratory syndrome. Droplets from an infectious individual are how the pathogen is transmitted from one individual to another and occasionally, these particles can contain toxic textures that could also serve as an entry point for the pathogen. We formed a discrete fractional-order COVID-19 framework for this investigation using information and inferences from Thailand. To combat the illnesses, the region has implemented mandatory vaccination, interpersonal stratification and mask distribution programs. As a result, we divided the vulnerable people into two groups: those who support the initiatives and those who do not take the influence regulations seriously. We analyze endemic problems and common data while demonstrating the threshold evolution defined by the fundamental reproductive quantity R 0 . Employing the mean general interval, we have evaluated the configuration value systems in our framework. Such a framework has been shown to be adaptable to changing pathogen populations over time. The Picard Lindelöf technique is applied to determine the existence-uniqueness of the solution for the proposed scheme. In light of the relationship between the R 0 and the consistency of the fixed points in this framework, several theoretical conclusions are made. Numerous numerical simulations are conducted to validate the outcome.

A fractional-order mathematical model based on vaccinated and infected compartments of SARS-CoV-2 with a real case study during the last stages of the epidemiological event.

In 2020 the world faced with a pandemic spread that affected almost everything of humans' social and health life. Regulations to decrease the epidemiological spread and studies to produce the vaccine of SARS-CoV-2 were on one side a hope to return back to the regular life, but on the other side there were also notable criticism about the vaccines itself. In this study, we established a fractional order differential equations system incorporating the vaccinated and re-infected compartments to a S I R frame to consider the expanded and detailed form as an S V I I v R model. We considered in the model some essential parameters, such as the protection rate of the vaccines, the vaccination rate, and the vaccine's lost efficacy after a certain period. We obtained the local stability of the disease-free and co-existing equilibrium points under specific conditions using the Routh-Hurwitz Criterion and the global stability in using a suitable Lyapunov function. For the numerical solutions we applied the Euler's method. The data for the simulations were taken from the World Health Organization (WHO) to illustrate numerically some scenarios that happened.

Numerical treatments for the optimal control of two types variable-order COVID-19 model.

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge-Kutta method and the non-standard generalized fifth-order Runge-Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies' simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.

The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method.

Introduction: This study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure. Objectives: The present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically. Methods: The variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for (0<α(t)<1) . An efficient and accurate meshfree method based on the singular boundary method (SBM) and dual reciprocity method (DRM) in concomitant with finite difference scheme is proposed on three-dimensional arbitrary geometry. To discrete of the temporal term, the finite diffract method (FDM) is utilized. In the spatial variation domain; the proposal method is constructed two part. To evaluating first part, fundamental solution of (VO-TFDE) is transformed into inhomogeneous Helmholtz-type to implement the SBM approximation and other part the DRM is utilized to compute the particular solution. Results: The stability and convergent of the proposed method is numerically investigated on high dimensional domain. To verified the reliability and the accuracy of the present approach on complex geometry several examples are investigated. Conclusions: The result of study provides a rapid and practical scheme to capture the behavior of diffusion process.

A new general integral transform for solving integral equations.

Introduction: Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms. Objectives: In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc. Methods: A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation. Results: It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems. Conclusion: It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

A fractional order Covid-19 epidemic model with Mittag-Leffler kernel.

In this article, we are studying fractional-order COVID-19 model for the analytical and computational aspects. The model consists of five compartments including; ` ` S c ″ which denotes susceptible class, ` ` E c ″ represents exposed population, ` ` I c ″ is the class for infected people who have been developed with COVID-19 and can cause spread in the population. The recovered class is denoted by ` ` R c ″ and ` ` V c ″ is the concentration of COVID-19 virus in the area. The computational study shows us that the spread will be continued for long time and the recovery reduces the infection rate. The numerical scheme is based on the Lagrange's interpolation polynomial and the numerical results for the suggested model are similar to the integer order which gives us the applicability of the numerical scheme and effectiveness of the fractional order derivative.

Review of fractional epidemic models.

The global impact of corona virus (COVID-19) has been profound, and the public health threat it represents is the most serious seen in a respiratory virus since the 1918 influenza A(H1N1) pandemic. In this paper, we have focused on reviewing the results of epidemiological modelling especially the fractional epidemic model and summarized different types of fractional epidemic models including fractional Susceptible-Infective-Recovered (SIR), Susceptible-Exposed-Infective-Recovered (SEIR), Susceptible-Exposed-Infective-Asymptomatic-Recovered (SEIAR) models and so on. Furthermore, we propose a general fractional SEIAR model in the case of single-term and multi-term fractional differential equations. A feasible and reliable parameter estimation method based on modified hybrid Nelder-Mead simplex search and particle swarm optimisation is also presented to fit the real data using fractional SEIAR model. The effective methods to solve the fractional epidemic models we introduced construct a simple and effective analytical technique that can be easily extended and applied to other fractional models, and can help guide the concerned bodies in preventing or controlling, even predicting the infectious disease outbreaks.

A new fractional mathematical modelling of COVID-19 with the availability of vaccine.

The most dangerous disease of this decade novel coronavirus or COVID-19 is yet not over. The whole world is facing this threat and trying to stand together to defeat this pandemic. Many countries have defeated this virus by their strong control strategies and many are still trying to do so. To date, some countries have prepared a vaccine against this virus but not in an enough amount. In this research article, we proposed a new SEIRS dynamical model by including the vaccine rate. First we formulate the model with integer order and after that we generalize it in Atangana-Baleanu derivative sense. The high motivation to apply Atangana-Baleanu fractional derivative on our model is to explore the dynamics of the model more clearly. We provide the analysis of the existence of solution for the given fractional SEIRS model. We use the famous Predictor-Corrector algorithm to derive the solution of the model. Also, the analysis for the stability of the given algorithm is established. We simulate number of graphs to see the role of vaccine on the dynamics of the population. For practical simulations, we use the parameter values which are based on real data of Spain. The main motivation or aim of this research study is to justify the role of vaccine in this tough time of COVID-19. A clear role of vaccine at this crucial time can be realized by this study.

A fractional-order model describing the dynamics of the novel coronavirus (COVID-19) with nonsingular kernel.

In this paper, we investigate an epidemic model of the novel coronavirus disease or COVID-19 using the Caputo-Fabrizio derivative. We discuss the existence and uniqueness of solution for the model under consideration, by using the the Picard-Lindelöf theorem. Further, using an efficient numerical approach we present an iterative scheme for the solutions of proposed fractional model. Finally, many numerical simulations are presented for various values of the fractional order to demonstrate the impact of some effective and commonly used interventions to mitigate this novel infection. From the simulation results we conclude that the fractional order epidemic model provides more insights about the disease dynamics.

Stability analysis and optimal control of Covid-19 pandemic SEIQR fractional mathematical model with harmonic mean type incidence rate and treatment.

In the present work, we investigated the transmission dynamics of fractional order SARS-CoV-2 mathematical model with the help of Susceptible S ( t ) , Exposed E ( t ) , Infected I ( t ) , Quarantine Q ( t ) , and Recovered R ( t ) . The aims of this work is to investigate the stability and optimal control of the concerned mathematical model for both local and global stability by third additive compound matrix approach and we also obtained threshold value by the next generation approach. The author's visualized the desired results graphically. We also control each of the population of underlying model with control variables by optimal control strategies with Pontryagin's maximum Principle and obtained the desired numerical results by using the homotopy perturbation method. The proposed model is locally asymptotically unstable, while stable globally asymptotically on endemic equilibrium. We also explored the results graphically in numerical section for better understanding of transmission dynamics.

Projections and fractional dynamics of COVID-19 with optimal control strategies.

When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, unprecedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention strategies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Bangladesh and India. Control reproduction numbers ( R c ) have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. Nevertheless, as the pandemic is not over in any country, it is highly recommended to use efficacious face coverings and maintain strict physical distancing in public gatherings. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control strategies for fractional system have been designed and the existence of unique solution has also been showed using Picard-Lindelof technique. Finally, unconditional stability of the fractional numerical technique has been proved.

Fractional model of COVID-19 applied to Galicia, Spain and Portugal.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models.

Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India.

Fractional-order derivative-based modeling is very significant to describe real-world problems with forecasting and analyze the realistic situation of the proposed model. The aim of this work is to predict future trends in the behavior of the COVID-19 epidemic of confirmed cases and deaths in India for October 2020, using the expert modeler model and statistical analysis programs (SPSS version 23 & Eviews version 9). We also generalize a mathematical model based on a fractal fractional operator to investigate the existing outbreak of this disease. Our model describes the diverse transmission passages in the infection dynamics and affirms the role of the environmental reservoir in the transmission and outbreak of this disease. We give an itemized analysis of the proposed model including, the equilibrium points analysis, reproductive number R 0 , and the positiveness of the model solutions. Besides, the existence, uniqueness, and Ulam-Hyers stability results are investigated of the suggested model via some fixed point technique. The fractional Adams Bashforth method is applied to solve the fractal fractional model. Finally, a brief discussion of the graphical results using the numerical simulation (Matlab version 16) is shown.

Theoretical and numerical analysis of novel COVID-19 via fractional order mathematical model.

In the work, author's presents a very significant and important issues related to the health of mankind's. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam's type stability for proposed system by using the tools of non-linear analysis. The author's thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author's interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

Fractal-Fractional Mathematical Model Addressing the Situation of Corona Virus in Pakistan.

This work is the consideration of a fractal fractional mathematical model on the transmission and control of corona virus (COVID-19), in which the total population of an infected area is divided into susceptible, infected and recovered classes. We consider a fractal-fractional order SIR type model for investigation of Covid-19. To realize the transmission and control of corona virus in a much better way, first we study the stability of the corresponding deterministic model using next generation matrix along with basic reproduction number. After this, we study the qualitative analysis using "fixed point theory" approach. Next, we use fractional Adams-Bashforth approach for investigation of approximate solution to the considered model. At the end numerical simulation are been given by matlab to provide the validity of mathematical system having the arbitrary order and fractal dimension.

Study of transmission dynamics of COVID-19 mathematical model under ABC fractional order derivative.

The current research work is devoted to address some results related to the existence and stability as well as numerical finding of a novel Coronavirus disease (COVID-19) by using a mathematical model. By using fixed point results we establish existence results for the proposed model under Atangana-Baleanu-Caputo (ABC) derivative with fractional order. Further, using the famous numerical technique due to Adams Bashforth, we simulate the concerned results for two famous cities of China known as Wuhan and Huanggang which are interconnected cities. The graphical presentations are given to observe the transmission dynamics from February 1 a=2020 to April 20, 2020 through various fractional order. The concerned dynamics is global in nature due to the various values of fractional order.

Event-based fractional order control.

The present study provides a generalization of event-based control to the field of fractional calculus, combining the benefits brought by the two approaches into an industrial-suitable control strategy. During recent years, control applications based on fractional order differintegral operators have gained more popularity due to their proven superior performance when compared to classical, integer order, control strategies. However, the current industrial setting is not yet prepared to fully adapt to complex fractional order control implementations that require hefty computational resources; needing highly-efficient methods with minimum control effort. The solution to this particular problem lies in combining benefits of event-based control such as resource optimization and bandwidth allocation with the superior performance of fractional order control. Theoretical and implementation aspects are developed in order to provide a generalization of event-based control into the fractional calculus field. Different numerical examples validate the proposed methodology, providing a useful tool, especially for industrial applications where the event-based control is most needed. Several event-based fractional order implementation possibilities are explored, the final result being an event-based fractional order control methodology.

Forecasting of COVID-19 pandemic: From integer derivatives to fractional derivatives.

In this work, a new compartmental mathematical model of COVID-19 pandemic has been proposed incorporating imperfect quarantine and disrespectful behavior of citizens towards lockdown policies, which are evident in most of the developing countries. An integer derivative model has been proposed initially and then the formula for calculating basic reproductive number, R 0 of the model has been presented. Cameroon has been considered as a representative for the developing countries and the epidemic threshold, R 0 has been estimated to be  ~ 3.41 ( 95 % CI : 2.2 - 4.4 ) as of July 9, 2020. Using real data compiled by the Cameroonian government, model calibration has been performed through an optimization algorithm based on renowned trust-region-reflective (TRR) algorithm. Based on our projection results, the probable peak date is estimated to be on August 1, 2020 with approximately 1073 ( 95 % CI : 714 - 1654 ) daily confirmed cases. The tally of cumulative infected cases could reach  ~ 20, 100 ( 95 % CI : 17 , 343 - 24 , 584 ) cases by the end of August 2020. Later, global sensitivity analysis has been applied to quantify the most dominating model mechanisms that significantly affect the progression dynamics of COVID-19. Importantly, Caputo derivative concept has been performed to formulate a fractional model to gain a deeper insight into the probable peak dates and sizes in Cameroon. By showing the existence and uniqueness of solutions, a numerical scheme has been constructed using the Adams-Bashforth-Moulton method. Numerical simulations have enlightened the fact that if the fractional order α is close to unity, then the solutions will converge to the integer model solutions, and the decrease of the fractional-order parameter (0  <  α  <  1) leads to the delaying of the epidemic peaks.