Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study

Key risk factors associated with fractal dimension based geographical clustering of COVID-19 data in the Flemish and Brussels region, Belgium

Introduction

COVID-19 remains a major concern globally. Therefore, it is important to evaluate COVID-19's rapidly changing trends. The fractal dimension has been proposed as a viable method to characterize COVID-19 curves since epidemic data is often subject to considerable heterogeneity. In this study, we aim to investigate the association between various socio-demographic factors and the complexity of the COVID-19 curve as quantified through its fractal dimension.

Methods

We collected population indicators data (ethnic composition, socioeconomic status, number of inhabitants, population density, the older adult population proportion, vaccination rate, satisfaction, and trust in the government) at the level of the statistical sector in Belgium. We compared these data with fractal dimension indicators of COVID-19 incidence between 1 January - 31 December 2021 using canonical correlation analysis.

Results

Our results showed that these population indicators have a significant association with COVID-19 incidences, with the highest explanatory and predictive power coming from the number of inhabitants, population density, and ethnic composition.

Conclusion

It is important to monitor these population indicators during a pandemic, especially when dealing with targeted interventions for a specific population.

Laplace Residual Power Series Method for Solving Three-Dimensional Fractional Helmholtz Equations

In the present study, the exact solutions of the fractional three-dimensional (3D) Helmholtz equation (FHE) are obtained using the Laplace residual power series method (LRPSM). The fractional derivative is calculated using the Caputo operator. First, we introduce a novel method that combines the Laplace transform tool and the residual power series approach. We specifically give the specifics of how to apply the suggested approach to solve time-fractional nonlinear equations. Second, we use the FHE to evaluate the method’s efficacy and validity. Using 2D and 3D plots of the solutions, the derived and precise solutions are compared, confirming the suggested method’s improved accuracy. The results for nonfractional approximate and accurate solutions, as well as fractional approximation solutions for various fractional orders, are indicated in the tables. The relationship between the derived solutions and the actual solutions to each problem is examined, showing that the solution converges to the actual solution as the number of terms in the series solution of the problems increases. Two examples are shown to demonstrate the effectiveness of the suggested approach in solving various categories of fractional partial differential equations. It is evident from the estimated values that the procedure is precise and simple and that it can therefore be further extended to linear and nonlinear issues.

Remarks on Fractal-Fractional Malkus Waterwheel Model with Computational Analysis

In this paper, we investigate the fractal-fractional Malkus Waterwheel model in detail. We discuss the existence and uniqueness of a solution of the fractal-fractional model using the fixed point technique. We apply a very effective method to obtain the solutions of the model. We prove with numerical simulations the accuracy of the proposed method. We put in evidence the effects of the fractional order and the fractal dimension for a symmetric Malkus Waterwheel model.

Caputo Fractal Fractional Order Derivative of Soil Pollution Model Due to Industrial and Agrochemical

This paper narrates a non-linear and non-local Caputo fractal fractional operator of eco epidemic model with the advance of soil pollution considered in five compartments. The qualitative analysis of solutions such as existence and uniqueness of the model is carried out by using the standard condition of Schauder's fixed point theorem and Banach Contraction principle. The local and global stability are characterized with the help of basic reproduction number. The Ulam-Hyer stability is analyzed for the small perturbation. The Power law kernel is used to get a reliable result for the soil pollution model. Analytical solution studied by means of Modified Euler method. Numerical simulation of Euler scheme algorithm is performed to show the effects of various fractional orders ( 0.5 < η < 1 ) and validating the theoretical parameter values of real time data by the support of MATLAB.

A New Numerical Approach for the Analysis of Variable Fractal and Fractional Order Differential Equations

The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams-Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.

A (2+1)-Dimensional Fractional-Order Epidemic Model with Pulse Jumps for Omicron COVID-19 Transmission and Its Numerical Simulation

In this paper, we would like to propose a (2+1)-dimensional fractional-order epidemic model with pulse jumps to describe the spread of the Omicron variant of COVID-19. The problem of identifying the involved parameters in the proposed model is reduced to a minimization problem of a quadratic objective function, based on the reported data. Moreover, we perform numerical simulation to study the effect of the parameters in diverse fractional-order cases. The number of undiscovered cases can be calculated precisely to assess the severity of the outbreak. The results by numerical simulation show that the degree of accuracy is higher than the classical epidemic models. The regular testing protocol is very important to find the undiscovered cases in the beginning of the outbreak.