Mathematical modeling of coronavirus disease COVID-19 dynamics using CF and ABC non-singular fractional derivatives.

In this article, Coronavirus Disease COVID-19 transmission dynamics were studied to examine the utility of the SEIR compartmental model, using two non-singular kernel fractional derivative operators. This method was used to evaluate the complete memory effects within the model. The Caputo-Fabrizio (CF) and Atangana-Baleanu models were used predicatively, to demonstrate the possible long-term trajectories of COVID-19. Thus, the expression of the basic reproduction number using the next generating matrix was derived. We also investigated the local stability of the equilibrium points. Additionally, we examined the existence and uniqueness of the solution for both extensions of these models. Comparisons of these two epidemic modeling approaches (i.e. CF and ABC fractional derivative) illustrated that, for non-integer τ value. The ABC approach had a significant effect on the dynamics of the epidemic and provided new perspective for its utilization as a tool to advance research in disease transmission dynamics for critical COVID-19 cases. Concurrently, the CF approach demonstrated promise for use in mild cases. Furthermore, the integer τ value results of both approaches were identical.

A mathematical study on a fractional COVID-19 transmission model within the framework of nonsingular and nonlocal kernel.

In this work, a mathematical model consisting of a compartmentalized coupled nonlinear system of fractional order differential equations describing the transmission dynamics of COVID-19 is studied. The fractional derivative is taken in the Atangana-Baleanu-Caputo sense. The basic dynamic properties of the fractional model such as invariant region, existence of equilibrium points as well as basic reproduction number are briefly discussed. Qualitative results on the existence and uniqueness of solutions via a fixed point argument as well as stability of the model solutions in the sense of Ulam-Hyers are furnished. Furthermore, the model is fitted to the COVID-19 data circulated by Nigeria Centre for Disease Control and the two-step Adams-Bashforth method incorporating the noninteger order parameter is used to obtain an iterative scheme from which numerical results for the model can be generated. Numerical simulations for the proposed model using Adams-Bashforth iterative scheme are presented to describe the behaviors at distinct values of the fractional index parameter for of each of the system state variables. It was shown numerically that the value of fractional index parameter has a significant effect on the transmission behavior of the disease however, the infected population (the exposed, the asymptomatic infectious, the symptomatic infectious) shrinks with time when the basic reproduction number is less than one irrespective of the value of fractional index parameter.

Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia.

Governments and health officials are eager to gain a thorough understanding of the dynamics of COVID-19 transmission in order to devise strategies to mitigate the pandemic's negative effects. As a result, we created a new fractional order mathematical model to investigate the dynamics of Covid-19 vaccine transmission in Ethiopia. The nonlinear system of differential equations for the model is represented using Atangana-Baleanu fractional derivative in Caputo sense and the Jacobi spectral collocation method is used to convert this system into an algebraic system of equations, which is then solved using inexact Newton's method. The fundamental reproduction number, R 0 for the proposed model is determined using the next generation matrix approach.

Analysis of Atangana-Baleanu fractional-order SEAIR epidemic model with optimal control.

We consider a SEAIR epidemic model with Atangana-Baleanu fractional-order derivative. We approximate the solution of the model using the numerical scheme developed by Toufic and Atangana. The numerical simulation corresponding to several fractional orders shows that, as the fractional order reduces from 1, the spread of the endemic grows slower. Optimal control analysis and simulation show that the control strategy designed is operative in reducing the number of cases in different compartments. Moreover, simulating the optimal profile revealed that reducing the fractional-order from 1 leads to the need for quick starting of the application of the designed control strategy at the maximum possible level and maintaining it for the majority of the period of the pandemic.

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 system of delay differential equation with nonsingular kernels in modeling hand-foot-mouth disease.

In this article, we examine a computational model to explore the prevalence of a viral infectious disease, namely hand-foot-mouth disease, which is more common in infants and children. The structure of this model consists of six sub-populations along with two delay parameters. Besides, by taking advantage of the Atangana-Baleanu fractional derivative, the ability of the model to justify different situations for the system has been improved. Discussions about the existence of the solution and its uniqueness are also included in the article. Subsequently, an effective numerical scheme has been employed to obtain several meaningful approximate solutions in various scenarios imposed on the problem. The sensitivity analysis of some existing parameters in the model has also been investigated through several numerical simulations. One of the advantages of the fractional derivative used in the model is the use of the concept of memory in maintaining the substantial properties of the understudied phenomena from the origin of time to the desired time. It seems that the tools used in this model are very powerful and can effectively simulate the expected theoretical conditions in the problem, and can also be recommended in modeling other computational models in infectious diseases.

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains.

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.

Fractional Herglotz variational problems with Atangana-Baleanu fractional derivatives.

The purpose of this paper is to solve fractional calculus of variational Herglotz problem depending on an Atangana-Baleanu fractional derivative. Since the new Atangana-Baleanu fractional derivative is non-singular and non-local, the Euler-Lagrange equations are proposed for the problems of Herglotz. Fractional variational Herglotz problems of variable order are considered and two cases are shown. The Noether-type theorem with this new fractional derivative is proved. Several typical examples of the results of this paper are expressed in this paper.