This book provides a thorough conversation on the underpinnings of Covid-19 spread modelling by using stochastics nonlocal differential and integral operators with singular and non-singular kernels. The book presents the dynamic of Covid-19 spread behaviour worldwide. It is noticed that the spread dynamic followed process with nonlocal behaviours which resemble power law, fading memory, crossover and stochastic behaviours. Fractional stochastic differential equations are therefore used to model spread behaviours in different parts of the worlds. The content coverage includes brief history of Covid-19 spread worldwide from December 2019 to September 2021, followed by statistical analysis of collected data for infected, death and recovery classes.
Author(s): Abdon Atangana, Seda İgret Araz
Series: Industrial and Applied Mathematics
Edition: 1
Publisher: Springer Nature Singapore
Year: 2022
Language: English
Pages: 540
City: Singapore
Tags: fractional calculus, stochastic differential equation, Covid-19 ,fractional integration, fractional differentiation, numerical methods
Preface
Contents
About the Authors
1 History on Covid-19 Spread
1.1 Introduction
1.2 Statistical Representation of WHO Data of Infected, Recoveries …
1.3 Motivation: Modeling with Nonlocal Operators
References
2 Fractional Differential and Integral Operators
2.1 Existence of Fractional Differential Operator
2.2 Existence of Integrals
2.3 Lipschitz Condition of Fractional Integrals
2.4 Numerical Approximation of the Fractional Derivatives
References
3 Existence and Uniqueness for Stochastic Differential Equations
3.1 Stochastic Differential Equations with Global Derivative
3.2 Stochastic Differential Equations with the Riemann–Liouville Derivative
3.3 Stochastic Differential Equations with the Caputo–Fabrizio Fractional Derivative
3.4 Stochastic Differential Equations with the Atangana–Baleanu Fractional Derivative
References
4 Numerical Scheme for a General Stochastic Equation with Classical and Fractional Derivatives
4.1 Numerical Scheme with Newton Polynomial Interpolation …
4.2 Predictor–Corrector Scheme with Newton Polynomial for a General …
4.3 Error Analysis for a General Stochastic Equation with Global Derivative
4.3.1 Error Analysis with Caputo Case
4.3.2 Error Analysis with the Generalized Mittag-Leffler Kernel
4.3.3 Error Analysis with the Exponential Kernel
References
5 A Simple SIR Model of Covid-19 Spread
5.1 Positivity and Boundness of the Solutions
5.2 Local and Global Stability of the Disease-Free Equilibrium
5.3 Local and Global Stability of the Endemic Equilibrium
5.4 Positive Solutions with Nonlocal Operators
5.5 Optimal Control for Covid-19 Model
5.6 Applications of Covid-19 Stochastic Models from Classical to Nonlocal Operators
5.6.1 Existence and Uniqueness of the SIR Stochastic Model
5.6.2 Existence of a Unique Global Positive System of Solution
5.6.3 Extinction of Infection
5.7 Numerical Scheme for SIR Stochastic Model
5.7.1 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
5.7.2 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
5.7.3 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
5.8 Numerical Simulation
5.9 Comparison Between the Suggested Model and Experimental Data
References
6 An Application of SEIRD Approach
6.1 Positivity and Boundness of the Solutions
6.2 Local and Global Stability of the Disease-Free Equilibrium
6.3 Local and Global Stability of the Endemic Equilibrium
6.4 Positive Solutions with Nonlocal Operators
6.5 Optimal Control for Covid-19 Model
6.6 Applications of Covid-19 Stochastic Models from Classical to Nonlocal Operators
6.6.1 Existence and Uniqueness of the SEIRD Stochastic Model
6.6.2 Existence of a Unique Global Positive System of Solution
6.6.3 Extinction of Infection
6.7 Numerical Scheme for SEIRD Stochastic Model
6.7.1 Numerical Solution of the Model with Caputo–Fabrizio Fractional Derivative
6.7.2 Numerical Solution of the Model with Atangana–Baleanu Fractional Derivative
6.7.3 Numerical Solution of the Model with Caputo Fractional Derivative
6.7.4 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
6.7.5 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
6.7.6 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
6.8 Numerical Simulation
6.9 Comparison Between the Suggested Model and Experimental Data
References
7 Modeling the Transmission of Coronavirus with SEIR Approach
7.1 Positivity and Boundness of the Solutions
7.2 Local and Global Stability of the Disease-Free Equilibrium
7.3 Local and Global Stability of the Endemic Equilibrium
7.4 Positive Solutions with Nonlocal Operators
7.5 Optimal Control for Covid-19 Model
7.6 Applications of Covid-19 Models from Classical to Nonlocal Operators
7.6.1 Existence and Uniqueness of the SEIR Stochastic Model
7.6.2 Existence of a Unique Global Positive System of Solution
7.6.3 Extinction of Infection
7.7 Numerical Scheme for SEIR Stochastic Model
7.7.1 Numerical Solution of the Model with the Caputo–Fabrizio Fractal-Fractional Derivative
7.7.2 Numerical Solution of the Model with the Caputo Fractal-Fractional Derivative
7.7.3 Numerical Solution of the Model with the Atangana–Baleanu Fractal-Fractional Derivative
7.8 Numerical Simulation
7.9 Comparison Between the Suggested Model and Experimental Data
References
8 Modeling the Spread of Covid-19 with a SIA IR IU Approach: Inclusion of Unreported Infected Class
8.1 Local and Global Stability of the Disease-Free Equilibrium
8.2 Local and Global Stability of the Endemic Equilibrium
8.3 Positive Solutions for with Nonlocal Operators
8.4 Optimal Control for Model
8.5 Applications of Covid-19 Stochastic Model from Classical to Nonlocal Operators
8.5.1 Existence and Uniqueness of the Stochastic Model of Covid-19 Spread
8.5.2 Existence of a Unique Global Positive System of Solution
8.6 Numerical Scheme for the Stochastic Model of Covid-19 Spread
8.6.1 Numerical Solution of the Model with Caputo–Fabrizio Fractional Derivative
8.6.2 Numerical Solution of the Model with Atangana–Baleanu Fractional Derivative
8.6.3 Numerical Solution of the Model with Caputo Fractional Derivative
8.6.4 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
8.6.5 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
8.6.6 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
8.7 Numerical Simulation
8.8 Comparison Between the Suggested Model and Experimental Data
References
9 A Comprehensive Analysis of the Covid-19 Model
9.1 Positivity and Boundness of the Solutions
9.2 Local and Global Stability of the Disease-Free Equilibrium
9.3 Local and Global Stability of the Endemic Equilibrium
9.4 Positive Solutions with Nonlocal Operators
9.5 Optimal Control for the Covid-19 Model
9.6 Applications of Covid-19 Models from Classical to Nonlocal Operators
9.6.1 Existence and Uniqueness of the Stochastic Covid-19 Model
9.6.2 Existence of a Unique Global Positive System of Solution
9.6.3 Extinction of Infection
9.7 Numerical Scheme for the Stochastic Covid-19 Model
9.7.1 Numerical Solution of the Model with the Caputo–Fabrizio Fractional Derivative
9.7.2 Numerical Solution of the Model with the Atangana–Baleanu Fractional Derivative
9.7.3 Numerical Solution of the Model with the Caputo Fractional Derivative
9.7.4 Numerical Solution of the Model with the Caputo–Fabrizio Fractal-Fractional Derivative
9.7.5 Numerical Solution of the Model with the Atangana–Baleanu Fractal-Fractional Derivative
9.7.6 Numerical Solution of the Model with the Caputo Fractal-Fractional Derivative
9.8 Numerical Simulation
9.9 Comparison Between the Suggested Model and Experimental Data
References
10 Analysis of SEIARD Model of Coronavirus Transmission
10.1 Positivity and Boundness of the Solutions
10.2 Local and Global Stability of the Disease-Free Equilibrium
10.3 Local and Global Stability of the Endemic Equilibrium
10.4 Positive Solutions with Non-local Operators
10.5 Optimal Control for Covid-19 Model
10.6 Applications of Covid-19 Stochastic Model from Classical to Nonlocal Operators
10.6.1 Existence and Uniqueness of the SEIARD Stochastic Model
10.6.2 Existence of a Unique Global Positive System of Solution
10.6.3 Extinction of Species
10.7 Numerical Scheme for SEIARD Model
10.7.1 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
10.7.2 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
10.7.3 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
10.8 Numerical Simulation
10.9 Comparison Between the Suggested Model and Experimental Data
References
11 A Mathematical Model with Covid-19 Reservoir
11.1 Positivity and Boundness of the Solutions
11.2 Local and Global Stability of the Disease-Free Equilibrium
11.3 Local and Global Stability of the Endemic Equilibrium
11.4 Positive Solutions with Nonlocal Operators
11.5 Optimal Control for Covid-19 Model
11.6 Applications of Covid-19 Stochastic Model from Classical to Nonlocal Operators
11.6.1 Existence and Uniqueness of the Stochastic SEIARD Model
11.6.2 Existence of a Unique Global Positive System of Solution
11.6.3 Extinction of Infection
11.7 Numerical Scheme for Stochastic SEIARD Model
11.7.1 Numerical Solution of the Model with Caputo–Fabrizio Fractional Derivative
11.7.2 Numerical Solution of the Model with Atangana–Baleanu Fractional Derivative
11.7.3 Numerical Solution of the Model with Caputo Fractional Derivative
11.7.4 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
11.7.5 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
11.7.6 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
11.8 Numerical Simulation
11.9 Comparison Between the Suggested Model and Experimental Data
References
12 A New Model with Asymptomatic and Quarantined Classes
12.1 Positivity and Boundness of the Solutions
12.2 Local and Global Stability of the Disease-Free Equilibrium
12.3 Local and Global Stability of the Endemic Equilibrium
12.4 Positive Solutions with Nonlocal Operators
12.5 Optimal Control for Covid-19 Model
12.6 Applications to Covid-19 Stochastic Model from Classical to Nonlocal Operators
12.6.1 Existence and Uniqueness of the SEIARD Stochastic Model
12.6.2 Existence of a Unique Global Positive System of Solution
12.6.3 Extinction of Infection
12.7 Numerical Scheme for SEIQAR Stochastic Model
12.7.1 Numerical Solution of the Model with Caputo–Fabrizio Fractional Derivative
12.7.2 Numerical Solution of the Model with Atangana–Baleanu Fractional Derivative
12.7.3 Numerical Solution of the Model with Caputo Fractional Derivative
12.7.4 Numerical Solution of the Model with Caputo–Fabrizio Fractal-Fractional Derivative
12.7.5 Numerical Solution of the Model with Atangana–Baleanu Fractal-Fractional Derivative
12.7.6 Numerical Solution of the Model with Caputo Fractal-Fractional Derivative
12.8 Numerical Simulation
12.9 Comparison Between the Suggested Model and Experimental Data
References
