Fractional Stochastic Differential Equations - Applications to Covid-19 Modeling

Description

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. 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