This book discusses the numerical treatment of delay differential equations and their applications in bioscience. A wide range of delay differential equations are discussed with integer and fractional-order derivatives to demonstrate their richer mathematical framework compared to differential equations without memory for the analysis of dynamical systems. The book also provides interesting applications of delay differential equations in infectious diseases, including COVID-19. It will be valuable to mathematicians and specialists associated with mathematical biology, mathematical modelling, life sciences, immunology and infectious diseases.
Author(s): Fathalla A. Rihan
Series: Forum for Interdisciplinary Mathematics
Edition: 1
Publisher: Springer Nature Singapore Pte Ltd
Year: 2021
Language: English
Pages: 286
City: Singapore
Tags: Delay Differential Equations, Stochastic Delay Differential Equations, Fractional-Order Delay Differential Equations, Infectious Diseases, Predator-Prey Systems, COVID-19
Preface
Acknowledgments
Contents
About the Author
Part I Qualitative and Quantitative Features of Delay Differential Equations
1 Qualitative Features of Delay Differential Equations
1.1 Introduction
1.2 Delay Models in Population Dynamics
1.2.1 Logistic Equation with Discrete Delay
1.2.2 Logistic Equation with Distributed Delay
1.2.3 Delayed Lotka-Volterra System
1.3 Stability of DDEs
1.3.1 Stability of Linear Constant Coefficient DDEs
1.3.2 Asymptotical Stability Region for Linear DDEs
1.3.3 Stability of Linear NDDEs
1.3.4 Asymptotic Stability Region for Linear NDDEs
1.4 Stability of Non-linear DDEs and Contractivity Conditions
1.5 Stability of DDEs in Lyapunov Method
1.5.1 Lyapunov-Krasovskii Sense
1.5.2 Lyapunov-Razumikhin Sense
1.5.3 Stability of Linear Systems with Discrete Delays
1.6 Concluding Remarks
References
2 Numerical Solutions of Delay Differential Equations
2.1 Propagation and Location of Discontinuities in DDEs
2.2 Method of Steps for DDEs
2.3 Existence and Uniqueness Solution of DDEs
2.4 Numerical Approach for DDEs
2.4.1 General Approach
2.4.2 Θ-Methods for DDEs
2.4.3 Continuous One-Step Runge-Kutta Methods for ODE
2.4.4 Runge-Kutta Method for DDEs
2.5 More General Classes of DDEs
2.5.1 Neutral Delay Differential Equations (NDDEs)
2.5.2 Equations with State-Dependent Lags
2.5.3 Equation with a Small or Vanishing Lag
2.6 Stiffness Problems
2.7 Software Aspects
2.7.1 Discretization Error
2.7.2 Location of Jump Discontinuities
2.7.3 Stepsize Control
2.7.4 Interpolation to widetildey(t)
2.7.5 DDE Solvers and Available Software
2.8 Concluding Remarks
References
3 Stability Concepts of Numerical Solutions of Delay Differential Equations
3.1 Introduction
3.2 Stability of Numerical Methods for DDEs
3.2.1 Stability Regions for DDEs: P-stability and GP-stability
3.2.2 Stability Regions for Linear NDDEs
3.3 Contractivity Concepts and GPN-Stability
3.3.1 Contractivity Concepts and GRN-Stability
3.4 Concluding Remarks
References
4 Numerical Solutions of Volterra Delay Integro-Differential Equations
4.1 Introduction
4.2 Analytical Stability
4.3 Continuous Mono-Implicit RK Scheme for DDEs
4.4 Numerical Treatment of VDIDEs
4.4.1 CMIRK Scheme for VDIDEs
4.4.2 Numerical Integration Formula and Boole's Quadrature Rule
4.4.3 MIDDE Software Aspects
4.5 Numerical Stability
4.6 Numerical Results and Simulations
4.7 Concluding Remarks
References
5 Parameter Estimation with Delay Differential Equations
5.1 Introduction
5.2 Parameter Estimation with DDEs
5.2.1 Non-linearity of Model Predictions
5.3 Computation of Estimates
5.3.1 Basic Iteration
5.3.2 Acceptability
5.3.3 Convergence
5.4 Discontinuities Associated with Delay
5.5 Solving the Minimization Problem
5.6 Models and Goodness of Fit for Cell Growth
5.6.1 Problem 1: Fitting DDEs with Growth of Fission Yeast
5.6.2 Fitting DDEs with Growth of Tetrahymena Pyriformis
5.7 Concluding Remarks
References
6 Sensitivity Analysis of Delay Differential Equations
6.1 Introduction
6.2 Sensitivity Functions
6.2.1 Adjoint Equations
6.3 Variational Approach
6.4 Direct Approach
6.5 Sensitivity of Optimum Parameter p"0362p to Data
6.5.1 Standard Deviation of Parameter Estimates
6.5.2 Non-linearity and Indications of Bias
6.6 Numerical Results
6.7 Concluding Remarks
References
7 Stochastic Delay Differential Equations
7.1 Introduction
7.1.1 Preliminaries
7.2 Existence and Uniqueness of Solutions for SDDEs
7.3 Stability Criteria for SDDEs
7.4 Numerical Scheme for Autonomous SDDEs
7.4.1 Convergence and Consistency
7.5 Numerical Schemes for Non-autonomous SDDE
7.5.1 Taylor Approximation
7.5.2 Implicit Strong Approximations
7.6 Milstein Scheme for SDDEs
7.6.1 Convergence and Mean-Square Stability of the Milstein Scheme
7.7 Concluding Remarks
References
Part II Applications of Delay Differential Equations in Biosciences
8 Delay Differential Equations with Infectious Diseases
8.1 Introduction
8.2 Time-Delay in Epidemic Models
8.2.1 Development of SIR Model (8.1)
8.3 Delay Differential Models with Viral Infection
8.3.1 DDEs with HIV Infection of CD4+ T-cells
8.3.2 Steady States
8.3.3 Stability Analysis of Infected Steady State
8.3.4 Existence of Hopf Bifurcation
8.4 Physiology
8.5 Concluding Remarks
References
9 Delay Differential Equations of Tumor-Immune System with Treatment and Control
9.1 Introduction
9.2 Description of the Model
9.2.1 Non-negativity and Boundedness Solutions of Model (9.3)
9.2.2 Model with Chemotherapy
9.3 Drug-Free Steady States and Their Stability
9.3.1 Stability of Tumor-Free Steady State
9.3.2 Stability of Lethal Steady States
9.3.3 Stability of Coexisting Steady States
9.4 Optimal Control Problem Governed by DDEs
9.5 Existence of Optimal Solution
9.6 Optimality Conditions
9.7 Immuno-Chemotherapy
9.8 Numerical Simulations of Optimal Control System
9.9 Concluding Remarks
References
10 Delay Differential Equations of Ecological Systems with Allee Effect
10.1 Introduction
10.2 Delay Differential Model of Two-Prey One-Predator System
10.2.1 Positivity and Boundedness of the Solution
10.3 Local Stability and Hopf Bifurcation
10.3.1 Existence of Interior Equilibrium Points
10.3.2 Stability and Bifurcation Analysis of the Interior Equilibrium
10.4 Global Stability of Interior Steady State calE*
10.5 Sensitivity to Allee Effect
10.6 Numerical Simulations
10.7 Concluding Remarks
References
11 Fractional-Order Delay Differential Equations with Predator-Prey Systems
11.1 Introduction
11.1.1 Preliminaries
11.2 Fractional Delayed Predator-Prey Model
11.3 Local Stability Analysis and Hopf Bifurcation
11.3.1 Trivial and Semi-trivial Equilibria and Their Stabilities
11.3.2 Interior Equilibrium and Its Stability
11.4 Global Stability Analysis
11.5 Implicit Euler's Scheme for FODDEs
11.5.1 Stability and Convergence of Implicit Scheme for FODDEs
11.5.2 Numerical Simulations
11.6 Concluding Remarks
References
12 Fractional-Order Delay Differential Equations of Hepatitis C Virus
12.1 Introduction
12.2 Mathematical Model of HCV
12.3 Local Stability of Infection-Free and Infected Steady States
12.4 Global Stability of Infection-Free Steady State calE0
12.5 Numerical Simulations and Validity of Model
12.5.1 Parameter Estimation and Validity of Model
12.6 Concluding Remarks
References
13 Stochastic Delay Differential Model for Coronavirus Infection COVID-19
13.1 Introduction
13.2 Stochastic SIRC Epidemic Model
13.3 Existence and Uniqueness of Positive Solution
13.4 Existence of Ergodic Stationary Distribution
13.5 Extinction
13.6 Numerical Simulations and Discussions
13.7 Concluding Remarks
References
14 Remarks and Current Challenges
Appendix A Fifth-Order Dormand and Prince RK Method
Appendix B Adams-Bashforth-Moulton Method for Fractional-Order Delay Differential Equations
Appendix C Matlab Program for Stochastic Delay Differential Equations Using Milstein Scheme
References
