Delay Differential Equations and Applications to Biology

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 new edition introduces advanced numerical techniques for solving pantograph-type DDEs, including stability analyses. The book also integrates recent research on stochastic models of DDEs, particularly from the fields of immunology and cancer modeling. In addition, the book includes updated case studies from biological and medical sciences, such as glucose-insulin interaction models and infectious disease dynamics. This book is particularly valuable to mathematicians, researchers, and professionals in the fields of mathematical biology, life sciences, immunology, and epidemiology.

Discusses topics in delay differential equations, including theory, numerical methods, stability, control, and biology Combines both qualitative and quantitative features of delay differential equations with real-life problems Provides interesting applications in infectious diseases, including COVID-19, and glucose-insulin regulation

Autorentext

Fathalla A. Rihan is a Professor of Mathematics at both the United Arab Emirates University, UAE, and Helwan University, Egypt. He earned his Doctor of Science (DSc) in 2018 from the National University of Uzbekistan for his work on "Features of Delay Differential Equations and Their Applications", and received his Ph.D. in 2000 from the University of Manchester, UK, with a thesis on "Numerical Treatment of Delay Differential Equations in Biosciences". His research interests span numerical analysis, mathematical biology, and the modeling of real-life phenomena involving memory effects, such as cell division, population dynamics, and infectious diseases. He is also engaged in parameter estimation, sensitivity analysis, and both qualitative and quantitative studies of delay differential equations. Professor Rihan has published extensively in high-impact journals and presented his work at over 100 international conferences. He serves on the editorial boards of several international journals, acts as a peer reviewer, and is a member of the American Mathematical Society. He has supervised numerous Ph.D. students and led multiple research projects as principal investigator. With over 30 years of teaching experience at both undergraduate and postgraduate levels, Professor Rihan is also recognized for his expertise in higher education quality assurance and accreditation. He has been consistently ranked among the world’s top 2% of researchers by Stanford University (2019–2024). In 2025, Prof Rihan was awarded the Khalifa Award for Distinguished University Professor (Cycle 18) in recognition of his exceptional contributions to research and education.

Inhalt

1. Qualitative Features of Delay Differential Equations.- 2. Continuous Runge-Kutta Methods for Delay Differential Equations.- 3. Stability Concepts of Numerical Solutions of Delay Differential Equations.- 4. Mono-Implicit Runge-Kutta Methods for Volterra Delay Integro-Differential Equations.- 5. Parameter Estimation with Delay Differential Equations.- 6. Sensitivity Analysis of Delay Differential Equations.- 7. Stochastic Delay Differential Equations.- 8. Continuous Runge-Kutta Schemes for Pantograph Delay Differential Equations.- 9. Split-Step -Method for Stochastic Pantograph Differential Equations.- 10. Delay Differential Equations with Infectious Diseases.- 11 Delay Differential Equations of Tumor-Immune System with Treatment and Control.- 12 Delay Differential Equations of Ecological Systems with Allee Effect.- 13 Fractional-Order Delay Differential Equations with Predator-prey Systems.- 14 Fractional-Order Delay Differential Equations of Hepatitis C Virus.- 15 Stochastic Delay Differential Model for Coronavirus Infection COVID-19.- 16 Delay Differential Equations with Glucose-Insulin Dynamics.- 17 Remarks and Current Challenges.