Dear Colleagues,

Delay differential equations (DDEs) are a type of differential equation in which the derivative of a function depends not only on its current value, but also on its past values. In other words, the rate of change of the function at a given time depends on its values at previous times. This introduces a time delay into the system, hence the name "delay" differential equations.

DDEs find applications in various fields, including biology, physics, chemistry, economics, and engineering. They are particularly useful for modeling systems with memory effects or systems in which time delays play a crucial role. Some examples include population dynamics, with time delays in birth or death rates, chemical reactions with delayed effects, and control systems with communication delays.

Solving DDEs analytically can be challenging due to the presence of time delays. Numerical methods, such as the method of steps, collocation methods, and numerical continuation, are commonly employed to approximate solutions to DDEs. These methods discretize the time domain and approximate the delayed terms using appropriate interpolation techniques. 

The primary objective of this Special Issue is twofold: first, to provide a comprehensive review of the current state-of-the-art knowledge in Delay Differential Equations (DDEs) and their applications in engineering and biological systems. The second aim is to delve into the emerging challenges and open problems in mathematics that have arisen as a result of these novel and intricate models. These DDE models, spanning various classes, have been developed with the aim of unraveling the complexities of diverse phenomena, and their analysis has led to intriguing mathematical questions that require further investigation.

Keywords: bifurcation; fractional-order; epidemic models; Lyapunov functionals; predator–prey model; sensitivity; stability; stationary distribution; stochastic perturbations; time delays

In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following:

• Qualitative behaviors of DDEs;
• Fractional-order and Stochastic DDEs and applications;
• Optimal control in biological systems, medicine/spread of disease;
• Analysis and numerical implementation of models described by ODEs, DDEs, and PDEs systems;
• Numerical schemes for DDEs and stochastic DDEs;
• Dynamics, including stability, bifurcation, and chaos of DDEs.

I look forward to receiving your contributions.

Dr. Hebatallah J. Alsakaji
Prof. Dr. Snezhana Hristova
Guest Editors

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• bifurcation
• fractional-order
• epidemic models
• Lyapunov functionals
• predator–prey model
• sensitivity
• stability
• stationary distribution
• stochastic perturbations
• time delays