**Preface to "Models of Delay Differential Equations"**

Models of differential equations with delay have pervaded many scientific and technical fields in the last decades. The use of delay differential equations and partial delay differential equations to model problems with the presence of lags or hereditary effects have demonstrated a valuable balance between realism and tractability. Of special interest in recent years is the development and analysis of models with interactions between delay and random effects, through the use of stochastic and random delay differential equations.

In this Special Issue we gather quite a balanced picture of mainstreams topics in the realm of delay differential equations. Indeed, we can find contributions dealing with the construction of exact solutions, numerical methods, dynamical properties, and applications to mathematical modeling of phenomena and processes in biology, economics and engineering, in both deterministic and stochastic settings.

In the paper by Arenas et al. a mathematical model is proposed, based on a set of delay differential equations, that describes intracellular HIV infection. The model considers the time delay between viral entry into a target cell and the production of new virions. The study includes local stability analysis and the design of a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model. In his paper, Kashchenko studies the nonlocal dynamics of a system of delay differential equations with large parameters using the method of steps. This system simulates coupled generators with delayed feedback. The study shows that the dynamics of the system are significantly different in the case of positive coupling and in the case of negative coupling. In the paper by Debbouche and Fedorov, local unique solvability for a class of degenerate fractional differential equations and its application to study initial-boundary value problems for systems of equations with delays is proved. Hendy and Staelen introduce a high order numerical approximation method for convection diffusion wave equations with a multiterm time fractional Caputo operator and a nonlinear fixed time delay. In the paper by Matsumoto and Szidarovszky, the dynamic behavior of n-firm oligopolies is studied, assuming the companies are able to face both implementation and information delays. The analysis includes a classification of stability scenarios depending on the relationship between delays. Continuing within the economical setting, the paper by Abia et al. presents a new numerical method to obtain the solution to a size-structured population model that describes the evolution of a consumer feeding on a dynamical resource that reacts to the environment with a lag-time response. The model is formulated by combining a partial and an ordinary differential equation with delay. Two papers address the numerical and theoretical analysis of linear random delay differential equations. The first one, by Calatayud et al., proposes a mean square convergent non-standard numerical scheme while the second one, by Cortes and Jornet, ´ constructs, rigorously, a solution in the important case that the source term is a stochastic process. In the realm of applications, Majchrzak and Mochnacki, propose a second-order dual phase lag equation to model phase changes associated with heating and cooling of thin metal films. In the paper by Gomez-Valle et al. a partial differential equation for pricing an Asian-style option, termed a freight ´ option, derived from a stochastic delay differential equation is established. This partial differential equation permits attainment of lower and upper bounds for the prime of this type of derivative. The theoretical findings are nicely illustrated using real data from the Baltic Exchange. Zhe Ying et al. study the stability of an age-structured susceptible–exposed—infective–recovered–susceptible (SEIRS) model with time delay. After obtaining one disease-free equilibrium point and one endemic equilibrium point of the model, they establish sufficient conditions in order for the local stability to be guaranteed. In the paper by A. Ashyralyev and D. Agirseven, the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator from a theoretical standpoint is studied. This analysis is complemented with some numerical experiments in the case of semilinear hyperbolic equations with unbounded time delay term, since, in general, it is not possible to obtain the exact solution. The volume finishes with the study of exact and nonstandard finite difference schemes for a class of coupled linear delay differential systems. The study includes the analysis of consistency properties of the new nonstandard schemes and several illustrative examples.
