Delay differential equations have numerous applications in science and engineering. They arise from the mathematical modeling of time-dependent processes where the evolution of the system is not only determined by the present state of the process, but it also depends on certain past states. These equations are different from ordinary differential equations, as the derivative of the unknown function at any time is given by the values of the function at present and prior times. In most applications of delay equations, the delayed feedback function is given explicitly. In this Ph.D.~dissertation we propose various models from population dynamics and epidemiology, where the delay terms in the model equations arise as the solution of another dynamical system. The general form of initial value problems for nonautonomous functional differential equations with such dynamically defined delayed feedback function will also be considered in this work. We obtain the usual existence, uniqueness and continuous dependence result for the solution, and show some other, biologically relevant properties. The results derived for the general framework enable us to analyze the model equations coming from biological applications, and in particular, they also provide a powerful tool to investigate some questions of major public health concern, such as the spatial spread of infectious diseases.
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