Delay Differential Equations Models

Abstract

Time delays are used to model several different mechanisms in the dynamics of epidemics. Incubation periods, maturation times, age structure, seasonal or diurnal variations, interactions across spatial distances or through complicated paths, as well as other mechanisms have been modeled by the introduction of time delays in dynamic models. In fact, all models that employ difference equations, that is, where time or age is discretized, introduce a delay in a fundamental way, since the state of such a system at time t is determined, at least in part, by the state at time t — Δt, where Δt is the discrete time step of the difference equation. This represents a delayed effect in the dynamics which may be justified by the situation that is being modeled. Chapter 3 is devoted to the study of such difference equations models. In this chapter, we consider models which use differential equations with delays in order to account for one or more of the mechanisms we have described above. Even though it is relatively easy to justify the introduction of delays in particular models, the resulting equations often present new mathematical phenomena and difficulties which often require rather complicated tools for their analysis. These mathematical complications are justified by the resulting realism of the models, and by the additional biological phenomena that can be described through the use of time delays.