Applications of delay differential equations to the physics of complex systems

In the present Thesis we introduce Delay Differential Equations as an approach to model emergent states arising from the interaction of different time scales in Complex Systems. We devote our attention to models related to neuronal phenomena and epidemiological forecasting. Our work on neural models consists in demonstrating how the interplay of neural spiking and refractoriness time scales, and cyclic structures in a directed interaction network can give rise to self-sustained traveling waves. We also build a description of the bifurcation phenomenon and the dynamical steady state in terms of a single Delay Differential Equation, the solution of which can be interpolated to obtain single node trajectories. We subsequently formulate a normal form model to explain how the interplay of a delayed feedback and the time scale of the system can stabilize an orbit near privileged locations in phase space. Due to the interest in neural computation and information processing we also construct a discrete stochastic model that preserves the activation statistics of a noisy nonlinear neuron, focusing on the different time scales involved in the process. The results on epidemiological modeling are concerned with the usage of Distributed Delay Differential Equations in the forecast of epidemic events on the short and medium term on a metropolitan scale in Bologna. We show that these models can cover for the shortcomings of models based on Ordinary Differential Equations at the considered scales, despite the more complicated nature of Delayed Equations. In particular we show that road traffic data can be used as a proxy for the rate of contacts in a population, during periods when other factors are approximately unchanging. We also set a basis for quantitative predictivity analysis, by obtaining linear response laws for the model and in particular for the variables that could be used in regression tasks.