The field theoretical ABC of epidemic dynamics.

We go beyond a systematic review of several main mathematical models employed to describe the diffusion of infectious diseases and demonstrate how the different approaches are related. It is shown that the frameworks exhibit common features such as criticality and self-similarity under time rescaling. These features are naturally encoded within the unifying field theoretical approach. The latter leads to an efficient description of the time evolution of the disease via a framework in which (near) time-dilation invariance is explicitly realised. When needed, the models are extended to account for observed phenomena such as multi-wave dynamics. Although we consider the COVID-19 pandemic as an explicit phenomenological application, the models presented here are of immediate relevance for different realms of scientific enquiry from medical applications to the understanding of human behaviour.