  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.
