  Infectious diseases are a threat for human health with tremendous impact on
our society at large. The recent COVID-19 pandemic, caused by the SARS-CoV-2,
is the latest example of a highly infectious disease ravaging the world, since
late 2019. It is therefore imperative to develop efficient mathematical models,
able to substantially curb the damages of a pandemic by unveiling disease
spreading dynamics and symmetries. This will help inform (non)-pharmaceutical
prevention strategies. For the reasons above we wrote this report that goes at
the heart of mathematical modelling of infectious disease diffusion by
simultaneously investigating the underlying microscopic dynamics in terms of
percolation models, effective description via compartmental models and the
employment of temporal symmetries naturally encoded in the mathematical
language of critical phenomena. Our report reviews these approaches and
determines their common denominators, relevant for theoretical epidemiology and
its link to important concepts in theoretical physics. We show that the
different 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. As important test
of the relevance of symmetries we show how to mathematically account for
observed phenomena such as multi-wave dynamics. The models presented here are
of immediate relevance for different realms of scientific enquiry from medical
applications to the understanding of human behaviour. Our review offers novel
perspectives on how to model, capture, organise and understand epidemiological
data and disease dynamics for modelling real-world phenomena.
