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Infectious diseases are a threat for human health with tremendous impact onour 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, sincelate 2019. It is therefore imperative to develop efficient mathematical models,able to substantially curb the damages of a pandemic by unveiling diseasespreading dynamics and symmetries. This will help inform (non)-pharmaceuticalprevention strategies. For the reasons above we wrote this report that goes atthe heart of mathematical modelling of infectious disease diffusion bysimultaneously investigating the underlying microscopic dynamics in terms ofpercolation models, effective description via compartmental models and theemployment of temporal symmetries naturally encoded in the mathematicallanguage of critical phenomena. Our report reviews these approaches anddetermines their common denominators, relevant for theoretical epidemiology andits link to important concepts in theoretical physics. We show that thedifferent frameworks exhibit common features such as criticality andself-similarity under time rescaling. These features are naturally encodedwithin the unifying field theoretical approach. The latter leads to anefficient description of the time evolution of the disease via a framework inwhich (near) time-dilation invariance is explicitly realised. As important testof the relevance of symmetries we show how to mathematically account forobserved phenomena such as multi-wave dynamics. The models presented here areof immediate relevance for different realms of scientific enquiry from medicalapplications to the understanding of human behaviour. Our review offers novelperspectives on how to model, capture, organise and understand epidemiologicaldata and disease dynamics for modelling real-world phenomena.

The field theoretical ABC of epidemic dynamics