Epidemic spreading in an expanded parameter space: the supercritical scaling laws and subcritical metastable phases.

While the mathematical laws of uncontrolled epidemic spreading are well known, the statistical physics of coronavirus epidemics with containment measures is currently lacking. The modelling of available data of the first wave of the Covid-19 pandemic in 2020 over 230 days, in different countries representative of different containment policies is relevant to quantify the efficiency of these policies to face the containment of any successive wave. At this aim we have built a 3D phase diagram tracking the simultaneous evolution and the interplay of the doubling time,Td, and the reproductive number,Rtmeasured using the methodological definition used by the Robert Koch Institute. In this expanded parameter space three different main phases,supercritical,criticalandsubcriticalare identified. Moreover, we have found that in thesupercriticalregime withRt> 1 the doubling time is smaller than 40 days. In this phase we have established the power law relation betweenTdand (Rt- 1)-νwith the exponentνdepending on the definition of reproductive number. In thesubcriticalregime whereRt< 1 andTd> 100 days, we have identified arrested metastable phases whereTdis nearly constant.