The interplay between subcritical fluctuations and import: understanding COVID-19 epidemiology dynamics

The effective reproduction ratio r(t) of an epidemic, defined as the average number of secondary infected cases per infectious case in a population in the current state, including both susceptible and non-susceptible hosts, controls the transition between a subcritical threshold regime (r(t) < 1) and a supercritical threshold regime (r(t) > 1). While in subcritical regimes, an index infected case will cause an outbreak that will die out sooner or later, with large fluctuations observed when approaching the epidemic threshold, the supercritical regimes leads to an exponential growths of infection. The super- or subcritical regime of an outbreak is often not distinguished when close to the epidemic threshold, but its behaviour is of major importance to understand the course of an epidemic and public health management of disease control. In a subcritical parameter regime undetected infection, here called "imported case" or import, i.e. a susceptible individual becoming infected from outside the study area e.g., can either spark recurrent isolated outbreaks or keep the ongoing levels of infection, but cannot cause an exponential growths of infection. However, when the community transmission becomes supercritical, any index case or few "imported cases" will lead the epidemic to an exponential growths of infections, hence being distinguished from the subcritical dynamics by a critical epidemic threshold in which large fluctuations occur in stochastic versions of the considered processes. As a continuation of the COVID-19 Basque Modeling Task Force, we now investigate the role of critical fluctuations and import in basic Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) epidemiological models on disease spreading dynamics. Without loss of generality, these simple models can be treated analytically and, when considering the mean field approximation of more complex underlying stochastic and eventually spatially extended or generalized network processes, results can be applied to more complex models used to describe the COVID-19 epidemics. In this paper, we explore possible features of the course of an epidemic, showing that the subcritical regime can explain the dynamic behaviour of COVID-19 spreading in the Basque Country, with this theory supported by empirical data data.