The effects of heterogeneity and stochastic variability of behaviours on the intrinsic dynamics of epidemics

We report here on a campaign of weekly measurements of concentration of SARS-Cov-2 in wastewater in several treatment plants around the Thau lagoon in the Southwest of France over a nine month period of time. The use of Digital PCR yielded very precise measurements. The observations thus generated exhibit a rough stabilization on plateaus of the epidemic and other remarkable features. Such plateaus are widely reported in the setting of the Covid-19 pandemics. In this paper we raise the question of why such plateaus and other features of epidemics dynamics arise. Indeed, the classical SIR model and its extensions hardly provide an explanation for such behavior. To address this question we introduce here a new model, which takes into account heterogeneity and natural variability of behaviors in populations. Owing to this model, we show that features such as plateaus, rebounds, and "shoulders" are part of the intrinsic dynamics of an epidemic. In particular, in the context of the Thau lagoon, we argue that they are not generated by public health policy measures or psychological reactions of the population. We then show that this model fits very well the measures obtained around the Thau lagoon.