Modeling the role of clusters and diffusion in the evolution of COVID-19 infections during lock-down

Epidemics such as the spreading of the SARS-CoV-2 virus are highly non linear, and therefore difficult to predict. In the present pandemic as time evolves, it appears more and more clearly that a clustered dynamics is a key element of description. This means that the disease rapidly evolves within spatially localized networks, that diffuse and eventually create new clusters. We improve upon the simplest possible compartmental model, the SIR model, by adding an additional compartment associated with the clustered individuals. The so-obtained SBIR model compares satisfactorily with results on the pandemic propagation in a number of European countries, during and immediately after lock-down. Especially, the decay exponent of the number of new cases after the first peak of the epidemic, is observed to be very similar for countries in which a strict lock-down was applied. We derive an analytical expression for the value of this exponent.