Derivation of the effective reproduction number R for COVID-19 in relation to mobility restrictions and confinement

Alex Arenas; Wesley Cota; Jesus Gomez-Gardenes; Sergio Gomez; Clara Granell; Joan T. Matamalas; David Soriano-Panos; Benjamin Steinegger

In this paper we aim to solve this problem. We analytically relate the R of the evolution of COVID-19, in a demographically heterogeneous structured population, to the mobility patterns and the permeability of households under confinement. We derive a mathematical expression that allows us to anticipate the result of mobility restrictions and confinement on the effective reproduction number of the spreading of COVID-19 in any specific region. The dependence of R on the average mobility is nonlinear, and presents a sudden transition at R = 1. This transition separates a regime in which the mobility restrictions smoothly slow down the number of cases -the effect known as flattening the curve-, to a regime in which these restrictions sharply decelerate the number of cases -which we refer to as bending the curve-. The difference in the outcome of these two scenarios is very pronounced. First, the curve bending strategy provokes a drastic reduction in the infection speed, attaining a lower attack rate than the flattening strategy.