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

where ρ S (t) is the fraction of reachable susceptible individuals at time t and R 0 is the basic reproduction number [3] . In our case, the containment has an effect on both terms. First, κ 0 reduces the pool of individuals susceptible of contracting the disease, i.e. it reduces the term ρ S . And, second, κ 0 reduces the average degree of the individuals and thus it reduces R 0 . Eq. (S50) allows us to accurately determine the effective reproduction number and capture the impact of human mobility and containment measures on the spread of the disease. Computing Eq. (S50) by means of the NGM approach involves the calculation of the spectral radius of tensor Z, which hinders an understanding of the roots behind the transition from flattening to bending the epidemic curve triggered by an efficient confinement, as illustrated in the main text. In order to have an analytical estimation of the critical value of the needed confinement to observe this transition, κ c 0 , let us consider a single well mixed population. In this case, we have