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

The spread of COVID-19 is posing an unprecedented threat to health systems worldwide ¹ . The fast propagation of the disease combined with the existence of covert contagions by asymptomatic individuals make the controlling of this disease particularly challenging. The key parameter to track the progression of the epidemics is the effective reproduction number ℛ, defined as the number of secondary infections generated by an infected individual ² . The suppression of the epidemics is directly related to this value, and is attained when ℛ < 1. Here, we find an analytical expression for ℛ as a function of mobility restrictions and confinement measures, using an epidemic model tailored for COVID-19. This expression for ℛ is an extremely useful tool to design containment policies that are able to suppress the epidemics. We applied our epidemic model for the case of Spain, successfully forecasting both the observed incidence in each region and the overload of the health system. The expression for ℛ allowed us to determine the precise reduction of mobility κ ₀ needed to bend the curve of epidemic incidence, which turned out to be κ ₀ ∼ 0.7. This value, for the case of Spain, translates to a total lockdown with the exception of the mobility associated to essential services, a policy that was finally enforced on March 28.