The reproduction number R for COVID-19 in England: Why hasn't ''lockdown'' been more effective?

The reproduction number R, the average number of people that a single individual with a contagious disease infects, is central to understanding the dynamics of the COVID-19 epidemic. Values greater than one correspond to increasing rates of infection, and values less than one indicate that control measures are being effective. Here, we summarise how changes in the behaviour of individuals alter the value of R. We also use matrix models that correctly recreate distributions of times that individuals spend incubating the disease and being infective to demonstrate the accuracy of a simple approximation to estimate R directly from time series of case numbers, hospital admissions or deaths. The largest uncertainty is that the generation time of the infection is not precisely known, but this challenge also affects most of the more complex methods of calculating R. We use this approximation to examine changes in R in response to the introduction of "lockdown" restrictions in England. This suggests that there was a substantial reduction in R before large scale compulsory restrictions on economic and social activity were imposed on 23rd March 2020. From mid-April to mid-June decline of the epidemic at national and regional level has been relatively slow, despite these restrictions (R values clustered around 0.81). However, these estimates of R are consistent with the relatively high average numbers of close contacts reported by confirmed cases combined with directly measured attack rates via close interactions. This implies that a significant portion of transmission is occurring in workplaces; overcrowded housing or through close contacts that are not currently lawful, routes on which nationwide lockdown will have limited impact.