Surveillance-based estimates of the reproductive number may be biased in spatially structured populations

The early evolution of an outbreak of an infectious disease epidemic depends on its reproductive number R: the average number of secondary cases that a case generates. An accurate and timely estimate of the reproductive number is crucial to make projections on the near-future evolution of the epidemic, and to set up the appropriate public health response. Estimates of R often come from surveillance data, as it has been in the case of the COVID-19 pandemic. This means statistically inferring R from time series of daily reported cases, hospitalizations, or deaths. In this study, however, we argue that surveillance-based measures of the reproductive number may not always be accurate measures of the true reproductive number. We focus on structured populations, i.e., populations made up of spatially distinct communities. In the case of structured populations, one can define an operator Q that determines the spatiotemporal early evolution of the epidemic, and whose entry Qij encodes the average number of cases that a case in community i generates in j. We show that the true R can be computed from the spectral properties of Q. We also show that the reproductive number measured by surveillance approaches the true reproductive number only asymptotically in time. This means that early estimates may either underestimate or overestimate R. Also, we show that under particular conditions the surveillance-based estimate may oscillate around the true value, alternating between underestimates and overestimates. We also show that local (i.e., community-level) estimates of the reproductive number are inaccurate in describing the global epidemic dynamics, reaching R only asymptotically. We then analytically study the speeds at which both global and local surveillance based measures of R converge to the true R, and found that these speeds depend on both the topology of the spatial network encoded in Q, and the initial spatial distribution of cases. As a case study, we then apply our findings to the COVID-19 epidemic in France and Italy, in 2020. We build Q in these countries by defining communities at the level of department (France) and province (Italy), and coupling communities using human mobility as inferred from Colocation Maps and Movement Range Maps developed within Facebook’s Data for Good program. We reconstruct the spatial distribution of reported COVID-19 cases using data from data.gouv.fr (France) and pcm-dpc/COVID-19 (Italy). We derive the expected case distribution from Q, and compare it to the recorded distribution, finding evidence of the asymptotic convergence of the surveillance-based metrics towards the true Q-derived metrics, as predicted theoretically. Then, we build a metapopulation model of the COVID-19 epidemic to numerically study the magnitude of the initial bias in the estimate of R, the conditions allowing to quickly reduce the bias, and the speed at which measures become reliable. We found that country-level and local surveillance-based R estimates have all different convergence times. Also, equilibrium often cannot be reached. We also simulated strong mobility restrictions (i.e., lockdowns) and found surveillance-based measures of R are extremely inaccurate in the aftermath of disruptions in mobility patterns. Our study shows that the spatial structure of human populations induced by mobility may make standard surveillance unable to accurately estimate the reproductive number. Our finding will help to correct this bias, and to generate more robust measures.