Predicting the SARS-CoV-2 effective reproduction number using bulk contact data from mobile phones

Abstract

Over the last months, cases of SARS-CoV-2 surged repeatedly in many countries but could often be controlled with nonpharmaceutical interventions including social distancing. We analyzed deidentified Global Positioning System (GPS) tracking data from 1.15 to 1.4 million cell phones in Germany per day between March and November 2020 to identify encounters between individuals and statistically evaluate contact behavior. Using graph sampling theory, we estimated the contact index (CX), a metric for number and heterogeneity of contacts. We found that CX, and not the total number of contacts, is an accurate predictor for the effective reproduction number R derived from case numbers. A high correlation between CX and R recorded more than 2 wk later allows assessment of social behavior well before changes in case numbers become detectable. By construction, the CX quantifies the role of superspreading and permits assigning risks to specific contact behavior. We provide a critical CX value beyond which R is expected to rise above 1 and propose to use that value to leverage the social-distancing interventions for the coming months.

Keywords: COVID-19; epidemiology; network science.

Fig. 1.

(A) Our method of identifying encounters by GPS coordinates rests on the colocation of two devices on a geolocation tile within a time interval of 2 min. In the contact graph, pairs of devices are linked if a cospace–time location was found at least once per day. The exemplary network shown in B is only the part of our total network of phones that is located in Leipzig on 29 February 2020. The size of the dots corresponds to the degree, or number of contacts, of the node. The layout of nodes is obtained from a spring-force algorithm (25).

Fig. 2.

(A) The effective reproduction number $R$ for Germany based on a summation of case numbers for 7 d. (B) Mean number of contacts per day from cell phone records (blue) and 7-d moving average (red). (C) Contact index per day (blue) and 7-d moving average (red).

Fig. 3.

Illustration of the friendship paradox in heterogeneous contact networks. (A) An infected individual (red) has the same probability of giving the infection to each of the susceptible contacts. However, individuals with large number of contacts, such as the purple one, have a higher chance to be connected to the infected person. Therefore, the infection is given with a higher chance to those that have high contact number. As a result, the mean number of contacts (B) does not represent the effective $R$ (blue curves) value linearly. Instead, the contact index CX (C) is needed to correct for the broadness of contact number distribution so that the expected linear relation is obtained. Blue curves in B and C are obtained by simulations of the SEIR model on graphs with variable degree distribution (SI Appendix).

Fig. 4.

(A) The evolution of CX and $R$ during and after the first surge in Germany in 2020. We have added time points of notable political interventions (vertical lines). (B) Evolution after the first wave shows a substantial and related increase of $R$ and contact index. (C) $R$ and the contact index exhibit an almost linear relation if plotted with a shift of 17 d. Colors denote different weeks as indicated. Inset shows the Pearson correlation coefficient against the number of days in delay: $R$ versus CX in green, $R$ versus mean $k$ in red.