Reduction in mobility and COVID-19 transmission

In response to the COVID-19 pandemic, countries have sought to control SARS-CoV-2 transmission by restricting population movement through social distancing interventions, thus reducing the number of contacts. Mobility data represent an important proxy measure of social distancing, and here, we characterise the relationship between transmission and mobility for 52 countries around the world. Transmission significantly decreased with the initial reduction in mobility in 73% of the countries analysed, but we found evidence of decoupling of transmission and mobility following the relaxation of strict control measures for 80% of countries. For the majority of countries, mobility explained a substantial proportion of the variation in transmissibility (median adjusted R-squared: 48%, interquartile range - IQR - across countries [27–77%]). Where a change in the relationship occurred, predictive ability decreased after the relaxation; from a median adjusted R-squared of 74% (IQR across countries [49–91%]) pre-relaxation, to a median adjusted R-squared of 30% (IQR across countries [12–48%]) post-relaxation. In countries with a clear relationship between mobility and transmission both before and after strict control measures were relaxed, mobility was associated with lower transmission rates after control measures were relaxed indicating that the beneficial effects of ongoing social distancing behaviours were substantial.


Supplementary Figure 2:
Incidence of deaths and mobility data available in each country. Each stream of mobility raw data is represented with dots and with colored lines represent our smoothed mobility measure. Dashed light blue lines represent either Apple or Google (A or G) mobility measures combine all/some of the respective data streams, while the black dashed lines represent the combined (C) Apple-Google mobility estimates.

Correlation in mobility estimates
Supplementary Figure 3: Comparison of each mobility data stream against the Apple-Google data stream. The Google stream reflecting 'park' and 'residential' mobility show some clear discrepancy compared to other streams. This is likely reflecting the allowance for essential movement linked to subsistence and exercise. Figure 1 in main text. Relationship between mobility and transmission. a) Apple-Google mobility (purple line) and daily mobility (reflecting other data streams). b) Estimated daily reproduction number for new infections (red) and deaths by date of reported death (blue) estimated using the best-fitting model and mobility data. Instantaneous reproduction number estimated from deaths data alone using a daily 7-day sliding window (grey). In each case shading represents the 95% credible interval. Horizontal orange dot and line show the median and 95% CrI for the timing of the change in the relationship between mobility and transmission. c) Estimates of the reproduction number against changes in mobility using our best model (5 estimated parameters): Turquoise/orange lines showing the median predictions pre-post change in relationship, with shading indicating the 95% credible interval. The 'EpiEstim' delayed effective reproduction numbers using 'EpiEstim'-like method are shown as error-bars in turquoise/orange for approximate pre-post change in relationship with 95% credible interval.

Supplementary note 5: Correlation between R 0 and the mobility threshold
We evaluated the correlation between estimated mobility thresholds and basic reproduction number across countries to ensure the variation in the estimated thresholds was not driven by the variation in estimated basic reproduction number.
We found no evidence of a correlation between the estimated mobility threshold and the estimated basic reproduction number.
Supplementary Figure 6: Relationship between the estimated (medians) basic reproduction number and the estimated (medians) mobility thresholds. As we evaluate the relation with the basic reproduction number, this is based on data up to the 10 th May 2020. For both Apple and Google mobility, we found no significant correlation between the estimated R 0 and mobility thresholds.

Supplementary note 6: Sensitivity of estimated parameters to early epidemic dynamics
As the reporting of deaths might have changed during the country-specific early phase of the epidemic, we re-estimated the mobility-transmission relationship excluding from the likelihood all days previous to the two consecutive weeks reporting each at least 10 deaths (the criteria for sustained epidemic). However, the estimated parameters were robust to excluding the very early dynamic from the likelihood (i.e. before our sustained epidemic criteria is met).

Supplementary Figure 7:
Estimated parameters(median and 95%CrI) when excluding the early epidemic phase (black), or including the early epidemic phase (blue) from the likelihood calculation.
As we evaluate the relation with the basic reproduction number, this is based on data up to the 10 th May 2020.
Page 54 of 64

Supplementary note 7: Sensitivity of estimated mobility thresholds to the serial interval distribution
We found that the estimated mobility thresholds and how those relate to observed mobility are consistent across assumptions of serial interval distribution and likelihood formulation.

Supplementary note 8: Modelling over-dispersion
Over-dispersion using a negative binomial likelihood is typically modelled in two ways 35 , which differ in how the variance is linked to the mean: While, arguably, NB2 has somehow more solid statistical foundation with this formulation being formally derived from following a binomial distribution with heterogeneous following a gamma distribution, the NB1 is also very popular (generalized linear models) due to its simple interpretation (scaling the Poisson variance).
There are no clear rule on using one formulation or another, and if the range of observed is not too wide, both formulations will give similar results.
However, when the range of spans multiple orders of magnitude, the formulation will affect the regression parameter estimates (see Supplementary Figure 8  This in turns has implications in term of forecasts-i.e. projecting forward based on the same relationship and assumption of future mobility patterns (figure below) Supplementary Figure 9: Forecasted incidence of deaths using NB1 or NB2 for Argentina (lowincidence setting) and the UK (high-incidence setting). Results based on Apple mobility . Dots and lines: medians; error bars and bands: 95%CrI.
From the estimated R t 's and forecasts, the NB1 structure tends to over-estimate R t and the variance in forecasts when the incidence is low (i.e. Argentina). The NB2 structure tends to over-estimate the uncertainty in R t and the variance in forecasts when the incidence is high (i.e. UK).
This suggests that: -When incidence is low, ( ) is over-estimated with NB1, but reasonable for NB2, -When incidence is high, ( ) is over-estimated with NB2, but reasonable for NB1.
We therefore propose that instead of 'scaling' the variance by 2 (NB2) or (NB1), we use an alternative formation where the increase variance slows down as increases (Supplementary Table  3): The NBsqrt model outlined seems to better capture the link between variance and mean (Supplementary Figure 10 below). This could reflect the fact that the variance is reflecting 2 processes generating heterogeneity: -Heterogeneity in transmissibility (i.e. super spreading), which would be linked to NB2, see 36 , and would impact more when incidence is small. -Heterogeneity in reporting, which would be linked to NB1 as the variance would not necessarily dramatically increase as incidence increased.
Based on DIC, we confirmed the improved fit (NB1 DIC = 39,115. NB2 DIC = 36,060, NBsqrt DIC = 35,916). Figure 10: Estimated R t 's and forecasted incidence rates of deaths using NBsqrt for Argentina (low-incidence setting) and the UK (high-incidence setting). Results based on Apple mobility. Dots and lines: medians; error bars and bands: 95%CrI.

Supplementary
In the final models, we estimate the median over-dispersion at 0.

Supplementary note 9: Linking current transmission to transmission measured at deaths
We first characterise incidence of reported deaths: If we do the same for incidence of infections: Relating deaths to infections, we have: Re-arranging the sum at the bottom, we have: is the overall infectivity on day s, so: can be interpreted as an average overall infectivity (weighted by the infection- The simplified equation is therefore an approximation. When, epidemic is growing, more weight should be put on recent mobility, so our approximation under-estimates slightly recent changes in mobility. When the epidemic is declining, more weight should be put on past mobility, so our approximation over-estimates slightly recent changes in mobility. Effectively when estimating , at time t: when R=1, the largest weight is put on , when s=t-14 days (i.e. 14 days being mode of the h distribution); when R=1.5, the largest weight is put on , when s=t-11 days; when R=0.8, the largest weight is put on , when s=t-16 days.
Given uncertainty surrounding transmission, the unknown (we use an estimate of the overall infectivity linked to deaths but not infections), the delay between infection and deaths, and the serial interval, we believe we can confidently use the approximation. The only situation where the approximation would significantly fail would be if we were observing rapid fluctuations (i.e. decrease followed by increase) in mobility with a 2-3 days period. What we observed are smooth changes in mobility, and while the reduction in mobility may be sharp, we never observe a reversion of the mobility reductions on such timescale. where B is a matrix, of size _ , _ ,with each column equal to .
The reproduction number relevant at the time of death becomes: ,2 = .
The likelihood is computed from and * ,2 (with * the element by element product) For short-term forecasts and longer-term scenarios, we augment the mobility matrix above for future dates, get the effective reproduction matrices (using the joint posterior distribution of estimate R 0 and ), obtain the new augmented matrices of reproduction number at time of death ,2 , and finally compute the expected numbers of daily deaths in the future.
Using a Poisson/negative binomial random number generator, we get short-term forecasts or longer-term scenarios.