Estimating the effect of mobility on SARS-CoV-2 transmission during the first and second wave of the COVID-19 epidemic, Switzerland, March to December 2020

Introduction Human mobility was considerably reduced during the COVID-19 pandemic. To support disease surveillance, it is important to understand the effect of mobility on transmission. Aim We compared the role of mobility during the first and second COVID-19 wave in Switzerland by studying the link between daily travel distances and the effective reproduction number (Rt ) of SARS-CoV-2. Methods We used aggregated mobile phone data from a representative panel survey of the Swiss population to measure human mobility. We estimated the effects of reductions in daily travel distance on Rt via a regression model. We compared mobility effects between the first (2 March–7 April 2020) and second wave (1 October–10 December 2020). Results Daily travel distances decreased by 73% in the first and by 44% in the second wave (relative to February 2020). For a 1% reduction in average daily travel distance, Rt was estimated to decline by 0.73% (95% credible interval (CrI): 0.34–1.03) in the first wave and by 1.04% (95% CrI: 0.66–1.42) in the second wave. The estimated mobility effects were similar in both waves for all modes of transport, travel purposes and sociodemographic subgroups but differed for movement radius. Conclusion Mobility was associated with SARS-CoV-2 Rt during the first two epidemic waves in Switzerland. The relative effect of mobility was similar in both waves, but smaller mobility reductions in the second wave corresponded to smaller overall reductions in Rt . Mobility data from mobile phones have a continued potential to support real-time surveillance of COVID-19.

same day, scaled by their current infectiousness. [5] Moreover, the delay between infection and reporting of cases was taken into account through a preceding deconvolution step. [4] Hence, the estimates of R t on day t represent the average transmission per person precisely on that respective day t, as is recommended for studying the effects of extrinsic factors such as human mobility on transmission. [6] In surveillance practice, these estimates can only be obtained with a delay of several weeks. The estimates were provided as a 3-day moving average.

Policy measures
The policy measures considered in this study were selected in a three-step procedure as follows. First, we collected all national COVID-19 policy measures in Switzerland and their corresponding implementation dates from the official regulations of the Swiss Federal Council. Second, we filtered for policy measures that have been considered as relatively influential by recent studies, [7,8] that is, restrictions concerning private or public gatherings, businesses, or educational facilities. Third, we checked the dates of the policy measures obtained against the dates from the Oxford Government Response Tracker and the Swiss National COVID-19 Science Task Force and found them to be in agreement.

Main model
We linked mobility to the effective reproduction number of COVID-19 via a Bayesian regression model. A Bayesian modelling approach was chosen because it allows to specify weakly informative priors that provide regularization of unrealistic, extremely large positive or negative effects of mobility, thereby ensuring conservative parameter estimates. Moreover, this approach yields estimates of the full posterior probability distribution of parameters and credible intervals which better agree with common-sense interpretations of uncertainty than frequentist confidence intervals, [10] and which are thus more useful for decision-making in public health.
The effective reproduction number on day t, denoted by R t , was modelled to follow a gamma distribution, i. e., Here, α and β t is the shape and rate parameter of the gamma distribution, respectively, that define the expected value of the effective reproduction number on a given day t as E[R t ] = µ = α βt . We modelled the expected value with the following regression function of the explanatory variables using a log-link: The term δ wd(t) is an intercept allowed to vary by weekday in order to capture differences between weekdays in reporting and mobility (e. g., due to less reporting and mobility on weekends) that could confound the relationship between mobility and the effective reproduction number. To control for confounding by policy measures, the dummy variable p j,t was included, which equals 1 if policy measure j was in effect on day t, and 0 otherwise.
The parameter γ j can be interpreted as the log expected change in R t when policy measure j was implemented, conditional on the day of the week, mobility, and the other policy measures. The explanatory variablem t is the sample average of the daily travel distance in the study population on day t. A property of the gamma distribution is that the variance is proportional to the mean. Specifically, if the effective reproductive number is gamma distributed, its variance function on day t is given by This shows that the estimated variance of R t on a given day is proportional to the square of the expected value predicted by the regression function.
The property enables us to capture that R t may vary more, and, hence, be more uncertain as it takes larger values.

Models for subgroups
We further analysed how the estimated mobility effect differs when using mobility data only of a certain mode of transport or travel purpose, or from participants of a certain sociodemographic subgroup (age group, employment status).. Here, we used a separate model for each mode of transport, travel purpose, and sociodemographic subgroup. The respective average daily travel distance was used as explanatory variable, i. e. the regression function was defined as wherem k,t denotes the average daily travel distance for mode of transport, travel purpose, or sociodemographic subgroup k. Apart from the changed explanatory variable, the model specification was the same as for the main model.

Models for movement radius
We also analysed the effects of the movement radius. Five different categories for the movement radius were distinguished, i. e., residential (<500 m), local (500 m-2 km), municipal (2 km-10 km), regional (10-50 km), and longrange (>50 km). Here, the model used was similar to the main model, but, instead of the average travel distance, the share of the population that travelled within a certain movement radius was defined as explanatory variable.
The model is where d l,t is the share of the population travelling within movement radius l on day t. Note that, in this model, no logarithmic transformation of mobility was used as the share is already on a relative scale. Thus, θ l can be interpreted as the expected percentage change in R t associated with an increase in the share of the population travelling in movement radius l by one percentage point, conditional on the day of the week and the policy measures. A separate model was fitted for each movement radius. Table B.2 provides an overview over the choices of priors for the parameters in our models. We used weakly informative priors for all parameters. The priors on the parameters related to policy measures and mobility reflect weak assumptions about the range of realistic effect sizes. All other priors were informed by the general recommendations from the Stan development team. [11] Similar to the approach taken by Brauner et al., [7] we modelled the effects of policy measures using an asymmetric Laplace prior on the parameters γ j . The parameter for the effect of mobility was also given an asymmetric Laplace prior. Again, both positive and negative effects were allowed. This implies that decreases in mobility may also lead to increases in the effective reproduction number, but 90 % of the probability mass was placed on positive effects and the highest probability was assigned to an effect size of zero.

Model priors
The scale parameter was chosen such that for a 1 % decrease in mobility, a 1 % reduction in the effective reproduction number is expected, and reductions of up to 3 % are probable. In the models for the different modes of transport, travel purposes, and sociodemographic subgroups, the parameter for the effect of mobility is given the same prior as in the main model.
The effects of movement radius were given a normal prior centred at zero, since it cannot be a priori known whether the increase in the share of a certain movement radius would increase or decrease the effective reproduction number. The standard deviation chosen reflects our expectation that, for a one percentage point increase in the share of a movement radius, an increase or decrease in R t of more than 20 % is unlikely.
The priors on the weekday-specific intercepts are weakly informative and encompass potential relative differences between weekdays due to reporting and mobility patterns. The shape parameter α of the gamma distribution for R t was also given a weakly informative prior. The rate parameter β t of the gamma distribution for R t is however not assigned a prior since it is calculated as α µ(Rt) . Therefore, its prior is implicitly given by the priors on α and µ(R t ). Similarly, the variance of R t is a function of µ(R t ) and α, hence its prior is also defined implicitly.

Parameter Description
Prior Models

C. Model diagnostics
The parameter estimates were analysed with common Bayesian model diagnostics to ensure model fit, a sufficient effective sample size, convergence of the chains, and absence of highly influential observations. Here, we followed recommendations by Gelman et al. [12] and the Stan development team [13].
Posterior predictive checks

Effective sample size
The ratio of the effective sample size to the total sample size [13] was computed for each model parameter (Fig. C.8). The ratio was around 0.5 for most parameters and above 0.1 for all parameters, indicating a sufficient number of independent draws from the posterior distribution.

Influential observations
We checked for influential observations in the data using the tail shape

D. Robustness checks
The robustness of the estimated mobility effects was analysed with respect to alternative study periods, alternative estimates of the effective reproduction number, and an alternative model specification with autocorrelated errors.
We also compared our estimates with unadjusted estimates obtained from models not accounting for potential confounding through policy measures.
Moreover, we tested the sensitivity of estimated effects to the use of mobility data from additional sociodemographic subgroups. Furthermore, the model was extended to incorporate changes in testing intensity.

Study periods
We checked the robustness of our results regarding the specific choice of study periods. In our study, we analysed the time periods from March 2-April 7, 2020 (wave 1) and October 1-December 10, 2020 (wave 2). To assess potential sensitivity to this choice, we also tested how the effect estimates changed when slightly extending or shortening the study periods.
As shown in Fig. D.11, the differences in estimates from alternative study periods were small. The estimates for study periods with a start date one week later were slightly smaller. This finding was expected because, in both waves, the first weeks of the study periods show the effect of mobility reductions when policy measures were not yet implemented, so their exclusion should slightly reduce the estimated effect of mobility. The estimates for study periods with an end date one week earlier have slightly wider credible intervals, suggesting that the variation of mobility under policy measures observed at the end of the study periods was informative for our estimates. We also estimated effects of mobility for the time between the first and second wave. It is important to note that, for the time period in between the waves, estimates of the effective reproduction number were highly uncertain.
This can be explained by the overall small number of reported cases. Moreover, the proportion of infectious persons was likely lower than during the waves, as demonstrated by a lower test positivity rate with a minimum of 0.2 % (June 18, 2020) over the summer period. We fitted a separate model for each month between the first and second wave. No significant relationship between mobility and R t was found for any of the six months between the waves (Fig. D.12).

Autocorrelated errors
To assess the robustness of our main estimates against autocorrelated model errors, which may be present in time series regressions, we used the same regression function for the main model, i. e., and added multiplicative errors exp(ϵ t ), where ϵ t = Rt−µ(Rt) √

Var[Rt]
are Pearson residuals. We model the residuals to follow an AR(2) process, i. e., The lag order of 2 was chosen by inspecting the autocorrelation function of the posterior predictive errors from the original model. The autocorrelation parameters τ 1 and τ 2 were both given a regularizing normal prior, i. e., τ 1 , τ 2 ∼ Normal(0, 0.4). The term u t is the so-called innovation of the AR process and is implicitly defined for any given day t through the other parameters of the model and the residuals of previous days. The specification of the model was otherwise analogous to the main model. The model with autocorrelated errors estimates slightly stronger effects of mobility, and the estimates are slightly more uncertain for the second wave.
Nevertheless, the qualitative findings remain the same.

Testing intensity
Changes in testing intensity over time may influence the effective reproduction number or estimates thereof. On the one hand, an abrupt surge in testing may distort estimates of the effective reproduction number when these are based on confirmed cases, because more testing can lead to a higher proportion of infections that are detected. On the other hand, as detected cases potentially lead to contact tracing and isolation of infectious individuals, a high testing intensity also has the potential to reduce the number of future infections and thus lower the effective reproduction number.
In our study, the direct influence of changes in testing intensity is mostly avoided through the use of R t estimates that are inferred from the number of newly hospitalized patients. Regarding the indirect effect of testing intensity, we wanted to assess whether testing may confound the effect of mobility in our model. We thus extended the model by adding the share of positive PCR tests for COVID-19 as another predictor. This share, also called test positivity rate, often serves as an indicator of the intensity of testing relative to the current number of confirmed cases. A low positive rate means that many tests are performed in comparison to the number of new cases detected; therefore, testing is rather comprehensive. On the contrary, a high positive test rate may indicate many further cases which have not been detected and confirmed. Similar to our modelling of mobility effects, we smoothed the test positivity rate with a 3-day moving average and applied a logarithmic transformation to measure changes in testing intensity on a relative scale. We furthermore added a lag of 6 days, because tests performed on a given day can only avoid subsequent infections through contact tracing and isolation.
Hence, a change in testing should only change the effective reproduction number with a delay of several days. The chosen lag of 6 days roughly equals the mean generation time of COVID-19. [22,23] We used a broad gamma prior for the effect of testing intensity, here Gamma(µ = 1, σ = 0.5).
The prior reflects the assumption that for a 1 % decrease in the positive rate (which translates to an increase in testing intensity), the effective reproduction number can decrease by up to 6% but will not increase. All other priors are chosen as in the main model.
As shown in Fig. D.18, the estimates for the effect of mobility on the effective reproduction number are almost identical for the models with and without the effect of testing intensity. We thus found no evidence that testing intensity confounded our results.