[1,3]Sune Lehmann

1]DTU Compute, Technical University of Denmark 2]Department of Political
Science, Aarhus University 3]Center for Social Data Science, University of
Copenhagen

# Monitoring Public Behavior During a Pandemic Using Surveys: Proof-of-Concept
Via Epidemic Modelling

Andreas Koher  Frederik Jørgensen  Michael Bang Petersen<EMAIL_ADDRESS>[  [  [

###### Abstract

Implementing a lockdown for disease mitigation is a balancing act: Non-
pharmaceutical interventions can reduce disease transmission significantly,
but interventions also have considerable societal costs. Therefore, decision-
makers need near real-time information to calibrate the level of restrictions.
We fielded daily surveys in Denmark during the second wave of the COVID-19
pandemic to monitor public response to the announced lockdown. A key question
asked respondents to state their number of close contacts within the past 24
hours. Here, we establish a link between survey data, mobility data, and,
hospitalizations via epidemic modeling. Using Bayesian analysis, we then
evaluate the usefulness of survey responses as a tool to monitor the effects
of lockdown and then compare the predictive performance to that of mobility
data. We find that, unlike mobility, self-reported contacts track the
immediate behavioral response after the lockdown’s announcement, weeks before
the lockdown’s national implementation. The survey data agree with the
inferred effective reproduction number and their addition to the model results
in greater improvement of predictive performance than mobility data. A
detailed analysis of contact types indicates that disease transmission is
driven by friends and strangers, whereas contacts to colleagues and family
members (outside the household) only played a minor role despite Christmas
holidays. Our work shows that an announcement of non-pharmaceutical
interventions can lead to immediate behavioral responses, weeks before the
actual implementation. Specifically, we find that self-reported contacts
capture this early signal and thus qualify as a reliable, non-privacy invasive
monitoring tool to track the implementation of non-pharmaceutical
interventions.

###### keywords:

Epidemic monitoring, Mobility data, Survey data, Epidemic modelling

##  1 Introduction

Pandemic management is a balancing act. When an outbreak of infections flares
up, governments and authorities need to impose restrictions and
recommendations on society that are carefully calibrated to the situation. On
the one hand, during the COVID-19 pandemic, such non-pharmaceutical
interventions have considerable benefits by changing the dominant transmission
route – close contacts between individuals – via the incentives and
information they provide soltesz2020effect ; brauner2021inferring . On the
other hand, these interventions have considerable costs in the form of
negative externalities relating to the economy and mental health
banks2020mental ; petersen2021pandemic ; clemmensen2020will .

This balancing act puts authorities and governments in need of information to
continuously calibrate the level of restrictions. It is not a matter of simply
sending out a single set of instructions regarding restrictions and
recommendations. Rather, authorities need to continuously receive information
about the effectiveness of those restrictions and recommendations and adjust
accordingly. An obvious source of information is directly related to the
epidemic such as the number of infection cases, hospitalizations, and deaths.
Yet infection cases are difficult to monitor and e.g. changes in the public’s
motivation to participate in testing programs may create problems with respect
to comparisons over time fernandez2021wastewater . Furthermore, there is a
significant lag between the onset of interventions and hospitalizations and
death counts, which imply that it is difficult to calibrate the interventions
on the basis of such information. Consequently, researchers, authorities and
governments worldwide have complemented epidemiological information with
information on the direct target of the interventions: behaviour
kogan2021early ; benita2021human .

Figure 1:  Panel A: inferred reproduction number from national
hospitalizations. Panel B: Comparison between thresholds that define risk-
taking behaviour: The percentile gives a critical number of contacts $n$ that
defines risk-taking behaviour. The time-series present the daily fraction of
individuals $P(\\#\text{total contacts}\geq n)$ that report at least $n$
contacts. Panel C: Comparison between risk-taking behaviour with a threshold
at the 85th percentile (_self-reported survey data_), Google mobility, Apple
mobility, and telecommunication data (_Telco_).

In this manuscript, we assess the predictive validity of a particular source
of information about behavior during lockdowns: Population-based surveys on
social contact, fielded daily to representative samples of the Danish
population during the COVID-19 pandemic (see Methods for details on this
dataset). This assessment aligns with recommendations about the use of surveys
as epidemic monitoring tools on the basis of experiences during the SARS
epidemic in Hong Kong leung2005longitudinal  and recommendations from the
World Health Organization during the COVID-19 pandemic world2020survey . From
a public health policy perspective, these particular data are a unique test
case as they were, in fact, reported to the Danish government for this purpose
on a twice-weekly basis during the second wave of the coronavirus in December
2020.

Furthermore, these data are unique in another respect: They constitute an open
and ‘citizen science’ bonney2014next  alternative to the most used source of
information on pandemic behavior: Mobility data. As we detail below, mobility
data as a source of information may be problematic from both a methodological
and policy perspective. Mobility data provides a proxy for close contacts
between people and has been heavily utilized by researchers and public health
institutions benita2021human ; buckee2020aggregated ; ALE22 . Mobility data
quantifies the population’s movement patterns and is unobtrusively obtained in
a number of ways, for example, via people’s smart phones and provided to
researchers and governments via private companies such as Google AKT20 . This
reliance, however, can and has, raised concerns. First, in many cases, it
implies that pandemic management and research relies on the willingness of
private companies to share information during a critical crisis. Second,
citizens themselves may be concerned about real or perceived privacy issues
related to the sharing of data with authorities hu2021human ; jung2020too .
Given the importance of public trust for successful pandemic management
bollyky2022pandemic , such concerns – if widespread – can complicate pandemic
control. Third, data from companies such as Google, Facebook and local phone
companies may not be representative of the population of interest: The entire
population of the country. Rather than being invited on the basis of
traditional sampling methods, people opt-in to the services of different
companies and, hence, the data from any single company is likely a biased
sample. Fourth, the movements of people in society as captured by mobility
data is only a proxy of the quantity of interest: Actual close encounters
between individuals that drive the pandemic.

For these reasons, it is key to assess alternative sources of information
about public behavior such as nationally representative surveys of the adult
population. In principle, surveys could alleviate the problems identified
relating to the collection and validity of mobility data. Survey research is a
centuries old low-cost methodology that can be utilized by public actors and
that relies on well-established procedures for obtaining representative
information on private behaviours in voluntary and anonymous ways
krosnick1999survey .

At the same time, data from surveys come with their own methodological
complications. As documented by decades of research, people may not accurately
report on their own behaviour schuman1996questions . Survey answers during the
pandemic may be biased by, for example, self-presentational concerns and
inaccurate memory. While research on survey reports of behaviour during the
pandemic suggests that self-presentational concerns may not affect survey
estimates larsen2020survey , memory biases may (although such biases are
likely small for salient social behavior) hansen2021reporting . Even with such
biases, however, surveys may be fully capable to serve as an informative
monitoring tool. The key quantity to monitor is changes in aggregate behaviour
over time. If reporting biases are randomly distributed within the population,
aggregation will provide an unbiased estimate. Even if this is not the case,
changes in the survey data will still accurately reflect changes in population
behaviour as long as reporting biases are sSupplementary Table within the
relevant time period.

On this basis, the purpose of the present manuscript is, first, to examine the
degree to which survey data provide useful diagnostic information about the
trajectory of behavior during a lockdown and, second, to compare its
usefulness to information arising from mobility data. To this end, we focus on
a narrow period around Denmark’s lockdown during the second wave of the
COVID-19 epidemic in the fall of 2020, prior to vaccine roll-out when it was
crucial for authorities to closely monitor public behavior.

Specifically, we ask whether a) daily representative surveys regarding the
number of close social contacts and b) mobility data allow us to track changes
in the observed number of hospitalizations in response to the lockdown. In
addition, to further probe the usefulness of survey data, we provide a fine-
grained analysis of how different types of social contacts relate to
hospitalizations. Our results shed new light on the usefulness of survey data.
Previous studies during the COVID-19 pandemic have documented high degrees of
overlap between self-reported survey data on social behavior and mobility
data, but have not assessed whether these data sources contain useful
information for predicting transmission dynamics gollwitzer2020connecting ;
kalleitner2021varieties . One study did compare the predictive power of
mobility data to survey data on the psychosocial antecedents of behavior
jirsa2020integrating  and found that mobility data was more predictive than
the survey data of COVID-19 transmission dynamics. Here, we provide a more
balanced test by comparing the predictive value of mobility data and survey
data when directly focused on self-reported behavior rather than simply its
psychosocial antecedents.

##  2 Results

We establish the link between survey data, mobility data, and hospitalizations
via state-of-the-art epidemic modeling, which uses the behavioural survey and
mobility data as an input to capture underlying infectious activity NOU21 ;
UNW20 . Specifically, to jointly model the epidemic spreading within the five
regions of Denmark, we extend the semi-mechanistic Bayesian model from Flaxman
et al. FLA20 ; UNW20 . Where possible, we use partial pooling of parameters to
share information across regions and thus reduce region specific biases. As a
result, we are able to determine effect sizes for each region and establish a
pooled effect size for all of Denmark. See Methods for full details on the
model. Because we are interested in the use of behavioural data as a guide for
decision-making, our inference focuses on the key period of the second wave
from 1-December-2020, i.e., about one week before Denmark’s lockdown
announcement, to 1-February-2021 when vaccinations accelerated across the
country. The period captures a sharp increase in reported cases during the
second wave of Denmark’s Covid-19 pandemic (see Supplementary Fig. S1).

###  2.1 Self-reported survey data versus mobility data

As a monitoring tool, we first consider self-reported survey data on the daily
number of contacts, defined as close encounters with less than 2 meters
distance for at least 15 minutes jorgensen2020does . The reported numbers are
highly skewed, with $15.6\%$ of all counts concentrated on zero with some
reporting over $1\,000$ contacts (see Supplementary Fig. S2). As a result,
taking the mean over daily reported numbers is highly sensitive to outliers,
while reporting quantile-based measures obscure most of the variation.
Instead, we form the following robust measure of risk-taking behaviour: We
label a participant in the survey as risk-taking if they report contacts above
a certain threshold and propose the daily fraction of risk-taking individuals
as a predictor to the effective reproduction number. The intuition is that
infections tend to be linked to large clusters via super-spreading events.
Therefore, we focus on the fraction of the population that reports an above-
average number of contacts. However, what is a reasonable threshold? To this
end, we will compare the predictive power of different choices and define
risk-taking behaviour using percentiles of all reported contacts within a
reference period (see Supplementary Fig. S2 for details). As discussed below,
we estimate an optimal threshold for prediction, but our findings are robust
to varying the specific definition for risk-taking behaviour.

Figure 2: Self-reported survey data has the largest effect on the reproduction
number $R_{t}$. In detail, we present the effect of a $50\%$ reduction in all
predictors on $R_{t}$ relative to $R_{t}=1.5$. The latter reflects our
estimate just before the lockdown’s announcement in early December (see
Supplementary Fig. S5). The predictors include (a) self-reported survey data
(_Survey_), (b) Google mobility, (c) Apple mobility, and (d) telecommunication
data (_Telco_). Panel A: Pooled effect sizes across all regions. We present
the posterior density, 95 % HPDI (highest probability density interval), and
median as filled curves, horizontal bars and white circles, respectively.
Panel B: regional effects with the same summary statistics as in panel A.

Figure 1, panel B illustrates the dynamics of risk-taking behaviour, referred
to as _self-reported survey data_ , for thresholds ranging from the median to
the $90$th percentile, corresponding to $4$ and $24$ contacts, respectively.
For thresholds above the $70$th percentile, risk-taking behaviour shows the
strongest response to the announced lockdown and increases little during the
Christmas period. Qualitatively, this behaviour matches the time-varying
reproduction number $R_{t}$ (see Fig. 1 panel A) that we inferred from
national hospitalizations using a latent random-walk model (details in Sec.
4.2).

For a quantitative analysis, we parametrize $R_{t}$ using a single predictor
and compare the performance for different definitions of risk-taking behaviour
using leave-one-out cross-validation (LOO). LOO works by fitting the model to
the observed hospitalizations excluding a single observation and comparing the
prediction of the unseen observation against the observed real-world data.
Repeating this process over all observations, allows one to estimate the model
performance on out-of-sample data, which does not suffer from over-fitting
(such as explained variance ($R^{2}$), root mean square error (RMSE) or other
in-sample measures of accuracy) GEL95 . Here, we use an efficient estimation
of LOO based on pareto-smoothed importance sampling (PSIS-LOO) that does not
require refitting the model for every excluded observation VEH16 .

A region-level analysis with PSIS-LOO reveals that the $85$th percentile,
provides the best predictor in terms of the observed hospitalizations (see
Supplementary Table S2). However, the result is not sensitive to this value as
all models with a threshold between the $60$th and $90$th percentile perform
within the significance level. In the following, we will use the $85$th
percentile as a defining threshold unless stated otherwise.

Crucially, by considering self-reported survey data, we capture the sharp
decline in the reproduction number after the lockdown’s announcement, i.e.,
about two weeks before its nationwide implementation (see Supplementary Table
S1 for a detailed timeline). This early signal is not as pronounced in the
combined mobility time series from Google and Apple that have been proposed in
NOU21 , nor in the telecommunication data from Danish mobile network operators
(see Fig. 1 panel C and Sec. 4.1 for detailed information about the data).
Indeed, a cross-validation analysis confirms that self-reported survey data
gives the best out-of-sample predictions (see Supplementary Table S4).

There are limitations to PSIS-LOO, especially when sample sizes and
differences in predictive performance are small SIV20 . Therefore, we consider
the relative usefulness of survey vs. mobility data via direct comparison as
an alternative approach. In this case, we parametrize the reproduction number
as a linear combination of predictors and compare posterior effect sizes (see
Fig. 2). Consistently for all regions, a reduction of risk-taking behaviour
has the strongest effect on $R_{t}$ compared to the same relative change in
Google, Apple, and telco mobility data. This observation confirms our cross-
validation analysis and in Sec. S5, we present a detailed comparison with the
individual data streams that underlie Google and Apple mobility.

Figure 3: Risk-taking behaviour towards friends and strangers have the largest
effect on the reproduction number $R_{t}$. As in Fig. 2, we present the effect
of a $50\%$ reduction in context-dependent risk-taking behaviour on $R_{t}$.
Panel A: Pooled effect sizes across all regions with the posterior density,
the 95 % HPDI (highest probability density interval), and median as filled
curves, horizontal bars and white circles, respectively. Panel B: regional
effects with the same summary statistics as in panel A.

###  2.2 Understanding the role of contact-types

In our survey, we assessed the daily number of contacts separately for (a)
family members outside the household, (b) friends and acquaintances, (c)
colleagues and (d) strangers, i.e. all other contacts. Therefore, we can
evaluate the impact of context-depending risk-taking behaviour on $R_{t}$ and
observed hospitalizations, respectively. As above, we choose the 85th
percentile as a threshold for risk-taking behaviour for each contact type.
Then, we parametrize $R_{t}$ with a single predictor and compare the
performance in terms of the PSIS-LOO score. Cross-validation shows that risk-
taking behaviour towards friends and strangers is significantly more
predictive than family members and colleagues (see Supplementary Table S6).
For a direct comparison – similar to the comparison between survey and
mobility data above – we also parametrize the reproduction number as a linear
combination of all four contact types. The inferred effect sizes confirm the
cross-validation analysis: Reducing risk-taking behaviour towards friends and
strangers has the greatest impact on the reproduction number (see Fig. 3).
However, the posterior uncertainty is relatively large because risk-taking
behaviour towards friends, strangers and colleagues are highly co-linear (see
Supplementary Fig. S9) and thus effect sizes are strongly negatively
correlated (see Supplementary Fig. S12) MCE20 .

A notable edge-case is contacts with colleagues. Here the predictive
performance with PSIS-LOO is only marginally below the significance level and
though effect sizes in Fig. 3 have a significant overlap with zero, the
uncertainty permits also relatively large effects. Indeed, the predictive
performance improves above the significance threshold if we define risk-taking
behaviour using the 90th percentile (see Sec. S6). Hence, only reported
contacts with family members shows no significant effect on the reproduction
number for all thresholds that we considered. For an intuitive explanation of
this finding, compare the reproduction number $R_{t}$ with risk-taking
behaviour towards family members in Supplementary Fig. S9: Risk-taking spikes
during the Christmas period with no apparent effect on $R_{t}$. Thus, the data
suggests that family gatherings during the Christmas period did not drive up
hospitalizations. This finding can be explained by cautious behaviour,
including extended testing and isolation prior to Christmas, or simply due to
fact that spreading within families is limited to a small group unlike in the
context of strangers, friends and, to some extent, colleagues (see
Supplementary Fig. S9).

##  3 Discussion

During a lockdown, decision-makers need high-fidelity, real-time information
about social behavior in order to carefully calibrate restrictions to both the
epidemic wave and levels of public compliance. Interventions that are too
lenient will not sufficiently reduce the wave, while too severe interventions
(e.g., curfews) may have significant negative externalities on, for example,
public trust and mental health petersen2021pandemic ; clemmensen2020will .

To this end, researchers and authorities worldwide have relied on mobility
data, which have been cheaply available as they were already unobtrusively
collected by, for example, private tech companies. At the same time, such
reliance entails a dependency on data collected by company actors and data
which may raise privacy issues.

In the present analysis, we have provided evidence for the usefulness of daily
surveys of nationally representative samples as an alternative source of
information during a lockdown. While the use of surveys has been recommended
during the COVID-19 pandemic by WHO world2020survey  and on the basis of the
SARS epidemic in Hong Kong leung2005longitudinal , the present analysis
provides one of the first attempts to quantify the predictive validity of
surveys of self-reported behavior during a lockdown. In contrast, prior
research has focused on the behavioral antecedents of behavior such as self-
reported fear of COVID-19 jirsa2020integrating . While understanding the
impact of such antecedents is a theoretically important endeavour, more direct
measures of behavior may be preferable for a monitoring purpose (see also
Supplementary Fig. S4 and Supplementary Table S3 for a comparison with
indirect measures from our survey).

The analyses provides proof-of-concept that self-reported measures of behavior
can be superior to mobility. Given the widespread use of mobility data it is
relevant to ask why survey data fared better. Unlike mobility data, the survey
data was able to capture behavioural changes immediately after the partial
lockdown’s announcement, i.e., before it was actually implemented. We see
similar decreases of activity also in regions that were not targeted with the
initial partial lockdown. This finding implies that part of the problem of
mobility data may be that it is too coarse and, hence, does not capture the
micro-adjustments in social behavior that people make when they are concerned
with infection risk such as standing further away from others in public
queues, not mingling with co-workers at the workplace and so forth.

A related observation is that we find little association between contacts to
family members and the epidemic activity despite the Christmas holidays in the
period covered by the analyses. This behavior could be explained by (a) other
fine-grained behavioral adjustments not captured by our survey data such as
testing, prior isolation or simply by (b) the relatively small size of family
gatherings that limit potential super-spreading events. Our inability to
distinguish above, however, suggests that there are multiple possibilities of
improving the measures used for monitoring public behavior. When knowledge has
been gathered about the main pathways of transmission, researchers and
authorities can more directly ask questions about social interactions in
situations that enhances or inhibits transmission risk. In the context of
social contact during the COVID-19 pandemic, for example, it would be relevant
to know whether the contact occurred inside or outside.

In summary, the present analysis has provided proof-of-concept regarding the
usefulness of survey data as public policy tool for monitoring compliance with
the announcement and implementation of lockdowns. While the analyses we
present are narrowly focused on a single lockdown, they support WHO’s
recommendation to integrate social science methods such as surveys into
pandemic surveillance and management.

##  4 Materials and Methods

###  4.1 Data

We use survey data from the HOPE (‘How Democracies Cope With COVID-19’)
research project (www.hope-project.dk). Specifically, the HOPE-project fielded
daily nationally representative survey in Denmark in mid-May 2020. Kantar
Gallup conducts the data collection until the end of April 2022. Each day a
nationally representative sample (with a daily target of $500$ complete
interviews) reports on their protective behaviour and perceptions of the
COVID-19 pandemic. Participants are Danish citizens aged 18 years or older.
They are recruited using stratified random sampling – on age, sex and
geographical location – based on the database of Danish social security
numbers. The mobility data comes from Apple APPLE , Google GOOGLE  and major
Danish mobile phone network operators MOL22  (for full description, see
section S4).

###  4.2 Model description

We observe regional COVID-19 related hospitalizations, which derive from an
initial number of infected and the time-varying reproduction number. We
parametrize the latter using behavioural survey data and mobility time series.
Our approach is a variant of the semi-mechanistic hierarchical Bayesian model
of Flaxman et al. FLA20  and Unwin et al. UNW20 , with the key difference that
we use daily COVID-19 related hospitalizations. In Denmark, hospitalizations
are a reliable proxy for pandemic activity available with a significantly
smaller delay than death counts and less confounding through varying test
intensity than PCR case counts. The schematic overview in Fig. 4 shows the
information flow in our epidemic model, which can be summarized into three
stages: (a) the observation model, (b) the infection model, and (c) the
transmission model.

The code is written in the Julia programming language JULIA  using the
Turing.jl package TURING  for Bayesian inference. The source code is fully
accessible on GitHub GITHUB  and next, we provide the mathematical details.

Figure 4: Schematic overview of the semi-mechanistic epidemic model that
connects behavioural data from the HOPE survey to observed hospitalizations in
Denmark.

Observation model: As observations, we take the daily number of
hospitalizations $H_{t,r}$ at time $t$ in region $r$ and assume these are
drawn from a Negative Binomial distribution with mean $h_{t,r}$ and over-
dispersion factor $\phi$:

$\displaystyle H_{t,r}$
$\displaystyle\sim\text{NegBinom}\left(h_{t,r},h_{t,r}+\frac{h^{2}_{t,r}}{\phi}\right)$
(1) $\displaystyle\phi$
$\displaystyle\sim\text{Gamma}(\text{mean}=50,\text{std}=20)$ (2)

From the expected number of hospitalizations $h_{t,r}$, we derive the latent,
i.e., unobserved number of new infections $i_{t,r}$. Two factors link
infections to hospitalizations: (a) The conditional probability $\alpha$ of
hospitalization following an infection and (b) the corresponding delay
distribution $\pi$:

$\displaystyle h_{t,r}$
$\displaystyle=\alpha\sum_{\tau=0}^{t-1}i_{\tau,r}\pi_{t-\tau}$ (3)
$\displaystyle\alpha$ $\displaystyle\sim\text{Normal}^{+}(0.028,0.002)$ (4)

We estimate the infection hospitalization rate $\alpha$ in Eq. 4 from a sero-
prevalence study ERI20 . The results are, however, not sensitive to this value
as we don’t account for the depletion of susceptible. The delay $\pi$ is a sum
of two independent random variables, i.e. the incubation period and the time
from onset of infection to hospitalization FAE20 . We take the corresponding
distributions from previous studies and parametrize the incubation period by a
Gamma distribution with a mean of $5.1$ days and a coefficient of variation of
$0.86$ LAU20  and the infection to hospitalization delay by a Weibull
distribution with shape parameter $0.845$ and scale parameter $5.506$ FAE20 :

$\begin{split}\pi\;\sim\;&\text{Gamma}(\text{mean}=5.1,~{}\text{CV}=0.86)\;+\\\
&\text{Weibull}(\text{shape}=0.845,~{}\text{scale}=5.506)\end{split}$ (6)

We then discretize the continuous distribution $\pi$ by
$\pi_{i}=\int_{0}^{1.5}g(\tau)d\tau$ for $i=2,3,...$ and
$\pi_{1}=\int_{0}^{1.5}g(\tau)d\tau$ for application in Eq. 3.

Infection model: The (unobserved) number of new infections, $i_{t,r}$, evolves
according to a discrete renewal process. This approach has been widely used in
epidemic modelling BHA20 ; FLA20 ; COR13 ; NOU18 , is related to the classical
susceptible-infected-recovered model KER27  and has a theoretical foundation
in age-dependent branching processes MIS20 ; BHA20 . New infections in region
$r$ at time $t$ are a product of the time-varying reproduction number
$R_{t,r}$ and the number of individuals that are infectious at time $t$. The
latter is a convolution of past infections and the generation interval
$g_{\tau}$:

$\displaystyle i_{t,r}$
$\displaystyle=R_{t,r}\sum_{\tau=0}^{t-1}i_{\tau,r}g_{t-\tau}$ (7)
$\displaystyle i_{0,r}$ $\displaystyle\sim\text{Exponential}(3I_{0,r})$ (8)

We estimate the initial number of infected $i_{0,r}$ from the observed test
positive cases $I_{0,r}$ assuming roughly an under estimation factor of three.
Again, the precise number of initially infected is not crucial as we can
neglect depletion of susceptible for our observation window. The generation
interval $g$ translates past infections to the present number of infectious
individuals and following previous studies, we assume a Gamma distribution
density $g(\tau)$ with mean $5.06$ and SD $2.11$ FER20 :

$g~{}\sim\text{Gamma}(\text{mean}=5.06,~{}\text{SD}=2.11)$ (10)

Again, we discretize the continuous distribution by
$g_{i}=\int_{0}^{1.5}g(\tau)d\tau$ for $i=2,3,...$ and
$g_{1}=\int_{0}^{1.5}g(\tau)d\tau$ to be used in Eq. 7.

Transmission model: At the heart of the analysis is the instantaneous
reproduction number $R_{t,r}$ for region $r$ at time $t$. It determines the
number of secondary transmissions, i.e. the number of newly infected
$i_{t,r}$, given the number of infectious individuals
$\sum_{\tau=0}^{t-1}i_{\tau,r}g_{t-\tau}$ in Eq. 7. We parametrize the
reproduction number as follows:

$R_{t,r}=f\left(\sum_{c}e_{c,r}X_{t,r,c}+\rho_{t,r}\right)$ (11)

The scaled logistic link function $f=K/(e^{-x}+1)$ BHA20  bounds $R_{t,r}$ to
an epidemiologically realistic range of values $0<R_{t,r}<K=4$. The argument
to the link function is a sum of two components and reflects our two modes of
fitting:

(a) The ”non-parametric” mode NOU21  applies outside the main analysis window
from 2020-12-01 to 2021-02-01. Here, we track weekly changes in $R_{t,r}$
using a latent random walk model $\rho_{t,r}$ and do not inform the effect
sizes $e_{c,r}$ by setting the corresponding predictors $X_{t,r,c}$ to a
constant value:

$\displaystyle X_{t,r,c}=\begin{cases}0\text{ for }t<\text{2020-12-01}\\\
X_{T,r,c}\text{ for }t>T=\text{2021-02-01}\end{cases}$ (12)

Hence, we have a flexible model of $R_{t,r}$ from 2020-08-01 to 2020-12-01 and
from 2021-02-01 to 2021-02-14, i.e. before and after the main analysis window,
respectively. The idea is to minimize potential biases due to (i) the
simplified assumptions that initialize the model (see Eq. 8) and (ii)
increasing uncertainty at the end of the observation. Hence, the latent random
walk model $\rho_{t,r}$ captures all changes in the observed hospitalizations
and is defined as follows:

$\displaystyle\rho_{t,r}$
$\displaystyle\sim\text{Normal}(\rho_{t-1,r},\sigma_{r})$ (13)
$\displaystyle\rho_{0,r}$ $\displaystyle=f^{-1}(R_{0,r})$ (14) $\displaystyle
R_{0,r}$ $\displaystyle\sim\text{Normal}^{+}(1.0,0.1)$ (15)
$\displaystyle\sigma_{r}$ $\displaystyle\sim\text{Normal}(0.8,.02)$ (16)

Here, the latent variable $\rho_{t,r}$ performs a random walk with a typical
step size of $\sigma_{r}$. The latter determines the smoothness of the
resulting reproduction number and we choose the same corresponding prior
distribution as in UNW20 . For the initial condition, we assume that the
reproduction number $R_{0,r}=f(r_{0,r})$ lies within $0.8-1.2$ $95$%CI, thus
reflecting our believe that the pandemic was under control on 2020-08-01, i.e.
before to the second wave of Covid-19 infections in Denmark.

(b) We use the ”parametric” mode NOU21  for the analysis window from
2020-12-01 to 2021-02-01. During this period, the value of $\rho_{t,r}$
remains fixed and changes in $R_{t}$ have to be explained by the regional
predictors $X_{t,r,c}$ (see Eq. 11). The predictors are normalized such that
$X_{t,r,c}$ gives the change in behaviour or mobility relative to the first
day of the inference window, i.e. 2020-12-01. Thus, the effect sizes $e_{c,r}$
in Eq. 11 translate a relative change $X_{t,r,c}$ in the predictor $c$ to a
change in the reproduction number $R_{t,r}$. We pool information in order to
reduce regional biases and to give a robust country-level effect estimate
$e_{c}$ akin to multi-level models GEL95 :

$\displaystyle e_{c,r}$ $\displaystyle\sim\text{Normal}(e_{c},\sigma_{e})$
(17) $\displaystyle e_{c}$ $\displaystyle\sim\text{Laplace}(0,0.2)$ (18)
$\displaystyle\sigma_{e}$
$\displaystyle\sim\text{Gamma}(\text{mean}=0.03,\text{SD}=0.02)$ (19)

Assuming no prior information towards the direction of the effect, we place a
symmetric Laplace prior over the pooled effect parameter $e_{c}$. The prior
choice can be understood when we look at their effect on the reproduction
number $R_{t,r}$: If we parametrize $R_{t,r}$ with four predictors (as in
Supplementary Fig. S5 and Fig. 3), then we expect that a 50 % reduction in for
all predictors results in a change of $R_{t}$ by [-32 %, +37 %]. Hence, Eq.
4.2 assumes a moderate overall effect and consequently it is a priori unlikely
that a single predictor explains all variations in $R_{t,r}$.

Furthermore, we also choose a restrictive prior on $\sigma_{e}$. Thus, our
choice of priors limits both regional differences and variations between
predictors in order to reduce potential overfitting given the limited data.
Note, however, that substantial effects differences are still inferrable if
the data provides sufficient evidence.

##  5 Data availability

All data necessary for the replication of our results is collated in
https://github.com/andreaskoher/Covid19Survey. The hospitalization data
originated from Statens Serum Institute https://covid19.ssi.dk/.

##  6 Code availability

All code necessary for the replication of our results is collated in
https://github.com/andreaskoher/Covid19Survey

##  7 Acknowledgements

We would like to thank the Turing.jl development team and in particular Tor
Erlend Fjelde for their technical support.

##  8 Author contributions

AK, FJ, MBP, and SL conceived the study and wrote the text. AK carried out
modeling and analyses. FJ and MBP collected the survey data.

##  9 Competing interests

The authors declare no competing interests.

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Supplementary Material
Monitoring Public behaviour During a Pandemic Using Surveys: Proof-of-Concept
Via Epidemic Modelling

##  S1 MCMC sampling

We implement the epidemiological model in the Julia programming language JULIA
using the Turing.jl package TURING  for Bayesian inference. In particular, we
use the No-U-Turn sampler HOF14 , i.e. a variant of the Hamilton Monte-Carlo
sampler with a target acceptance rate of $0.99$ and a maximum tree-depth of
$8$. We draw $5000$ samples from $5$ chains each and discard the first $1000$
for warm-up.

All inference results report no divergent transitions. Also, the maximum
Gelman–Rubin diagnostic and $\hat{R}$ statistics is below $1.1$ for all
simulations, thus indicating sufficient mixing and convergence of the Monte-
Carlo chains.

Further implementation details and a step-by-step tutorial to reproduce the
main results are available on GitHub GITHUB .

##  S2 Covid-19 restrictions timeline

date |  action
---|---
2020-12-07 |  partial lockdown announced∗: significant tightening of Covid-19 restrictions in 38 municipalities across Denmark, including the country’s three largest cities, Copenhagen, Aarhus, and Odense.
2020-12-09 |  partial lockdown in effect
2020-12-16 |  full lockdown announced∗∗. Nation-wide Restrictions are gradually increased, starting from 2020-12-17 until the full lockdown on 2020-12-25
2020-12-17 |  shopping malls closed
2020-12-21 |  school closure & shut down of businesses involving close contact such as
2020-12-24 |  private events over Christmas are encouraged not to exceed 10 people
2020-12-25 |  all non-essential retail businesses closed / ”full lockdown”
Table S1: The above timeline of Denmarks second Covid-19 lockdown follows
online media announcements:
∗ https://www.thelocal.dk/20201207/latest-denmark-announces-partial-
covid-19-lockdown-until-2021/, accessed: 2022-04-20
∗∗ https://www.thelocal.dk/20201216/new-denmark-announces-national-lockdown-
from-christmas-day/, accessed: 2022-04-20.

##  S3 Extended data discussion

Figure S1: top panel: national hospitalizations from 2020-08-01 to 2021-02-21.
We highlight the analysis window from 2020-12-01 to 2021-02-01, which is the
center of our analysis in Sec. 2, the lockdown’s announcement and the
lockdown’s roll-out period. Middle panel: inferred reproduction number $R_{t}$
from national hospitalizations. Bottom panel: fraction of Denmark’s vaccinated
population. In this report, we use regional-level hospitalizations from the
entire time window, i.e. we start on 2020-08-01 when the pandemic was under
control and stop on 2021-02-21 when vaccinations began to accelerate. However,
only for the analysis window from 2020-12-01 to 2021-02-01, we parametrize the
reproduction number using behavioural data as described in Sec. 4.2.

###  S3.1 Survey data

We contact participants via eBoks, the official electronic mail system of
public authorities, and provide no financial incentives. Notably, about 8 % of
the Danish population, mainly older people, are exempted from eBoks. Despite
this limitation and a response rate of 25 %, the participants are
representative of the broad Danish population regarding the stratified
characteristics jorgensen2020does . Further details on sampling and questions
are available in jorgensen2020does .

From our survey, we select the self-reported number of contacts within a two-
meter distance for at least 15 minutes and differentiate between contacts to
family members, friends, colleagues and strangers, where the latter refers to
all remaining contact types. Our analysis focuses on the period from
2020-12-01 to 2021-02-01 with 15.595 participants split into the five regions
of Denmark: Region Hovedstaden (Capital), Region Midtjylland (Center), Region
Nordjylland (North), Region Sjælland (Zealand), Region Syddanmark (South). We
remove unreasonable outliers that include negative numbers and values above
50, 100, 100 and 1000 for contacts to family members, friends, colleagues and
strangers, respectively, thus dropping $0.3\%$ of all responses. By summing up
the reported number of context-dependent contacts, we obtain every survey
participant’s total number of contacts. Next, we mark a participant as risk-
taking, either context-depending or in terms of total contacts, and derive the
daily fraction of risk-takers. This quantity is robust to outliers and
reflects our understanding that super-spreading events drive Covid-19
infections ADA20 . Finally, we take a 7-day moving average, centered on day
four and calculate the change in behaviour relative to the first observation
day, i.e., 2020-12-01. We thus have five data streams for each of the five
regions of Denmark: risk-taking behaviour given overall contacts and four
context-depending time series (see Fig. 1 panel B and appendix Fig. S9),
respectively.

###  S3.2 Mobility data

Apple provides three data streams, namely, _driving_ , _walking_ , and
_transit_. The latter is not available in all regions of Denmark, and
therefore we exclude it from the analysis.

The Google data includes six time-series: _grocery & pharmacy_, _retail &
recreation_, _transit stations_ , _workplaces_ , _parks_ , and _residential_.
We exclude _parks_ because data is too sparse on a regional level. In addition
to the individual data streams, we combine _driving_ and _walking_ to a single
_Apple_ time series, and equally for _Google_ , we use _grocery & pharmacy_,
_retail & recreation_, _transit stations_ , and _workplaces_.

The telecommunication (telco) time series derives from aggregated mobility
flows within and between municipalities. Denmark’s leading mobile network
operators provided the data to the Statens Serum Institut (SSI), covering
2020-02-01 to 2021-06-30. The SSI officially requested the data to improve
national Covid-19 models and understand population behaviour in response to
non-pharmaceutical interventions. Detailed information on the data is
available in MOL22  and the complete data set can be downloaded from TELCO .

As a final preprocessing step to the mobility data from Apple, Google and the
telco companies, we take a 7-days moving average and calculate the change in
mobility relative to the first observation day on 2020-08-01.

Figure S2: Histogram over the reported contacts from 2020-08-01 to 2020-12-01.
Left column: linear scaling with normalization to probabilities. Right column:
Log scale without normalization, i.e. bins represent the number of survey
participants that reported the corresponding number of close contacts. The
linear scaling highlights the large fraction of individuals that report zero
close contacts in the past 24h, whereas the log-scaling demonstrates the broad
distribution of contacts, even after removing outliers as described in Sec.
4.1. We use these statistics to define risk-taking behaviour in the main text
as follows: Given a threshold in terms of a percentile, we derive the
corresponding number of contacts from the above distributions. Then, we mark
individuals as risk-taking (towards the total number of contacts or context-
dependent) if they report more than the threshold number of contacts and
report the daily fraction of risk-taking individuals. The resulting time-
series captures subtle behavioural changes in the population and is robust
with respect to outliers.

##  S4 Extended information on risk-taking behaviour

Figure S3: Comparison between $R_{t}$ and risk-taking behaviour. Top panel: reproduction number $R_{t}$, derived from national hospitalizations. Lower panel: changes in risk-taking behaviour given the total number of contacts and different thresholds in terms of percentiles (see Sec. 4.1 for details). Similar to Fig. 1 in the main text, but includes more thresholds. Risk-taking behaviour that is derived from a larger threshold, shows an increased response to the lockdown’s announcement and a smaller Christmas-related peak. Visually, the dynamics of highly risk-taking behaviour corresponds well to $R_{t}$ and we confirm the improved predictive performance quantitatively in Table S2 using the PSIS-LOO score. rank | contacts | percentile | score difference | std error
---|---|---|---|---
0 | $\boldsymbol{>18}$ | 85th | 0.0 | 0.0
1 | $\boldsymbol{>6}$ | 60th | 1.16619 | 4.50549
2 | $\boldsymbol{>9}$ | 70th | 1.31973 | 3.9657
3 | $\boldsymbol{>14}$ | 80th | 1.81696 | 2.02611
4 | $\boldsymbol{>24}$ | 90th | 2.2114 | 3.45834
5 | $\boldsymbol{>11}$ | 75th | 2.57881 | 3.07653
6 | $\boldsymbol{>4}$ | 50th | 13.1546 | 6.69073
7 | $>2$ | 40th | 17.5915 | 6.39302
8 | $>0$ | 15th | 24.9685 | 6.82091
Table S2: Predictive performance for different definitions of risk-taking behaviour. In detail, we compare thresholds that define risk-taking behaviour given the total number of contacts. As a threshold, we use the percentile of all reported contacts before the lockdown’s announcement, i.e. from 2020-08-01 to 2020-12-01 (see first panel in Fig. S2), and provide the corresponding number of contacts in a separate column. We calculate the PSIS-LOO score, which approximates the out-of-sample predictive performance and rank the results from highest to lowest performing. We consider the score difference significant if it is larger than twice the associated standard error and highlight rows with bold letters that show a non-significant performance difference to the best performing model. Here, the 85th percentile performs best but, all thresholds above the median are only insignificantly worse. Figure S4: Comparison betweeen national-level $R_{t}$ and additional behavioural time-series from the HOPE survey in the upper and lower panel, respectively. The latter represent changes in the mean response to a number of additional survey questions. The dynamics reflects a similar, though inverted, patterns to $R_{t}$ and risk-taking behaviour (see Fig. S3). Therefore, these time-series support our argument that our survey captures early behavioural changes around the lockdown’s announcement. The details: Participants responded on a 1-to-7 scale from ”not at all” / ”completely disagree” to ”to a high degree” / ”completely agree”. The labels in the second panel correspond to the following questions / statements from the survey: (1) The Corona virus is a threat to Danish society. (2) To what extent did you yesterday avoid contacts? (3) To what extent did you yesterday keep 1-2 meters distance to other people? (4) To what extent did you yesterday avoid going to crowded places? (5) To what extent did you yesterday minimize activities where you have contact to other people? We took a seven-day moving average of the mean response value. Table S3 evaluates the predictive performance of the above time-series in terms of PSIS-LOO scores. rank | predictor | score difference | std error
---|---|---|---
0 | risk-taking behaviour | 0.0 | 0.0
1 | avoid crowded places | 3.13429 | 4.65039
2 | avoid contact activities | 8.8516 | 4.99238
3 | avoid contacts | 24.7277 | 7.55522
4 | social distancing | 40.9303 | 9.84551
5 | threat to society | 56.3773 | 8.36957
Table S3: Predictive performance for risk-taking behaviour and additional
behavioural time-series from our survey. In detail, we compare risk-taking
behaviour given the total number of contacts and a threshold at the 85th
percentile against the mean response to additional survey questions. The
latter are presented in Fig. S4 with details about the questions in the
corresponding caption. We calculate the PSIS-LOO score, which approximates the
out-of-sample predictive performance and rank the results from highest to
lowest performing. We consider the score difference significant if it is
larger than twice the associated standard error and highlight rows with bold
letters that show a non-significant performance difference. The PSIS-LOO score
demonstrates that risk-taking behaviour outperforms the predictors derived
from the additional questionnaire responses, though only marginally for _avoid
crowded places_ and _avoid contact activities_. In addition, this result
confirms that our survey captures early behavioural changes in different
aspects of daily life that have an impact on the disease transmission.

##  S5 Extended comparison with mobility data

Figure S5:  Regional-level comparison between $R_{t}$, self-reported survey
data (_Survey_), and mobility data. 1st row: regions of Denmark. 2nd row:
inferred reproduction number from regional hospitalizations with mean and 95%
CI. 3rd-6th row: Regional predictors including (3rd row) survey data, i.e.
risk-taking behaviour on overall contacts and a threshold at the 85th
percentile, (4th row) Google mobility, (5th row) Apple mobility, and (6th row)
telecommunication data (_Telco_). We mark the lockdown’s first announcement,
it’s partial and national implementation with a solid vertical line, a dashed
vertical line and shaded vertical area, respectively. Similar to the national-
level comparison in Fig. 1 of the main text, we observe that the self-reported
survey data shows a significant behavioural change around the lockdown’s
announcement with a similar qualitative dynamics as $R_{t}$. Importantly, this
early signal is not as pronounced in the mobility data. Note also that the
capital region has the largest population and is therefore most influential to
our inference model.

Figure S6: Self-reported survey data outperforms mobility data consistently across all regions of Denmark. 1st column: regions of Denmark. 2nd column: inferred reproduction number $R_{t}$ with 95% CI for the analysis window from 2020-12-01 to 2021-02-01. In the analysis period, we parametrize the reproduction number using a linear combination of (a) self-reported survey data (_Survey_), i.e. risk-taking behaviour on overall contacts and a threshold at the 85th percentile, (b) Google mobility, (c) Apple mobility, and (d) telecommunication data (_Telco_). A solid line, a dashed line and shaded vertical area mark the lockdown’s first announcement, it’s partial and national implementation, respectively. 3rd column: posterior predictive check with observed hospitalizations as scattered black dots. We present the mean and 95% CI of expected hospitalizations as a solid blue line and shaded blue area, respectively. 4th column: inferred regional effect sizes as in Fig. 2 of the main text, i.e., we present the effect of a 50% decrease in the predictors on $R_{t}$, relative to $R_{t}=1.5$. Effect sizes are summarized by their mean, $50\%$, and $95\%$ HPDI. We observe that the inferred reproduction number (2nd column) captures the trend in hospitalizations (3rd column) and is mostly informed by the self-reported survey data (4th column). rank | predictor | score difference | std error
---|---|---|---
0 | Survey | 0.0 | 0.0
1 | Google | 14.496 | 6.5086
2 | Telco | 37.7459 | 9.06595
3 | Apple | 47.7796 | 9.17988
Table S4: Self-reported survey data (_Survey_) demonstrates highest predictive performance compared to Google mobility, Apple mobility and telecommunication data (_Telco_). The details: Survey data refers to risk-taking behaviour on the total number of contacts with a threshold at the 85th percentile. We calculate the PSIS-LOO score, which approximates the out-of-sample predictive performance and rank the results from highest to lowest performing. We consider the score difference significant if it is larger than twice the associated standard error and highlight rows with bold letters that show a non-significant performance difference. See Fig. 1 and S8 for a visual comparison of the time-series data. Figure S7: Negative cross correlation between pooled effect sizes highlights co-linearity of predictors. We compare (a) self-reported survey data (_Survey_), i.e. risk-taking behaviour w.r.t the total number of contacts and a threshold at the 85th percentile, to mobility data from (b) Google, (c) Apple and (d) telecommunication providers (_Telco_). The diagonal shows raw posterior effect sizes, i.e., unlike Fig. 2 and S6, we present the values as they are, without transforming them into a more interpretable form. The upper non-diagonal fields give the Pearson’s correlation coefficient, whereas the lower non-diagonal fields present more details: a scatter plot of sampled effect sizes from two different predictors with contours of constant density and a linear regression line, which visualizes the correlation. Figure S8: National-level comparison between $R_{t}$ and individual data streams from Google and Apple. 1st row: inferred reproduction number from national hospitalizations. 2nd row: Individual data streams from Google mobility trends GOOGLE . We excluded the time-series ”parks” because of too many missing values on the regional level. The combined time-series _Google_ NOU21  includes _transit station_ , _workplaces_ , _retail & recreation_, _grocery & pharmacy_. 3rd row: Individual data streams from Apple mobility trends APPLE . We exclude _transit_ because of too many missing values on the regional level. The combined time series _Apple_ NOU21  both remaining data streams. The comparison reveils that individual data streams from Google vary substantially whereas _driving_ and _walking_ from Apple show a similar dynamics. In Table S5, we compare the predictive performance of individual mobility data streams with risk-taking behaviour from our survey. rank | predictor | score difference | std error
---|---|---|---
0 | Survey | 0.0 | 0.0
1 | Google (retail & recreation) | 4.96134 | 5.47153
2 | Google (grocery & pharmacy) | 12.108 | 5.75795
3 | Google (transit stations) | 12.2169 | 5.70765
4 | Google | 14.496 | 6.5086
5 | Google (residential) | 23.4672 | 7.35413
6 | Apple (driving) | 37.7415 | 8.23638
7 | Apple | 47.7796 | 9.17988
8 | Apple (walking) | 58.9249 | 9.99233
9 | Google (workplaces) | 72.2505 | 11.2305
Table S5:  Self-reported survey data (_Survey_) demonstrates highest
predictive performance compared to individual data streams from Google and
Apple mobility, though, the performance difference to Google’s _retail &
recreation_ is non-significant. The details: Survey data refers to risk-taking
behaviour on the total number of contacts with a threshold at the 85th
percentile. We calculate the PSIS-LOO score, which approximates the out-of-
sample predictive performance and rank the results from highest to lowest
performing. We consider the score difference significant if it is larger than
twice the associated standard error and highlight rows with bold letters that
show a non-significant performance difference to the best performing model.
See Fig. S8 for a visual comparison of the time-series data. Interestingly,
the score difference to Google’s _retail & recreation_ is non-significant.
This observation appears plausible given (a) the increased risk for super-
spreading events in retail and recreation spaces and (b) our results on risk-
taking behaviour in different social contexts: Risk-taking behaviour towards
friends and strangers predict hospitalizations best (see Fig. 3 and Table S6)
and these contacts are also likely the dominant transmission routes in
Google’s _retail & recreation_ setting.

##  S6 Extended comparison between contact types

Figure S9: Visual national-level comparison between $R_{t}$ and context-depending risk-taking behaviour. 1st row: Reproduction number $R_{t}$ inferred from national hospitalizations. 2nd column: risk-taking behaviour towards friends, strangers, and colleagues with a threshold at the 85th percentile. 3rd row: risk-taking behaviour towards family members. Except for the latter, risk-taking behaviour show high co-linearity, which explains their comparable predictive performance in Table S6,S7 and negatively correlated effects in Fig. S12. In contrast, contacts to family member spike around the Christmas holidays without notable effect on $R_{t}$. This low effect on disease transmission can be in part explained by (a) increased awareness and safety measures such testing and isolation before family visits and (b) the low overall group size: family gathering remain comparably small and thus the potential for super-spreading remains limited. rank | predictor | score difference | std error
---|---|---|---
0 | friends | 0.0 | 0.0
1 | strangers | 0.685311 | 8.01945
2 | colleagues | 21.3289 | 10.0893
3 | family | 104.258 | 12.0443
Table S6: Risk-taking behaviour towards friends and strangers predict the
observed hospitalizations best and colleagues performs only marginally worse.
In detail, we define risk-taking behaviour with a threshold at the 85th
percentile, calculate the PSIS-LOO score, which approximates the out-of-sample
predictive performance, and rank the results from highest to lowest
performing. We consider the score difference significant if it is larger than
twice the associated standard error and highlight rows with bold letters that
show a non-significant performance difference to the best performing model.
Here, risk-taking behaviour towards colleagues and family members outside the
household perform significantly worse. The former, however, is only marginally
significant and a slightly higher threshold already improves performance (see
Table S7). The latter, i.e. risk-taking behaviour towards family members,
performs significantly worse for all considered thresholds, which can be
explained by (a) increased awareness and safety measures such testing and
isolation before family visits and (b) the low overall group size: family
gathering remain comparably small and thus the potential for super-spreading
remains limited (see Fig. S9 for a visual comparison).

Figure S10:  Regional-level comparison between $R_{t}$ and risk-taking behaviour in different social contexts. 1st row: regions of Denmark. 2nd row: inferred reproduction number from regional hospitalizations with mean and 95% CI. 3rd-6th row: Regional predictors including risk-taking behaviour towards friends, strangers, colleagues, and family members outside the household, respectively, with a threshold at the 85th percentile. The solid vertical line, dashed vertical line and shaded area mark the lockdown’s first announcement, it’s partial implementation and national implementation, respectively. Similar to the national-level comparison in Fig. S9, we observe that all predictors, excluding risk-taking behaviour towards family members, show a significant behavioural change around the lockdown’s announcement with a similar qualitative dynamics as $R_{t}$. In contrast, family gatherings around Christmas appear to have little effect on $R_{t}$, possibly due to (a) increased awareness and safety measures such testing and isolation before family visits and (b) the small overall group size: family gathering remain comparably small and thus the potential for super-spreading remains limited. Note that the capital region has the highest population and thus dominates inference result. Figure S11:  Regional-level inference results. 1st column: regions of Denmark. 2nd column: inferred reproduction number $R_{t}$ with 95% CI for the analysis window from 2020-12-01 to 2021-02-01. In the analysis period, we parametrize the reproduction number using a linear combination of risk-taking behaviour towards friends, strangers, colleagues, and family members with a threshold at the 85th percentile. The solid vertical line, dashed vertical line and shaded area mark the lockdown’s first announcement, it’s partial implementation and national implementation, respectively. 3rd column: posterior predictive check with observed hospitalizations as scattered black dots. We present the mean and 95% CI of expected hospitalizations as a solid blue line and shaded blue area, respectively. 4th column: inferred regional effect sizes as in Fig. 3 of the main text, i.e., we present the effect of a 50% decrease in the predictors on $R_{t}$, relative to $R_{t}=1.5$. We summarize effect sizes by their mean, $50\%$, and $95\%$ HPDI. The inferred reproduction number (2nd column) captures the hospitalization dynamics (3rd column) and is mostly informed by risk-taking behaviour towards friends and strangers. Figure S12: Negative cross correlation between pooled effect sizes highlights co-linearity of predictors. We compare risk-taking behaviour with a threshold at the 85th percentile towards different social groups: contacts to strangers, family members outside the household, friends, and colleagues. The diagonal shows raw posterior effect sizes, i.e., unlike Fig. 3 and S10, we present the values as they are, without transforming them into a more interpretable form. The upper non-diagonal fields give the Pearson’s correlation coefficient, whereas the lower non-diagonal fields present more details: a scatter plot of sampled effect sizes from two different predictors with contours of constant density and a linear regression line, which visualizes the correlation. The figure shows that pooled effect sizes for risk-taking behaviour towards strangers, friends, and colleagues are negatively correlated and thus these predictors are co-linear. Note that family-related effect sizes are highly concentrated around zero and thus cross-correlation is not informative. rank | predictor | score difference | std error
---|---|---|---
0 | friends | 0.0 | 0.0
1 | strangers | 9.41977 | 8.27323
2 | colleagues | 14.2355 | 9.35753
3 | family | 102.801 | 12.1247
Table S7:  If we define risk-taking behaviour with a threshold at the 90th
percentile, i.e., a slightly higher value compared to Table S6, then contacts
to friends, strangers, and colleagues perform similarly well. In detail, we
calculate the PSIS-LOO score, which approximates the out-of-sample predictive
performance, and rank the results from highest to lowest performing. We
consider the score difference significant if it is larger than twice the
associated standard error and highlight rows with bold letters that show a
non-significant performance difference to the best performing model. Here,
only risk-taking behaviour towards family members outside the household has a
significantly lower PSIS-LOO score (see Table S6 for a comparison and addition
discussion). See also Fig. S13 for an alternative comparison in terms of
effect sizes.

Figure S13:  If we we define risk-taking behaviour with a threshold at the
90th percentile, i.e., a slightly higher value than in Fig. 3 of the main
text, then risk-taking behaviour towards colleagues appears also significant.
In detail: For the analysis window from 2020-12-01 to 2021-02-01, we
parametrize $R_{t}$ with a linear combination of predictors, namely: risk-
taking behaviour towards friends, strangers, colleagues, and family members
outside the household. Then, we infer the coefficients that give the best
model fit to the observed hospitalizations. Finally, we present the effect of
a $50\%$ reduction for all predictors on $R_{t}$ relative to $R_{t}=1.5$.
Panel A: Pooled effect sizes across all regions. We present the posterior
density, 95 % HPDI (highest probability density interval), and median as
filled curves, horizontal bars and white circles, respectively. Panel B:
regional effects with the same summary statistics as in panel A. We find that
only risk-taking behaviour towards family members is insignificant, which
confirms our analysis for out-of-sample predictions in Table S7

.