**1. Introduction**

4

The numbers of COVID-19 cases, both infections and casualties, are increasing daily all over the world, and concerns about their effects show no decrease. Even with the start of vaccination programs, it has not been possible to break the advance of the numbers, primarily because the speed of vaccination is asymmetric in different countries, but also because, contrarily to some respiratory diseases in the past, the spread between countries was higher [1–3]. With various negative economic and financial effects (see References [4–10]), COVID-19 also has several other consequences in people's lives, such as fear and depression [11,12], suicide trends [13] or in mental health [14,15].

The substantial effects of COVID-19 are related to the lockdowns that countries had to impose to control the spread of the disease. According to Reference [16], human behavior, among other factors, could contribute to respiratory viral infections, even more in a context where the superspreading conditions are not fully known [17]. However, it is crucial to reduce the number of social contacts, as complete vaccination programs are absent or not ye<sup>t</sup> fully developed, and social-distancing measures could be the key in helping to solve the problem [18].

The spread of COVID-19 could be related to several factors. For example, Reference [19] identified several of these factors in assessing community risk factors in Catalonia, Spain, such as air pollution, population density, demographic and socioeconomic conditions, or even land use. In addition to these factors, which could affect the incidence of the disease in a general way, the authors also identify other factors related to the possible individual prevalence of the disease, such as the existence of comorbidities.

The existence of social contacts could be proxied by mobility data [20], with frameworks such as Google's Community Mobility Reports (CMR) being able to measure that

**Citation:** Casa Nova, A.; Ferreira, P.; Almeida, D.; Dionísio, A.; Quintino, D. Are Mobility and COVID-19 Related? A Dynamic Analysis for Portuguese Districts. *Entropy* **2021**, *23*, 786. https://doi.org/10.3390/ e23060786

Academic Editors: Ryszard Kutner, Christophe Schinckus and H. Eugene Stanley

Received: 20 May 2021 Accepted: 19 June 2021 Published: 21 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

mobility, as it measures citizens' mobility according to different types (for more details about CMR, see References [21,22]).

The use of CMR and its effects in COVID-19 has already been made using different approaches; see, for example, the studies of References [20,23–28], which, at a country level, found that the reduction of the mobility has a direct impact on the decrease of the infections. Reference [29] also confirms these trends and adds that reducing cases due to mobility restrictions has a very significant effect on a 2-week basis.

At a regional level, we can find the studies of References [30,31], both for the US. Although both find relevance in the effect of mobility on controlling the disease, Reference [30] finds differences between urban and rural locations, while Reference [31] identifies that population density has different implications in the reduction of mobility (higher density has more impact on the reduction of mobility, for example, in stores). In Poland, Reference [32] concluded that the restrictions helped control COVID-19, although with the difference between regions, related to the strictness of state restrictions.

During January 2021, Portugal was constantly in the news, as it was considered the worst country in the world regarding the infections and death rate (see https://www. politico.eu/article/portugal-coronavirus-rate-surge/, accessed on 19 May 2021). The lifting of some restrictions during the Christmas season may have compounded this tragic scenario. In this context, our purpose is to analyze, in a dynamic way, and based on daily data, the relationship between citizens' mobility and new COVID-19 infections, using regional-level data, in this case, for Portuguese districts. Our main objective is to assess the relevant relationship between the number of new infections of COVID-19 and citizen mobility. Moreover, we also want to distinguish between the different types of mobility. Differentiating the analysis between regions could give important insights for possible future decisions about new lockdowns or lifting of restrictions.

The implementation of non-pharmacological measures has a relevant impact on the control of the dissemination of COVID-19. In Portugal, the introduction of mandatory personal protective equipment (PPE) such as masks, or the instructions for frequent use of alcohol gel and washing hands, among others, started with the beginning of the pandemic in March/April 2020. Since then, the use of PPE has remained mandatory, and the nonpharmacological measures have not changed significantly.

In this paper, the mobility is measured considering Google CMR reports, and the relationship between mobility and new cases is assessed through the detrended crosscorrelation analysis correlation coefficient. This non-linear framework has the ability to capture the relationship between variables for different timescales, which could give important information about the number of days needed to reduce infections. Moreover, we also propose the use of a sliding windows approach, which allows analysis of the evolution of the relationship over time.

Our main results corroborate that mobility is correlated with the number of new COVID-19 cases. However, the mobility correlation is not equal for the different typologies: for example, mobility in retail, recreation and groceries seems to have a higher correlation, while in general the mobility in workplaces shows little relationship. Despite the temporal evolution of the relationship, confirming that the lift of restrictions at Christmas had a highly significant impact on the increase of new COVID-19 cases, we also find that the impacts of the mobility are different across districts.

The remainder of the paper is organized as follows: in Section 2, both data and methodology are presented, with the results being present in Section 3, while Section 4 provides discussion and conclusions for the study.

#### **2. Data and Methodology**

Since the outbreak of COVID-19, and until 13 April 2021, almost 138 million cases were reported worldwide, with almost 3 million deaths. Portugal has about 828,000 cases and around 17,000 deaths. For cases of disease, information is available from the Portuguese Health Ministry, through Sistema Nacional de Vigilância Epidemiológica (SINAVE), with the complete set of registered cases until 28 February 2021 (due to data availability). Until this day, Portugal has had a total of 805,140 cases. Intending to analyze the relationship between mobility and COVID-19 in the different Portuguese districts, we considered only the information which is registered in Portuguese mainland districts due to the availability of data about mobility. In total, the number of cases of the districts is 775,954. All the data were transformed in daily incidence for each district to perform the correlational analysis with the information from Google CMR. In these reports, it is possible to retrieve information about six distinct mobility indices: (i) retail and recreation (I1); (ii) groceries and pharmacies (I2), (iii) parks (I3), (iv) transit stations (I4), (v) workplaces (I5) and (vi) residential areas (I6). For more information about the indices and the places where mobility is referred to, see https://www.google.com/covid19/mobility/index.html?hl=en (accessed on 19 May 2021).

Daily data for these indices were retrieved for Portuguese districts from 15 February 2020 to 28 February 2021, in a total of 380 observations. Some districts do not have information for the mobility indices in some days of August and September 2020, implying that the sample is smaller for those districts (355 observations). The information about the number of cases and the number of observations for each district are identified in Table 1. Moreover, as some districts present missing information for some indices, the correlations were calculated for the remainder, where data are available.

**Table 1.** Total number of COVID-19 cases for each district and the number of observations considered in the analysis.


To perform our correlational analysis, we use the detrended cross-correlation analysis coefficient (*ρDCCA*), proposed by Reference [33] and derived from the work of Reference [34]. The DCCA measures the long-range cross-correlation between two series *Yi* and *Xi* consisting on the sequence of *k* = 1, 2, ... , *N* observations. The first step of the DCCA consists of the calculation of the profiles:

$$Y\_k = \sum\_{i=1}^k (y\_i - \langle y \rangle) \text{ and } X\_k = \sum\_{i=1}^k (x\_i - \langle x \rangle) \tag{1}$$

with . as the mean operator. Those profiles are then divided into (*N* − *n*) overlapping boxes, from *n* = 4 to *n* = *N*/4 and for each box, based on the ordinary least squares, local trends *<sup>Y</sup>*<sup>1</sup>*k*,*<sup>i</sup>* and *<sup>X</sup>*<sup>1</sup>*k*,*<sup>i</sup>* are calculated, for future detrend of the profiles *Yk* and *Xk*. With the local trends, the covariance of the residuals of each box is calculated as follows:

$$f\_{xy}^{2}(n,i) = \frac{1}{(n+1)} \sum\_{k=1}^{i+n} \left(X\_k - \check{X}\_{k,i}\right) \left(\mathbb{Y}\_k - \check{\mathbb{Y}}\_{k,i}\right). \tag{2}$$

Considering the information of all the set of *N* − *n* boxes, the DCCA covariance is calculated as follows:

$$F\_{xy}^2(n) = \frac{1}{(N-n)} \sum\_{i=1}^{N-n} f\_{xy}^2(n, i), \tag{3}$$

which was used by Reference [33] to obtain the correlation coefficient given by the following:

$$
\rho\_{DCCA} = \frac{F\_{xy}^2(n)}{F\_x^2(n)F\_y^2(n)}.\tag{4}
$$

The denominator of *ρDCCA* consists of the fluctuation functions of the detrended fluctuation analysis of Reference [35], which analyzes the long-range behavior of each time series individually.

The *ρDCCA* is a non-linear correlation coefficient, robust to the presence of nonstationarity, and confirms the property of −1 ≤ *ρDCCA* ≤ 1 according to [36–39] and is testable according to [40]. Moreover, this is a multiscale correlation coefficient, allowing for the analysis of the behavior between variables in different time periods. Despite the statistical properties previously referred to, the robustness of the correlation coefficient is confirmed by its use in different research areas (see, for example, [41–46], among others).

In this analysis, the *ρDCCA* will be calculated using a sliding windows approach to analyze the evolution of the correlation over time, using windows of 250 observations. In Table 2 we present the critical values to test the null hypothesis of absence of correlation, considering 250 observations, as it is the dimension of the samples used in the analysis.

**Table 2.** Critical values to test the *ρDCCA* considering time series of 250 observations and different timescales, considering a confidence level of 95% (source: Reference [40]).

