Understanding spatiotemporal patterns of COVID-19 incidence in Portugal: a functional data analysis from August 2020 to March 2022

doi:10.21203/rs.3.rs-2972800/v1

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Understanding spatiotemporal patterns of COVID-19 incidence in Portugal: a functional data analysis from August 2020 to March 2022

https://doi.org/10.21203/rs.3.rs-2972800/v1

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Since the emergence of SARS-CoV-2 and the pandemic, massive amounts of daily data on incidence are being collected by governments and public health authorities, providing relevant information about the dissemination of pandemic in terms of its virological evolution and spatiotemporal distribution of cases, hospitalization, and deaths. We propose a novel approach combining functional data analysis and unsupervised learning algorithms to extract meaningful information about the main spatiotemporal patterns underlying SARS-CoV-2 incidence.

We modelled the daily COVID-19 confirmed cases by municipality as a function of time using functional principal component analysis to describe their temporal evolution in order to outline the main temporal patterns of variability. Municipalities were classified according to their spatiotemporal similarities through hierarchical clustering adapted to spatially correlated functional data. The proposed approach is applied to mainland Portugal with data collected between August 2020 and March 2022 by municipality.

The results obtained discriminate northern and coastal regions from southern and hinterland, and the effects in 2020-21 from the effects in 2021-22 autumn-winter seasons. Spatiotemporal patterns and classification of municipalities agree with results reported by other works and provides proof-of-concept that the proposed approach can be used to detect the main spatiotemporal patterns of disease incidence. The novel approach extends and refines existing exploratory tools for spatiotemporal analysis of public health data.

Health sciences/Diseases

Health sciences/Diseases/Infectious diseases

Earth and environmental sciences/Ecology/Ecological epidemiology

Physical sciences/Mathematics and computing/Applied mathematics

Physical sciences/Mathematics and computing/Statistics

COVID-19 mapping

spatiotemporal modelling

functional data analysis

geostatistics

spatial clustering

Due to the rapid spread of SARS-CoV-2 worldwide, with the number of positive cases and affected countries growing exponentially, a pandemic was declared on March 11, 2020 ¹. From then, more than 650 million cases were confirmed (https://covid19.who.int/, last accessed Dec 27, 2022), and forced governments and researchers around the world to collect, analyse and interpret COVID-19 data to understand the pandemic evolution at different places and times. At a population level, where the epidemic occurs, several factors do promote the spread of infection such as: climate ^2,3; socio-economic conditions ^4,5; or demographic characteristics (e.g., gender or age-structure). In addition, governmental interventions (e.g., social distancing, vaccination) are key to control and mitigate the spread of disease^6,7. As populations live in vast geographical domains with distinct dynamics, the distribution of COVID-19 incidence rates across time and space may differ substantially. At the same time, when modelling the spatiotemporal evolution of the disease we should consider the fact that regions close to each other, culturally, economically, or geographically, tend to have similar COVID-19 incidence dynamics as their populations tend to be exposed to similar conditions and lifestyles ^8–11.

As a result of this complex diversity of spatiotemporal patterns, it is difficult to describe and understand the underlying dynamics of COVID-19 incidence that are common across regions over time. Their spatiotemporal distribution has been usually recorded as time-series (sequentially structured data set) observed in different regions. Quantitative approaches using time-series data have been applied to model the temporal evolution of COVID-19 new cases (e.g., Carroll et al., 2020) or the mortality over different regions ¹³. These methods could be complemented with exploratory data analysis tools (e.g., dimension reduction techniques or unsupervised classification methods) to detect patterns or to discriminate differentiating features ^14,15. Alternatively, functional data analysis (FDA) transforms time-series observations into smoothing functions providing a model for the underlying process giving rise to them. Classical statistical tools like principal component analysis or hierarchical clustering methods have been adapted to functional data and provide an appealing option for the spatiotemporal modelling. These methods have been used for exploratory analysis, description, and classification of COVID-19 mortality, providing useful insights about patterns of mortality across different regions of Italy ¹⁶, by COVID-19. However, these exploratory tools rely on the assumption of independence between functional data and do not consider the spatial distribution observed in the data. In recent years promising approaches emerged in applied sciences to adapt exploratory tools considering the spatial structure of functional data ¹⁷.

In this paper we propose a novel approach combining a set of statistical tools from the fields of functional data analysis and unsupervised learning algorithms (i.e., functional principal component analysis and hierarchical classification of spatially correlated functional data) to unveil the main spatiotemporal patterns underlying COVID-19 incidence in mainland Portugal. The methods rely on combining spatial distribution of incidence rates across regions along with their temporal evolution while considering the spatial dependence structure in the data. This approach provides a comprehensive tool to identify major regional patterns of incidence. The analysis is illustrated with the daily number of newly COVID-19 cases notified for 570 consecutive days at municipality level in mainland Portugal. The period analysed was characterized by bundle of Non-Pharmacological Interventions (NPI), where restrictive measures were balanced against the impact on populations health, social and economic stability. We first detail the proposed methods, following with its application to mainland Portugal. The results are then discussed, before presenting the main conclusion of this study.

We analyzed, from a functional data perspective, 278 time-series with daily COVID-19 incidence data collected from all mainland Portuguese municipalities (i.e., 278 functions). The proposed modelling method assumes time-series data are derived from underlying processes that can be modeled with a set of smoothing functions (curves). Based on this, the unsupervised learning methods functional principal component analysis (FPCA) and hierarchical classification of spatially correlated functional data were combined to extract spatial patterns of COVID-19 from the set of incidence curves. With FPCA we analyzed the dominant types of variation among municipality curves, while their patterns were classified using a dissimilarity matrix which considers spatial covariance among functional data. Each one of these steps is described in the following subsections. All modelling and data visualization was performed in open-source software R ¹⁸ with packages fda (J. O. Ramsay et al., 2021)(J. O. Ramsay et al., 2021 for functional data and functional principal component analysis, geofd ²⁰ for hierarchical classification of spatially correlated functional data and ggplot2 ²¹ for data visualization.

2.1. Ethics declaration

The Portuguese Directorate-General of Health (DGS) provided the data to conduct this study in accordance with the Article 39 of Legislative Decree 2-B/2020 of 2 April 2020. The COVID-19 data used in this study is anonymized which implies the removal of any personal data, therefore informed consent was not required. The analysis was performed on aggregated-level data and in accordance with relevant guidelines and regulations.

2.2. Data collection

The time-series reporting the daily number of newly confirmed COVID-19 cases between August 14, 2020, and March 6, 2022 ($T$ = 570 days), at the municipality level (N = 278 municipalities) in mainland Portugal, were obtained from the Portuguese Epidemiological Surveillance System (SINAVE) database and provided by DGS.

Estimated population at risk by municipality were supplied by Statistics Portugal (https://www.ine.pt/). The daily indicator used for analysis was the 14-day incidence rate of SARS-CoV-2 by municipality, resulting from the quotient of number of new confirmed COVID-19 cases notified (numerator) in the previous 14 days and population at risk (denominator) per 100,000 population (hereinafter referred to as "daily incidence rates").

We analysed $N$ time-series of daily incidence rates ($r$) observed at $T$ time points, that can be represented as ${\left\{{r}_{i}\left({t}_{j}\right):j=1,\dots ,T\right\}}_{i=1}^{N}$ where ${r}_{i}\left({t}_{j}\right)$ represents incidence rate at day $j$ (with $j=1, ..., T)$ in municipality $i$ ($i=1, ..., N)$.

2.3. Functional data analysis

Functional data analysis (FDA) was used to model and describe variations of COVID-19 incidence rates by municipality as a function of time. Under this approach discrete data are converted into smooth continuous curves assuming the existence of underlying functions defined over a compact (continuous) temporal domain from which time-series data are observed. The approach provides a parsimonious representation of data ²² and a natural way to thinking through curve smoothing procedures that allows for data noise reduction ^23,24. The underlying smoothing function for the $i$-th municipality can be defined in the compact interval $\left\{{r}_{i}\left(t\right):t\in \left[0,T\right]\right\}$ and assumed to belong to a Hilbert space ²⁵ by means of a basis system (i.e., a linear combination of basis functions):

$${r}_{i}\left(t\right)=\sum _{j=1}^{J}{c}_{ji}{{\Phi }}_{j}\left(t\right)={\mathbf{c}}_{i}^{T}{\Phi }\left(t\right)$$

where ${\Phi }\left(t\right)$ is a vector of $J$ standard basis functions (e.g., polynomials, splines, sine, and cosine) and ${\mathbf{c}}_{{i}}$ is a vector of length J with the coefficients used to build functional data ${r}_{i}\left(t\right)$. The vector ${\mathbf{c}}_{{i}}$ can be estimated by minimization of the sum of squared errors (i.e., least squares estimator) with a roughness penalty (regularization) that looks for an optimal balance between data fit to original curves and smoothness. The degree of smoothness depends on the number of $J$ basis functions considered. On one hand, if the number $J$ is large the sum of squared errors will be small, but the risk of overfitting increases. On the other hand, if $J$ is small, relevant information about local features might be missed. According to the characteristics of the available time-series data, we have chosen to use a cubic B-spline basis every 12 days, resulting in a total of 48 knots equally spaced, to approximate the functional form. A smoothing parameter of 0.25 was applied to the estimated functional parameters, which minimized the average generalized cross-validation error across the 278 incidence rate curves.

2.4. Functional principal components analysis

In this paper we used functional principal component analysis (FPCA) to explore the variance among COVID-19 incidence curves. FPCA expands the application of classical principal component analysis (PCA), now in the context of functional data analysis ²⁴. Just like in PCA, FPCA looks for the dominant types of variation (maximizing variability, after subtracting the mean from each observation), by finding the decomposition of the covariance function that maximizes variance with minimal number of (functional) principal components. Mean corrected curves, ${{r}_{i}\left(t\right)}^{*}$, are obtained after adjustment for the (pointwise) estimated mean curve $\stackrel{-}{r}\left(t\right)= \frac{1}{n}\sum _{i=1}^{N}{r}_{i}\left(t\right)$:

$${{r}_{i}\left(t\right)}^{*}={r}_{i}\left(t\right) - \stackrel{-}{r}\left(t\right)$$

The covariance function can be estimated by discretising ${{r}_{i}\left(t\right)}^{*}$ into a regular grid of equally spaced values in the interval $t\in \left[0,T\right]$ to form a data matrix suited to compute the covariance as:

$$v(t,s)=\frac{1}{N-1}\sum _{i = 1}^{N}{{r}_{i}\left(t\right)}^{*}{{r}_{i}\left(s\right)}^{*}$$

Once the decomposition is solved, eigenvalues and eigenfunctions are obtained and new functional components can be interpreted as different types of variation - the implementation details of FPCA can be found in Ramsay and Silverman ²⁴.

The functional principal component scores are linear combinations of functional data, ${{r}_{i}\left(t\right)}^{*}$, where eigenfunctions are their coefficients. For municipality $i$ and principal component $\alpha$, the functional principal score, ${z}_{i\alpha }$, is given by the integral (4):

$${z}_{i\alpha }=\int {\xi }_{\alpha }\left(t\right){{r}_{i}\left(t\right)}^{*}dt,$$

where ${\xi }_{\alpha }\left(t\right)$ is the eigenfunction associated to $\alpha$.

Just like in classical PCA, the first principal components form a subspace representing the main types of variation found in the original data. Assuming the first components provide a satisfactory approximation, they allow new insights and clearer interpretations about the covariance structure among COVID-19 incidence curves at any instant of time. Once again, the strategies applied in classical PCA for the selection of main types of variation to interpret can also be applied in FPCA (i.e., scree plot or the proportion of variability explained). In our application example we have chosen to consider the first three components as they explained more than 70% of overall variability (see Fig. 3).

2.5. Clustering spatial functional data

Hierarchical clustering is traditionally used in multivariate data analysis for unsupervised classification. In this paper we used an extension of the algorithm that is appropriate to deal with spatially correlated functional data. As in classical approach, hierarchical clustering uses agglomerative or divisive strategies ²⁶ and a dissimilarity measure between functional data, which informs which clusters should be combined ²⁷. Euclidean distances (L2-norm) are considered to build a dissimilarity matrix and the dissimilarity/distance between two curves ${{r}_{i}\left(t\right)}^{*}$ and ${{r}_{j}\left(t\right)}^{*}$ can be given by (5):

$${d}_{ij}=\sqrt[2]{{\int }_{a}^{b}{\left[{{r}_{i}\left(t\right)}^{*}-{{r}_{j}\left(t\right)}^{*}\right]}^{2}dt}$$

To deal with spatial dependency, we used geostatistical methods allowing for clustering of contiguous municipalities with similar curves patterns ²². Trace-variogram (6) is an adaptation of the classical variogram ²⁸ and was used to capture the spatial dependence between functions, i.e., measure an average (expected) increase in functional data value for a given increase in the geographical domain:

$$\gamma \left(h\right)=\frac{1}{2}\mathbb{E}\left[{\int }_{a}^{b}{\left[{{r}_{i}\left(t\right)}^{*}-{{r}_{j}\left(t\right)}^{*}\right]}^{2}dt\right] with h=‖{{r}_{i}}^{*}-{{r}_{j}}^{*}‖$$

Municipalities are geographically referenced by the cartesian coordinates of their geometric centroids. Once the trace-variogram was estimated, we could fit a theoretical model (e.g., exponential, spherical or gaussian variogram model) to the experimental trace-variogram to characterize the spatial dependency between curves. The best fitted variogram was chosen to weight the dissimilarity matrix of functional data (7).

$${d}_{ij}^{w}={d}_{ij}*\gamma \left(h\right)$$

In this paper, we clustered municipalities using 3 functional principal components. The dissimilarity matrix was weighted by an exponential variogram model.

In autumn-winter periods (i.e., between October and February), especially during season 2021-22,) incidence rates exhibited higher variability in amplitude, when compared to the other seasons (see Fig. 1A).

The original time-series data were transformed into functional data composed of 278 curves, each corresponding to 570 days of incidence rates per municipality (Fig. 1B). The smoothed correlation and variance-covariance plot surfaces are illustrated in Fig. 2, showing higher correlations between consecutive or close days. The higher covariance values were observed in winter periods, as illustrated by the variance-covariance plot.

To find the dominant types of variation in the functional data, we first centered the curves (mean adjustment) and computed principal components. Over 70% of overall variability observed in the curves was explained by the first 3 principal components (see Fig. 3).

Figure 4 illustrates the results obtained for the first three functional principal components (accounting for 72% of overall variability), as they represent the main types of variation observed in the data.

The first functional principal component (PC) represents 47% of overall variability and defines the main type of variation shared by the 278 incidence curves. It reflects the major trends in the curves, as seen in Fig. 1, with higher incidence amplitudes in autumn-winter seasons and more stable evolutions in spring-summer seasons. We may refer to this component as the “global effect”. Looking at the first PC curve we can identify four main phases of pandemic evolution, the first phase (August to October 2020) shows low differences in pandemic evolution across municipalities, in the second (October 2020 to January 2021) differences between curves increased and reached a maximum around mid-November 2020 and, after that, began to decrease till mid-January 2021. The third phase (January 2021 to December 2021) is again characterized by low differences across municipalities. In the last phase (December 2021 to February 2022), considerable differences in incidence curves occurred across municipalities, as the pandemic curve increased severely and peaked in February 2022.

The second functional PC retained 16% of overall variability. The curves sharing this type of variation contributed to the decrease of the average incidence observed in the winter 2020-21 and between Christmas 2021 up to the end of January 2022. The municipalities with high negative scores in this component were the most relevant contributors for lowering the mean incidence rate observed in winter periods. This functional PC will be referred as “preventive measures effect”.

The third functional component accounts for 9% of overall variability and corresponds to the increase of incidence caused by curves with high incidence rates in autumn-winter periods, especially in autumn 2020-21. This functional PC was negative until December 2020 and since then, became positive and kept increasing to reach the peak on late-January 2021. After that, the curve started decreasing but stayed positive till the end of February 2021. Over the autumn-winter period 2021-22, it presents a similar pattern to the second functional component except for a smaller amplitude. This component will be referred as the “outbreak effect”. Remaining components represent small amounts of variation within the original 278 incidence curves (e.g., 6% and 4% for the fourth and fifth components, respectively), are more difficult to interpret. Consequently, from now on the results to be presented are restricted to the first three principal components.

From a spatial perspective, it is interesting to visualize the functional PC scores, as they may contribute to find geographical patterns, clusters or to detect outliers.

For example, on the first PC (viewed as the “global effect”) it is clear that the highest scores correspond to municipalities clustered along the northern coast and adjacent regions, where municipalities tend to be small and close to each other (Fig. 5A). Municipalities from those regions were the most relevant contributors for increasing the average incidence curve as they reported high incidence rates over autumn-winter seasons. On the other hand, the second functional PC (interpreted as the “preventive measures effect”) showed highest negative scores in municipalities dispersed along Portugal’s hinterland where municipalities are characterized by being separated by longer distances and having sparse populations, plus the municipalities from the region of Lisbon, that have been under severe restrictive measures (e.g., mobility restrictions, lockdown, and school closures) during pandemic (Fig. 5B). In this case, negative scores are linked to municipalities that contributed to lower average incidence rates during autumn-winter seasons. In the third component (Fig. 5C), interpreted as the “outbreak effect”, the municipalities with high positive scores are spread over the country in places where population sizes are small, and disease outbreaks (which occurred mainly in nursery houses) have been reported.

To simplify the interpretation of principal components scores, we further classify municipalities into homogenous groups using the hierarchical classification algorithm as described above. To weight the dissimilarity matrix we used an Exponential variogram model with a nugget-effect of 31131 (38%), an estimated partial sill equal to 51313 (62%) (units: 14-day cumulative number of new cases/10⁵ habitants)² and an estimated range parameter of 227 kilometers (Fig. 6), which is close to the width of mainland across east-west.

The estimated parameters indicates that functional data from municipalities separated up to 227 km are still correlated, but almost two fifths of the variability (38%) is not explained by the distances separating them (i.e., the proportion of variability with no spatial pattern). The dissimilarity matrix used is based on the Euclidean distances among the coefficients of the cubic-spline basis functions, while the ward-D2 linkage is applied as the agglomeration method. The dendrogram (not presented here) obtained was splitted in different ways to find an interpretable number of partitions. The tree cutting procedure (i.e., pruning) was guided by visual inspection of results and attempted to find a number of partitions neither too small to identify any spatial clustering nor too big to miss the effect of spatial dependence found in the trace-variogram of functional data. As a result, the final dendrogram was cut into six groups, or clusters, where it is possible to detect spatially correlated patterns of municipalities groups according to the clusters assigned (Fig. 7).

After classification of functional data, we found clusters of municipalities located along the northern coast and adjacent regions (cluster 1), central coast (cluster 2), hinterland regions from north to south plus southern coast (clusters 3, 4 and 5), and the metropolitan area of Lisbon plus municipalities lying deep along the hinterland (cluster 6). The curves from municipalities classified in each cluster are presented in Fig. 8 along with the estimated mean national curve. The differences between clusters are mainly defined by the average height of the curves (i.e., amplitude) in autumn-winter seasons. We note, for example, the differential incidence between the curves in cluster 1 and cluster 6 in autumn-winter season 2021-22.

The scatterplots of the first three principal components scores (Fig. 9) capture main patterns of variations found in incidence curves. In the scatterplots, municipalities scores are classified according to the groups returned by hierarchical clustering to facilitate the interpretation of results.

Figure 9A illustrates the projection of municipalities scores on the first two functional PCs. The first component scores (fPC 1) correspond to the “global-effect” and separates most of the municipalities from clusters 1 (red) and 5 (grey) with positive scores, from municipalities of clusters 4 (yellow) and 6 (light blue) with negative scores. Positive scores are associated with incidence curves above the overall average incidence, while negative scores are associated to incidence curves below the overall average. Most of municipalities from clusters 1 and 5 are located along the northern coast and adjacent regions, while municipalities from clusters 4 and 6 are mostly located in inland areas of Portugal.

The second component shows other type of variability that splits most municipalities of cluster 2 (dark blue) with positive scores from municipalities in clusters 5 (grey) and 6 (light blue) with negative scores. This component is interpreted as corresponding to the “preventive measures effect”, showing cluster 5 and 6 municipalities as the most relevant contributors for the lower incidence rates observed along autumn-winter periods.

The position of municipalities (scores) in the third component (corresponding to the “outbreak effect”) are illustrated by Fig. 9B (y-axis). It is possible to highlight four municipalities from cluster 4 with the highest scores on this component. These municipalities have small size populations and have all suffered very high incidence rates in winter 2020-21, mainly due to outbreaks occurring in nursing homes. The incidence curves of these four municipalities are shown in Fig. 10.

In this paper, we integrate FDA with unsupervised learning and spatial statistics (i.e., functional principal component analysis and hierarchical classification of spatially correlated functional data) to develop an innovative approach for detecting the main spatiotemporal patterns of COVID-19 incidence. As the methods used here are well established, the novelty relies on their use for analysis of public health data. In fact, despite being successfully used in many different scientific areas²⁹ such as biology (e.g., Cremona et al., 2019), economy(e.g., Elayouty & Abou-Ali, 2022) or medicine (e.g., Sørensen et al., 2013), functional data analysis is still relatively recent in the field of public health. With the proposed approach, we were able to describe a coherent spatiotemporal evolution of COVID-19 incidence in Portuguese municipalities between August 2020 and March 2022 and to identify curve patterns describing different features found in the data over time and space.

4.1. FDA and clustering

Using FDA we estimated not only the distribution of incidence variability in any period of time but we also reduced the data noise through curve smoothing. These were highly relevant to detect the major patterns of variability and a first step toward dimension reduction throughout the analysed period. By applying FPCA to the curves, we provided means for new insights and clearer interpretations about the covariance, or correlation structures, in the spatially correlated multidimensional functional datasets (i.e., municipalities functional curves), reducing the interpretation of initial 278 dimensions to only 3 dimensions explaining approximately 72% of total variability. By applying these functional approaches to the incidence rates, we provide an alternative and valid solution to minimize the number of curves to be analysed while maximizing the retention of relevant information (i.e., variability) to be derived from the original data. Another positive aspect about FDA was the possibility to analyse the evolution of incidence rates over a period of 1,6 years (570 days), without separating periods with distinct evolution patterns, such as seasonality (e.g., winter vs summer). While separating the incidence rate curves in different periods would allow a better understanding and description of particular seasons, this would limit the analysis of the pandemic as a whole.

In combination with clustering, we proposed to explicitly include spatial structure dependency in the analysis. This showed to be key to describe the major patterns of incidence curves across the country since municipalities close enough to each other tend to be clustered. In fact, the approach used here (hierarchical clustering of functional data with spatial dependence) is a recent topic that offers the possibility of combining knowledge from spatial statistics and FDA ²⁶. Using a trace-variogram allowed us to efficiently describe the spatial variation of incidence curves in terms of shape and range distance. Therefore, an obvious advantage of adding weights representing the spatial dependence of data to the dissimilarity matrix, was the ability to obtain spatially structured clusters of municipalities.

4.2. Spatiotemporal evolution

The temporal pattern found in the first functional PC (Fig. 4, top) match the curve shape found for the overall incidence evolution in Portugal (curve in black, Fig. 1A), and defines the municipalities that had higher and increasingly above the mean incidence rates, especially in autumn-winter 2021-22 season. The sharper increase in incidence and variability patterns observed between December 2021 and February 2022 is consistent with the dissemination of the Omicron variant ^33,34, ease of public health restrictive measures that occurred after a long second and third waves of the COVID-19 pandemic, and the sense of safety created by the implementation of vaccination programmes ³⁵. The vast majority of municipalities located in the northern coast and adjacent regions are those who had relatively high incidence rates in that period and are classified in clusters 1 and 5 (Fig. 7). A closer look into the shapes of the curves in these two clusters (see Fig. 8) shows that they display above-average incidence rates, especially in the autumn-winter season 2021-22. The agreement between the behaviour of the incidence curves and their geographical locations appears to be illustrating some unmeasured/unknown underlying processes (e.g., sociodemographic, cultural, environmental processes, previous exposure to SARS-CoV-2) taking place in that geographical area. Identifying these processes is beyond the scope of this work. Nonetheless, previous research from Almendra et al³⁶ outlined a geographical pattern compatible with our results, as they found sociodemographic and economic factors to have higher impact on COVID-19 incidence in that region due to stronger cultural relationships between neighbor municipalities. Duarte et al¹⁵ also identified sociodemographic and economic profiles as relevant risk factors for increased COVID-19 incidence observed in the same region during the winter 2020-21.

The temporal pattern found in the second PC (Fig. 4, middle) represents mostly the contribution of municipalities with lower than the mean incidence up to autumn-winter 2021–2022. The municipalities in this group are mostly classified in cluster 6 (Fig. 8) showing a spatial distribution pattern dispersed along Portugal’s hinterland plus the region of Lisbon (Fig. 7). While these results are difficult to interpret due to their spatial dispersion and to the small proportion of total variation contained, they provide a meaningful temporal pattern consistent with the (successful) adoption of preventive measures (e.g., social distancing), in particular the tier-lockdowns by municipality, impact of public health interventions (e.g., vaccination), or related with social, economic, or demographic underlying aspects. This functional PC also shows a change in the signal in February 2022 (from negative to positive), that can be attributed to the abrupt decrease of incidence in municipalities that had higher incidence rates till then. This could be due to some effect of previous exposure to the virus of a significant part of the population that would provide partial immunity for future exposures.

The temporal pattern found in the third functional PC reflects the contribution of municipalities with relatively higher than the mean incidence during the autumn-winter 2020-21 season. The most relevant contributions to this PC coincide with municipalities located in the hinterland, where population sizes are small, and COVID-19 outbreaks were reported on that period. We consider this result illustrates the ability of this approach to identify and to monitor outbreak clusters occurrences for health surveillance.

The use of variogram analysis to measure spatial correlation allowed for detection of spatially homogenous COVID-19 incidence curves. By first estimating a trace-variogram and later fitting an exponential model, we found that incidence curves are correlated up to 227 km. COVID-19 incidence is indeed known to be spatially correlated ^37,38. Accordingly, our results show that taking into consideration spatial correlation is crucial when analyzing disease incidence data. On the other hand, we found that 38% of spatial variability had no spatial pattern. This result might be regarded as the variability caused by the absence of data over small distances (e.g. sparse municipalities) or by unmeasured factors ³⁹. For instance, incidence rate estimates are sensitive to changes in population sizes ⁴⁰ and do increase spatial variability when population sizes are heterogenous or small ^41,42. At the same time, the areas of municipalities contribute to increase spatial variability due to their irregular shapes, sizes and orientations ⁴³. By combining all these factors, we are likely looking at main contributors to the estimated nugget-effect.

On the basis of the above, we consider that the results contribute to characterize the major spatiotemporal COVID-19 incidence patterns found in Portugal, using an innovative and straightforward solution. Yet, we must consider that their description is based on smoothed curves fitted to the original incidence data and require caution in interpretation. We are assuming that the features identified (i.e., global, preventive measures and outbreak effects) are well described by the smoothed curves representing the underlying dynamics of COVID-19 incidence. But incidence rate curves are influenced by factors like, e.g., mobility rates, that also show spatiotemporal variability and different intensities ⁴⁴, and testing availability. As a result, neighbour municipalities with different incidence curves and mobility dynamics could eventually be more similar if mobility and testing intensity had been included in the analysis. Another relevant consideration is that the description and interpretation of principal component and hierarchical clustering results are subjective and must be regarded as exploratory ⁴⁵. As shown in our results, spatial and temporal features separate the northern and coastal regions from southern and hinterland, and the effects in 2020-21 from the effects in 2021-22 autumn-winter season. While there is no objective measurable way to validate these findings (due to limitation of unsupervised learning methods), our results are coherent and compatible with the results from other authors.

In this paper we propose a novel approach integrating FDA with unsupervised learning and spatial statistics to detect the main spatiotemporal patterns of COVID-19 incidence and applied it to Portugal mainland. With FDA we were able to model the shape of Portuguese municipalities incidence curves and use them to extract the main temporal patterns of variability and cluster the municipalities by curve similarity, while considering their spatial proximity. We analyzed a period of 1,6 years (1 year plus 7 months) and found three main patterns of variability occurring in autumn-winter seasons in different spatial clusters of municipalities The results for the spatiotemporal classification are coherent and compatible with results reported by other researchers, showing that the proposed approach detects the main spatiotemporal patterns of COVID-19 incidence. Based on this, the novel approach extends and refines existing exploratory tools for analysis of spatiotemporal public health data.

Competing interests

The author(s) declare no competing interests.

Funding

Not applicable.

Authors' contributions

Manuel C. Ribeiro wrote, conducted formal analysis, and reviewed, Leonardo Azevedo wrote and reviewed the manuscript, Maria João Pereira, wrote and reviewed the manuscript; André Peralta Santos and Pedro Pinto Leite reviewed the manuscript and added epidemiological context.

Acknowledgements

This work was developed under the research project SCOPE - Spatial Data Sciences for COVID-19 Pandemic, funded by Fundação para a Ciência e a Tecnologia under the call AI 4 COVID‑19: Data Science and Artificial Intelligence in the Public Administration to strengthen the fight against COVID‑19 and future pandemics—2020 (DSAIPA/DS/0115/2020). The authors gratefully acknowledge the support of CERENA (strategic project FCT-UID/ECI/04028/2020). Manuel C. Ribeiro acknowledges FCT support for the research contract established under the transitional rule of Decree-Law 57/2016 (IST-ID/175/2018).

Data availability statement

COVID-19 data are not publicly available and can be obtained from the Portuguese Epidemiological Surveillance System (SINAVE), which records all laboratory-confirmed cases of SARS-CoV-2 (detected via PCR and RAT). Data analysed in this study were used under license and provided by the Portuguese Directorate-General of Health (DGS) to authors upon request and after signing a declaration of secrecy and confidentiality, in accordance with the Article 39 of Legislative Decree 2-B/2020 of 2 April 2020 - Access to anonymized data from the National Epidemiological Surveillance System for scientific research. Data are available from the authors upon reasonable request and with permission of DGS.

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Understanding spatiotemporal patterns of COVID-19 incidence in Portugal: a functional data analysis from August 2020 to March 2022

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Abstract

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1. Introduction

2. Materials and Methods

2.1. Ethics declaration

2.2. Data collection

2.3. Functional data analysis

2.4. Functional principal components analysis

2.5. Clustering spatial functional data

3. Results

4. Discussion

4.1. FDA and clustering

4.2. Spatiotemporal evolution

5. Conclusions

Declarations

Competing interests

Funding

Acknowledgements

Data availability statement

References

Additional Declarations

Supplementary Files

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