Contrasting pre-vaccine COVID-19 waves in Italy through functional data analysis

Abstract

This study analyzes mortality patterns during two pre-vaccine COVID-19 waves in Italy, using data from its 107 provinces. Mortality is examined alongside information on mobility, governmental restrictions, and socio-demographic, infrastructural, and environmental factors using Functional Data Analysis tools. Publicly available differential mortality data and local mobility data from Google are processed with smoothing splines and aligned through landmark registration. The resulting curves are clustered to identify mortality patterns, and regression models are used to evaluate the impact of mobility, restrictions, and other factors on mortality. We find significant differences between the two waves: the first had higher, more concentrated mortality peaks, while the second was more widespread and asynchronous. Our results also support the effectiveness of timely restrictions in curbing mortality, and a strong positive association between local mobility and mortality in both pre-vaccine waves. Despite data quality limitations, our findings strengthen the evidence for the role of government restrictions and mobility controls during the pandemic.

Introduction

On January 30, 2020, Italy recorded the first two cases of SARS-CoV-2; a couple of Chinese tourists visiting Rome. The first non-travel related case was recorded in the municipality of Codogno (Lombardia) on February 19, 2020. With outbreaks quickly emerging in Lombardia and Veneto, local lockdowns were imposed in ten municipalities in the province of Lodi, and in one municipality in the province of Padova – affecting a total of over 50,000 people. As cases and deaths mounted, local and central authorities took progressively more stringent measures to limit mobility and social gatherings, e.g., suspending university activities, educational trips, sporting events, and exhibitions. These restrictions culminated with a general nationwide lockdown (March 9, 2020) and the suspension of all non-essential production activities (March 23, 2020), and were not lifted until May – when free movement was restored within the whole national territory.

After a fairly quiet summer, many countries in Europe, including Italy, witnessed a new rise in COVID-19 cases and deaths, leading to new containment measures. Face masks became mandatory, and gathering places such as gyms, theaters, and cinemas closed again. On November 4, 2020, the Italian government created a classification system whereby regions could be labeled as yellow, orange, or red based on a number of indicators of epidemic spread and stress on the healthcare infrastructure. Each color corresponded to different curfew rules and limitations on mobility. As a precautionary measure, during the holidays, red-zone rules were imposed in all regions – including those where the epidemic was not raging at the time (these were the strictest rules, equivalent to the lockdown of the first wave). In January 2021, as the second wave started to subside, restrictions were loosened again to the pre-holiday color system.

Prior analyses by our group, which employed Functional Data Analysis (FDA) tools^1,2 on data from the first COVID-19 wave at the resolution of regions, characterized different epidemic patterns unfolding in different areas of the country, and suggested a relevant role for mobility and some socio-demographic, infrastructural and environmental factors as statistical predictors of mortality³. While our findings aligned with those of other studies^4–7, they were potentially affected by signal dilution due to data aggregation. Furthermore, the limited sample size (Italy comprises only 20 regions) constrained our ability to use more complex statistical models without overfitting⁸. The current study builds upon Boschi et al.³ introducing significant novelty in data, methods and analysis aims. Here, we contrast the first and second wave of the epidemic, analyzing data on the two 150-day intervals from February 25, 2020, to July 23, 2020, and from October 1, 2020, to February 27, 2021. Moreover, we substantially increase spatial resolution considering 107 Italian provinces, as opposed to 20 Italian regions.

The increased resolution allows us to achieve a more granular characterization of the epidemic and its associations with potential predictors. Additionally, while some statistical techniques in this work relate to Boschi et al.³, we employ new methodology and define new proxies (derived from differential mortality curves) to handle shortcomings in public data. Crucially, while very different in terms of restriction regimes imposed on the population and of its mobility behaviors, the first and second waves of COVID-19 in Italy were both pre-vaccine (the vaccination campaign in the European Union, including Italy, started – for healthcare professionals and selected fragile categories – on December 27, 2020). Thus, contrasting them offers an opportunity to refine our understanding of the effectiveness of restrictions and non-vaccine-based interventions.

At the outset of our analyses, and yet among our key findings, is evidence of poor data quality. Specifically, we observe a vast under-counting of deaths in the official reports. This concerned especially the first epidemic wave (see also^9,10) but persisted into the second. We also observe alarming discrepancies in case counts. Ignoring information on cases, properly processing information on deaths, and using FDA tools, we find marked differences between the two waves. The first was dramatic and spatially concentrated, hitting particularly hard and with fairly homogeneous timing a handful of provinces in Lombardia. In contrast, the second wave was less dramatic but more widespread, affecting more locations with less homogeneity in timing. Notably, provinces hardest hit during the first wave were among the least affected during the second; the contrast is particularly striking in the case of Bergamo, which experienced the heaviest mortality during the first wave, making headlines around the world^11,12, and the lightest during the second. This may be due to the fact that inhabitants of provinces hardest hit during the first wave adhered more diligently to restrictions and safety mandates during the second, to a reduced number of people at risk (as an effect of the large number of deaths among the most vulnerable individuals during the first wave), or possibly to some kind of herd immunity^13–15 (as an effect of the large number of cases during the first wave).

To evaluate how the timing of restrictions may affect epidemic unfolding, and more specifically, whether introducing restrictions early enough may help curb mortality, we create an ad-hoc variable measuring, for each province and separately for the two waves, the area under the mortality curve up to the point in time when restrictions were imposed. Such cumulative mortality should capture how much the epidemic had a chance to “build up” in the province prior to the implementation of restriction measures. In our analyses, this variable is far stronger than socio-demographic, infrastructural, and environmental factors as a predictor of the mortality unfolding henceforth – a result consistent with findings in other recent studies^16,17. Assessing and comparing the role of mobility during the two waves is less straightforward. Restriction policies were very different in the two waves (uniform across the country in the first, localized and linked to the evolution of epidemic and stress parameters in the second), and so were the behaviors of the population in response to them. Likely as a consequence of these differences, the variability observed within the mobility curves of any given province, as well as the variability observed across such curves, changed markedly between the two waves. Notwithstanding these changes, we detect a strong and significant positive association between mortality and mobility in both waves, highlighting the importance of restrictions imposed specifically on the latter as tools to control the epidemic.

Our study offers some important elements of novelty compared to the vast literature devoted to the analysis of COVID-19 in Italy in the last five years – both in terms of study design and in terms of methodology. Compared to our prior work in Boschi et al.³ we increase resolution (province-level), contrast two pre-vaccine waves and, as we shall see in detail, use additional FDA tools and variables (e.g., landmark registration, concurrent functional regression, differential mortality curves and areas computed under them). Notably, Vinceti et al.¹⁵ do provide a province-level comparison of first vs. second-wave mortality, but do so without employing FDA tools or linking mortality to potential predictors. Mobility datasets and early lockdown effects are investigated in Pepe et al.¹⁸ and socio-economic drivers of mobility in Gauvin et al.¹⁹, but these works do not analyze mortality outcomes or FDA regressions. Ferrante²⁰ examines associations between mobility restrictions and lethality, yet does not integrate additional covariates or FDA methodology. Finally, Scimone et al.²¹ employ FDA for autoregressive dynamics and spatial clustering of mortality but do not incorporate mobility or policy covariates in their regressions. Thus, through its scope, resolution and use of sophisticated statistical techniques, our study fills a gap not addressed by prior literature.

In the remainder of this article we present our results, provide a discussion of key findings and conclusions, and describe the methodology employed – including details on data retrieval and pre-processing. The data used for our analyses are available at https://github.com/tobiaboschi/fdaCOVID2.

Results

Here we present the salient outcomes of our analyses. First, we document inconsistencies in case and death counts across different sources and geographical resolutions. Next, we characterize mortality patterns during the two waves and evaluate the effects of various potential predictors – including socio-demographic, infrastructural, and environmental factors. Last, we study mobility curves, their dynamics, and their association with mortality curves during the two waves.

Inconsistencies in death and case data

Unlike our previous work on COVID-19³, which employed the low spatial resolution of Italian regions, here we target the higher resolution of provinces. To date, Italian authorities (Dipartimento della Protezione Civile; DPC) provide COVID-19 case counts at both the regional and provincial level, and COVID-19 death counts only at the regional level (deaths by province remain undisclosed). However, differential mortality, which can be computed at both the regional and provincial level based on all-cause death counts published by the Italian National Statistical Institute (ISTAT; see Methods for details), is often advocated as an effective proxy when COVID-19 death counts are not available^22–24.

A first, critical insight on data quality comes from comparing differential mortality as sourced from ISTAT and COVID-19 mortality as sourced from the DPC at the resolution of regions. These reveal a vast under-counting in the DPC official death records – markedly in the first wave, but also in the second^9,10,25. Figure 1A contrasts, for each region, totals over the 150 days comprised in each wave. Overall, ISTAT and DPC data do become more consistent in the second wave, especially in regions such as Emilia Romagna, Lazio, Lombardia, Veneto, and Friuli Venezia Giulia. This, by and large, suggests an improvement in the DPC records – even though they remain much lower than ISTAT differential mortality in regions such as Basilicata, Calabria, and Sardegna. Notably, in Emilia Romagna and Lazio, DPC records were slightly higher than ISTAT differential mortality during the second wave. This may be a consequence of the fact that the latter also reflects phenomena indirectly related to COVID-19, affecting mortality in both directions; e.g., increases in mortality due to untreated emergencies or untimely treatment of chronic ailments, but also reductions in mortality due to fewer accidents during lockdowns or curfews.

Fig. 1.

Inconsistencies in death and case data. Panel (A) displays, for each region and wave, the ratio between COVID-19 mortality as reported by the DPC and differential mortality based on ISTAT data. During the first wave, all regions show marked inconsistencies between the two sources, suggesting a vast under-counting of deaths in the official records (ratios much lower than 1). The situation improves during the second wave; even though some regions appear to still under-report deaths (e.g., Basilicata, Calabria, Sardegna). The ratios slightly above 1 in Emilia Romagna and Lazio may reflect reductions in mortality, perhaps indirectly related to COVID-19 (e.g., fewer accidents during lockdowns and curfews), exceeding the epidemic death toll. Panel (B) displays, for each region and wave, the ratio between DPC case counts obtained by aggregating province-level counts within each region, and DPC case counts provided at the regional level. The discrepancies (i.e., departures from 1) are staggering, especially during the first wave – when aggregated, provincial records exceeded regional records by over 12-fold in Molise, and around 7-fold in Sicilia. As for death counts, the situation improves during the second wave, but discrepancies remain strong enough to suggest that case data are not reliable.

A second insight on data quality comes from comparing the number of cases reported by the DPC at the resolutions of provinces and regions. These harbor massive and concerning discrepancies. As shown in Fig. 1B, aggregating reported province-level cases by region, one obtains totals that do not match reported region-level cases. This is strikingly demonstrated by Molise and Sicilia, where the sums of the cases reported in the provinces are, respectively, over 12 and around 7 times higher than the cases reported at the regional level.

Due to the strong inconsistencies that still exist in the official Italian COVID-19 data, we ignore cases and focus on mortality alone to characterize the epidemic. Moreover, we do so using a differential mortality proxy instead of recorded COVID-19 deaths. This proxy has its own limitations, as it may indeed comprise some deaths not related, or indirectly related, to COVID-19 – but it is still more likely to accurately reflect mortality than the officially released deaths records and, crucially, it allows us to perform analyses at the spatial resolution of provinces.

Mortality patterns

We smooth daily province-level differential mortality data over the time intervals corresponding to the two epidemic waves – each 150-day long. This generates two sets of 107 curves. Next, within each set (i.e., for each wave) we shift the curves as to align their peaks. Intuitively, this landmark registration synchronizes the curves and facilitates their comparison net of their “horizontal” (i.e., temporal) differences.

Figure 2A shows aligned mortality curves during the first and second wave. Figure S1, in addition, shows the curves before the shifts applied to align their peaks, as well as the shift distributions. Full detail on the processing techniques utilized to produce and align the differential mortality curves is provided in the Methods.

Fig. 2.

Clustering of mortality curves. Panel (A) shows aligned province-level mortality curves in the first (top) and second (bottom) waves of the epidemic. For each wave, the curves are color-coded according to a partition in three clusters – with dark red, orange, and yellow representing decreasing aggressiveness of the epidemic. Thicker gray lines trace the means of the three clusters. Panel (B) shows the three clusters on the Italian map, for the first (left) and second (right) wave. Panel C shows again the Italian map, with provinces color-coded based on the difference in the ranks a province occupied in the first and second wave – for each wave, we constructed rankings based on the magnitude of the curves’ largest peaks. Thus, shades of blue (red) indicate provinces that suffered the first (second) wave more than the second (first).

Consistent with prior studies^26,27, our curves delineate a dramatic and concentrated first epidemic wave, with similar timing across a small number of very hard-hit provinces, especially in Lombardia²⁸, and a milder, more diffused second wave, with smaller differences and weaker synchronization across provinces. Notably, the largest mortality peaks were much higher in the first wave (around 20 daily deaths per 100,000 inhabitants) than in the second (around 5 daily deaths per 100,000 inhabitants), highlighting the grim death toll Italy withstood during the former. However, such death toll was very concentrated; many curves remained low, and just a handful had very high peaks. In contrast, the mortality peaks of the second wave were less differentiated, indicating an epidemic more evenly spread across the country. In terms of timing, the first epidemic wave was markedly more synchronous than the second; this can be appreciated considering the shifts required to align peaks, which are larger (a median of 19 vs. 12 days) and more variable in the second wave (see Fig. S1).

To characterize mortality patterns in a more systematic fashion, and further highlight differences in the unfolding of the two epidemic waves, we group the province-level aligned mortality curves in Fig. 2A using hierarchical clustering algorithms for functional data. Full detail is again provided in the Methods. The resulting dendrograms, one for each wave, are shown in Fig. S2. Their structure, along with the classical Hartigan²⁹ index, suggest a cut into clusters in both waves. The curves in Fig. 2A are color-coded based on these clusters and Fig. 2B shows the clusters on the Italian map (cluster memberships for all provinces are reported in Fig. S3).

As previously documented at the regional level³, in the first wave Italy saw the unfolding of two very different epidemic patterns: a relatively mild one in the majority of the country, and a very strong one affecting the north-western areas. The current analysis, performed at the higher resolution of provinces, affords a more detailed description of this phenomenon: while the epidemic was more severe and more spatially concentrated in the first wave than in the second, both waves harbored two, possibly three different patterns; namely, a flatter, “mild” pattern in the majority of the provinces, a very steep, “exponential” one in a small number of provinces, and an intermediate, still “exponential” but less extreme pattern in the rest of the country.

The provinces that withstood the most dramatic losses during the first wave were Bergamo, Brescia, Cremona, Lodi, and Piacenza. These provinces (in dark red in the upper portion of Fig. 2A and in the left portion of Fig. 2B) are all contiguous, in an area between Lombardia and adjacent Emilia Romagna (the very first cases of COVID-19 in the country were reported in Codogno, a municipality in the province of Lodi). Interestingly, in the second wave, they all belong to the cluster with the lightest mortality (in yellow in the lower portion of Fig. 2A and in the right portion of Fig. 2B). This could be due to several factors. The inhabitants of these areas may have been more diligent in practicing social distancing rules and adhering to restrictions and safety mandates after the dramatic events of the first wave. In addition, the large number of deaths among vulnerable individuals during the first wave may have effectively reduced the number of people at risk in these areas. Finally, the large number of cases during the first wave may have induced some degree of herd immunity³⁰.

In the second wave, the hardest hit provinces were Alessandria, Aosta, Asti, Biella, Cuneo, Udine, and Vercelli. The change in most affected provinces with respect to the first wave, along with the fact that fewer provinces belong to the lightest mortality cluster (in the first wave the yellow, orange, and red clusters, from lightest to hardest hit, comprise 86, 16 and 5 provinces, respectively; these counts for the second wave are 72, 28 and 7), confirm that, while less severe in terms of mortality, the epidemic had a broader spread across the country in the second wave.

We further compare the two waves considering the ranking of the province-level mortality curves based on their peaks (maxima; from smallest to largest) and the differences between such rankings – specifically, ranking in the second wave minus ranking in the first. These differences are color-coded on the Italian map of Fig. 2C, and show very clearly how many among the provinces most affected in the first wave (highest ranks) were among the least affected in the second (lowest ranks). The province of Bergamo is the clearest example; it ranks 107 in the first wave and 1 in the second. In contrast, some provinces in Piemonte were hard hit (high ranks) in both the first and second waves – showing very little difference in ranking.

Role of restrictions and socio-demographic, infrastructural and environmental factors

Next, we analyze associations between mortality and six scalar covariates representing a number of potentially relevant socio-demographic, infrastructural, and environmental factors, namely: aging of the population ( Over 65); quality of distributed primary health care (Adults per family doctor); the potential of hospitals (Ave. beds per hospital), schools (Ave. students per classroom) and workplaces (Ave. employees per firm) to act as contagion hubs; and pollution levels (PM10) (see Table 1). The significance of this kind of covariates has been shown in recent studies^3,16 related to COVID-19, both focusing on the first wave of the Italian epidemic, and on inter-generational relationships using data from 24 countries³¹. They could all be retrieved at the province-level from public databases with reasonable data quality. Full detail on their calculation and sourcing is provided in the Methods.

Table 1.

Scalar covariates: six variables considered as proxies of potentially relevant socio-demographic, infrastructural, and environmental factors.

Covariate	Description	Year and source
% Over 65	Aging of the population	2020, ISTAT
Adults per family doctor	Quality of distributed primary care	2019, Ministry of Health
Ave. beds per hospital	Hospitals acting as contagion hubs	2019, Ministry of Health
Ave. students per classroom	Schools acting as contagion hubs	2019, Ministry of Education
Ave. employees per firm	Workplaces acting as contagion hubs	2018, ISTAT
PM10	Pollution levels (particulates)	2019, ISTAT

We also consider two additional variables calculated from the mortality curves themselves, which we call area before, , and area after, . These measure the areas under the mortality curve to the left and right, respectively, of the date when restrictions were introduced (March 9, 2020, for the first wave and November 4, 2020, for the second – the calculation is performed before aligning the curves; detail is again provided in the Methods). Hence, they represent the cumulative mortality at the time of restriction and after restriction, respectively. Intuitively, before the introduction of restrictions the epidemic accelerates, with contagions spreading and, metaphorically, filling a pool of “potential deaths.” Restrictions, ideally, stop this process – or rather, in practice, curb it with varying degrees of effectiveness. After their introduction, as the mortality pattern unfolds over time, the pool of potential deaths empties.

is our attempt at capturing how far the epidemic has accelerated before restrictions hit the brakes on it – and to do so based on mortality data, not cases, because data on the latter are so poor and unreliable. We use it as a potential predictor in our regression models, along with socio-demographic, infrastructural, and environmental factors. In a way, it can be thought of as an “intermediate” predictor, subsuming the effects of a multitude of observed and unobserved factors, and influencing the subsequent unfolding of the mortality pattern. is our way to define a scalar summary of this subsequent unfolding, capturing its cumulative severity. We use it as a simpler, scalar response – in alternative to the full mortality curve (functional response). is akin in spirit to other proxies used to emphasize the importance of timely restriction policies in contrasting the spread of the epidemic. For example, using the effective reproduction number and case data that are not publicly available, a study of the first COVID-19 wave in Italy¹⁷ showed a strong relationship between the time spent above a given epidemic threshold prior to the introduction of restrictions and the total number of confirmed infections. Similarly, another study¹⁶ pinpointed the onset of the epidemic in an area and showed that the delay in such onset (and thus the shortening of the time between onset and restrictions) is a significant negative predictor of mortality levels.

To evaluate the relative importance of different variables, we perform feature selection in regression models for the scalar and for the mortality curves (functional regression). Full detail is provided in the Methods, and results for the two regressions, each fitted separately on first and second wave data, are summarized in the top graphics of Fig. 3A (elastic net for ) and Fig. 3B (fgen functional elastic net for the mortality curves).

Fig. 3.

Feature selection results. Panel (A) Elastic net results for (area after, scalar response). On the top, for each wave and each feature, we show balls with radius proportional to the average -ratio at which the feature enters the model across 500 replications of the regression on subsets of provinces ( is the smallest penalty value associated with 0 selected features; the larger the -ratio, the earlier a feature is selected and the greater its relevance). On the bottom, for each wave, we show fitted line, coefficient estimate and for the marginal regression of on (area before) – the most relevant feature in both waves – run on all 107 provinces (marginal regression results for other predictors are provided in Fig. S5). Panel (B) Functional elastic net (fgen) results for mortality curves, in a format similar to Panel A. For the marginal regressions of mortality curves on , bottom plots show coefficient curve estimates as black solid lines, with bands built adding and subtracting point-wise standard errors (marginal regression results for other predictors are provided in Fig. S6).

For each regression and wave, we display the features’ average -ratios across 500 runs on subsets of provinces. As detailed in the Methods, these provide a robust measure of feature relevance in predicting the outcome. The predictor emerges as the most relevant by far in both regressions and in both waves, with the largest average -ratios. We note that when is used as a predictor in models where mortality curves are the response, it essentially summarizes the initial portion of the response itself. Nevertheless, this relationship does not limit the shape of the mortality curves nor force any constraint in the estimation of regression coefficients. Table S1 and Fig. S4 report results of the elastic net and fgen feature selection procedures obtained on the full set of 107 provinces and with the parameter chosen by 5-fold cross-validation.

As a parallel exercise, for each response and wave, we run marginal regressions on every predictor separately – using the full set of 107 provinces. Results concerning are shown in the bottom graphics of Figs. 3A and B. Specifically, on the bottom of panel (A) we display scatter plots with fitted lines, coefficient estimates and coefficients of determination for the marginal regressions of on in first and second wave. The coefficient of determination represents the share of variance explained, , where and are regression and residual sums of squares. On the bottom of panel (B) we display coefficient curve estimates and coefficients of determination for the marginal functional regressions of mortality curves on , again in first and second wave. The for a functional regression is a straightforward generalization of the coefficient of determination for a scalar regression¹.

Marginal results are consistent with those obtained from elastic net-based feature selection, confirming strong positive (i.e., aggravating) statistical effects of in both waves of the epidemic. In fact, has by far the largest marginal s among all predictors considered; results for all other predictors are shown in Fig. S5 () and in Fig. S6 (mortality curves). The primacy of in our analyses provides further evidence that the timing of restrictions may play a crucial role in modulating the epidemic. Comparing marginal and joint results between the two waves, its effect appears to be larger in magnitude in the second wave for the scalar response, and in the first wave for the functional response. The values are always larger in the second wave, likely due to both and the mortality curves exhibiting less variability.

In addition to being generally weaker, the statistical effects of socio-demographic, infrastructural, and environmental factors appear to vary depending on the response, as well as the wave. For instance, considering mortality curves, adults per family doctor and average students per classroom exhibit a strong signal in the first wave, along with PM10 – confirming previous findings³. However, the relevance of these predictors decreases in the second wave, where the percentage of inhabitants over 65 seems to play a stronger role. These differences may result in part from the existence of collinearity among features (see Figs. S7 and S8), with the variable possibly subsuming a number of other effects, but they may also reflect actual changes that occurred between the two waves. For instance, average students per classroom (as measured in 2019; see Table 1) may indeed have been a good proxy for the potential of schools to act as contagion hubs during the first wave, but may have become a much less meaningful predictor during the second wave – when classroom sizes were reduced, many activities were held online, and even when attending school in person students had to adhere to masking and social distancing protocols.

Finally, we compute pc1 – the first principal component of the standardized socio-demographic, infrastructural, and environmental factors in Table 1, which can be used as an overall summary to control for their effects ( is not included in this computation). pc1, which explains of the overall variance, combines adults per family doctor, average beds per hospital, average students per classroom and PM10 (see loadings in Fig. S5B), and presents only a very mild correlation with (see Fig. S9C). Therefore, we further gauge the role of by running both the scalar regressions (with as response) and the functional regressions (with the mortality curves as response) using and pc1 as joint predictors, and computing their partial s. Results from these joint fits (see Fig. S9) support our previous statements, highlighting once again the importance of . The partial of a given predictor in a multiple regression is defined as , where is the coefficient of determination of the complete model, and that of the model comprising all predictors but the one being evaluated.

Mobility patterns

Several studies found evidence that local, short-range mobility worked as a key modulator of COVID-19 spread^7,32,33. Here, we produce province-level mobility curves over the time intervals corresponding to the two epidemic waves smoothing daily values of the “Grocery & Pharmacy” and “Workplace” mobility indexes released by Google. These express percentage changes in mobility linked to shopping for necessities and commuting to work with respect to a pre-pandemic reference (the first five weeks of 2020; see Methods).

Before analyzing the association between mortality and these mobility curves (below), we compare mobility patterns between the two waves, highlighting some marked differences. During the first wave, mobility showed a dramatic reduction with respect to reference values, suggesting that the restrictions implemented by the government were generally effective in changing people’s behaviors. In more detail, the mobility curves in Fig. 4A show the “Grocery & Pharmacy” index reaching levels between and (depending on the province) during the first wave lockdown. This reduction occurred even though citizens were allowed to leave their homes for these necessities. In the same period, and in particular after the halting of all non-essential production activities on March 23, 2020, the “Workplace” index reached even lower levels, dipping to values between and .

Fig. 4.

Mobility and lags. Panel (A) “Workplace” and “Grocery & Pharmacy” mobility curves during the first and second wave (no shift applied). Solid black lines are functional means, with bands built by adding and subtracting point-wise standard deviations. Vertical lines mark dates when mobility restrictions were introduced in the two waves (March 9, 2020, and November 4, 2020). Panel (B) scatterplot and marginal distributions of mortality peak values and lags for provinces with a discernible epidemic (hard- and medium-hit clusters 2 and 3; see Fig.2B) in the first and second wave. Lags are defined as the number of days between the mobility restriction date and the mortality peak for each province. Least-squares regression lines on the scatterplot have negative estimated slopes of (significant) for the first wave and (non-significant) for the second. Vertical lines on the marginal distributions mark medians. Panel (C)  values, obtained by regressing each functional variable in the rows (“Workplace”, “Grocery & Pharmacy”) against each functional (“Grocery & Pharmacy”) and scalar (, pc1) variable in the columns, in the first and second wave. For the function-on-function regressions (“Workplace” on “Grocery & Pharmacy”) we fit a concurrent model^2,34 with no lags, after shifting the mobility curves of each province by the same shift used to align its mortality curves. The background color intensity in each block is proportional to the values. These help us gauge predictor collinearities prior to building regression models for the mortality curves.

In contrast, during the second wave, while mobility still contracted with respect to the reference, reductions were less dramatic. The “Grocery & Pharmacy” index reached minima between and . The “Workplace” index again dipped deeper, between and , but only around Christmas and New Year – when the entire country was red-coded to try and take advantage of the holidays to curb the second epidemic wave. Also strikingly, the variance across curves was very low during the first wave, and much higher during the second.

In summary, mobility decreased dramatically and uniformly across the country during the first wave lockdown, and only moderately, with more variability from province to province – likely as a consequence of the adaptive restrictions imposed by the color-coded system – during the second wave. Notably, these patterns did not just reflect differences in the severity and uniformity of the restrictions themselves, but also differences in citizens’ attitudes. While during the first wave people limited also those activities that were permitted by the lockdown rules, during the second wave they took fuller advantage of whatever mobility was allowed to them.

Association between mortality and mobility

We now turn to the analysis of the association between mortality and mobility curves. To begin, we evaluate the delay between changes in mobility and their potential impact on mortality. For each province in each wave, we compute a lag, defined as the number of days between the date in which mobility restrictions were introduced and the date in which the province experienced its mortality peak. Restricting attention to the provinces which did experience a discernible epidemic in each wave (those in the hard- and medium-hit clusters 2 and 3; see Fig. 2B), Fig. 4B shows a scatterplot of mortality peak values against lags, together with their marginal distributions, for the two waves. For these provinces, lags are much more spread in the second wave than in the first, but their medians are remarkably similar (19 and 17 days, respectively). In contrast, peak values are higher and much more spread in the first wave than in the second; in particular, the first wave peak distribution has a long right tail. Notably, there appears to be a distinct negative relationship between lag and peak size in the first wave, which provides yet more evidence for the importance of restrictions timing. In the provinces most severely affected by the epidemic, restrictions came too late to curb mortality; these provinces reached their very high mortality peaks shortly after the implementation of the national lockdown. In contrast, provinces that, while affected, were spared the worst impacts were those where restrictions came earlier during the epidemic progression; they had smaller peaks occurring at a later stage. Intriguingly, an analogous negative relationship between lags and peaks cannot be traced in the second wave, when even in the worst-hit provinces peak values remained relatively low.

Next, we evaluate the impact of mobility on mortality developing appropriate regression models. We consider as potential predictors of mortality curves both “Grocery & Pharmacy” and “Workplace” mobility curves, as well as two scalar variables from our prior analyses; (which summarizes the progression of mortality up to the introduction of restrictions), and pc1 (which summarizes the factors listed in Table 1). To gauge collinearities, we fit functional regressions of “Grocery & Pharmacy” and “Workplace” on each other, and on each of the scalar variables. Figure 4C shows the resulting s which, together with the correlations reported in Figs. S8 and S9C, provide an overall picture of the dependence structure among predictors. In the first wave, predictor collinearities are very strong. In particular, “Grocery & Pharmacy” and “Workplace” have an of 0.95, confirming the effectiveness of the first lockdown. Both mobility indices are also highly correlated with (the is 0.78 for “Grocery & Pharmacy” and 0.85 for “Workplace”). A possible explanation is that, while mobility plummeted everywhere, whatever variation existed in such reduction across the country saw places hardest hit by the epidemic up to the lockdown date contract mobility the most. Because of these strong linear associations, to avoid variance inflation phenomena, we consider a simple model for the first wave (Model A); namely, a concurrent functional regression of mortality curves on just one type of mobility curves, say “Workplace”, with pc1 added as a scalar control. In the second wave, predictor collinearities are much milder. The more nuanced and differentiated color-coded restrictions allowed “Grocery & Pharmacy” and “Workplace” mobility to vary more across the country and to capture different signals, with an decreasing to 0.39. Also the s with decrease (to 0.18 for “Grocery & Pharmacy” and 0.30 for “Workplace”). We therefore consider both the simple model used for the first wave (Model A), and a more complex model (Model B); a concurrent functional regression of mortality curves on both types of mobility curves jointly, with both and pc1 added as scalar controls. In both models, we also introduce a binary predictor d to separate provinces with discernible epidemics (hard- and medium-hit clusters in Fig.2B; , Group 2) from the others (mild-cluster; , Group 1). Further detail on the concurrent functional regressions is provided in the Methods. These models specify a time lag for the association between mortality and mobility curves; since stability³⁵ is of particular concern when modeling functional objects and estimating functional coefficients, we test a range of lags to determine whether coefficient curve estimates are heavily affected by the lag specification, or exhibit a consistent pattern. Based on the observed lag distributions, we consider values in the 15-to-24 day range (see Fig. 4B; the range includes the medians for both waves). This is in line with previous findings³⁶ and with established epidemiological frameworks concerning, e.g., incubation period (time from exposure to symptom onset) and hospitalization-to-death intervals for COVID-19^37–39.

Figure 5A summarizes the fit of Model A on each wave. As expected, coefficient curve estimates are generally larger in magnitude for the first vs. the second wave, and for hard/intermediate hit vs. mildly hit provinces. In both waves, “Workplace” mobility is the leading predictor variable (based on the partial ). Its effect is positive along most of the domain during the first wave, but only at the beginning of the second wave. Notably, coefficient curve estimates are fairly consistent in shape and magnitude across lags suggesting stability of our findings. Robustness to varying model specifications is also confirmed using “Workplace” mobility and the group dummy d as predictors (without pc1 as control; see Fig. S11B), or replacing “Workplace” with “Grocery & Pharmacy” mobility (with pc1 and d; see Fig. S11A, or without pc1; see Fig. S11B). Interestingly though, coefficient curve estimates for “Grocery & Pharmacy” are slightly negative in the first half of the second wave. A mild (and temporary) negative association between this type of mobility and mortality may in fact make sense; since these “first necessity” movements were never restricted, the public may have engaged in them more liberally in provinces with lower mortality and a perception of (relative) safety. Finally, Fig. 5B summarizes the fit of Model B on the second wave. Coefficient curve estimates for both mobility predictors (“Grocery & Pharmacy” and “Workplace”), which are here considered jointly, are consistent with those obtained from Model A. However, the predictor with the largest partial here is , which shows a strong positive association with mortality – further confirming the critical role of restriction timing. Interestingly, the effect size of is similar in hard/intermediate and mildly hit provinces during the first half of the second wave. This may be due to the definition of , which is the integral of the curve in the first part of the domain (before restrictions take place). As in Model A, coefficient curve estimates are consistent in shape and magnitude across lags in the 15-to-24 day range, indicating a strong and stable effect. Furthermore, total and partial s are comparable across lags. Thus, both the explanatory power of the model and the relationships between response and predictors are stable across the range of lags considered, lending reliability to our findings.

Fig. 5.

Lagged concurrent functional regression results. Panel (A) for first (left) and second (right) wave, coefficient curve estimates for the regression of mortality on “Workplace” mobility and pc1, with an additional binary predictor separating provinces with mild (, Group 1) vs. hard/intermediate (, Group 2) epidemic courses. The plots show “beams” comprising 10 curves, one for each of the lags considered in the model (). Gray portions in a curve correspond to time intervals where it did not significantly depart from 0 (the confidence interval obtained adding and subtracting to the estimate pointwise standard errors contained 0). For each regression we report the average total and partial s over the 10 fits with different lags. Panel (B) in a format similar to A, but for the second wave only, results for the regression of mortality on “Workplace” mobility, “Grocery & Pharmacy” mobility, and pc1, again including the binary predictor separating provinces with mild vs. hard/intermediate epidemic courses.

Discussion

Notwithstanding the limited availability and at times poor quality of publicly available data, we were able to leverage Functional Data Analysis (FDA) techniques to characterize and compare the initial two waves of the COVID-19 pandemic in Italy. Our analyses were conducted at the province-level; this finer resolution compared to prior studies^3,24,40,41 allowed us to shed additional light on critical associations and relationships – avoiding the loss of signal caused by spatial aggregations and enabling the use of more sophisticated statistical models. The two waves span a complete year, from February 2020 to February 2021, and both occurred before the beginning of the vaccination campaign. Therefore, the patterns and associations we observed were not influenced by vaccine availability and the progressive increase of vaccine-related immunity in the country.

Using functional hierarchical clustering, we identified three groups of provinces in both waves; one characterized by a mild mortality pattern, and two characterized by an intermediate strength and a harsh exponential pattern, respectively. Perhaps the most interesting observation obtained by contrasting clustering results for the two waves is that some of the provinces most impacted during the first wave (e.g., Bergamo, Como, and Lodi in Lombardia) were among the least impacted during the second. This could have been due to a reduced number of vulnerable individuals after the deaths in the first wave, to behavioral adaptations that led the population in these areas to adhere more strictly to recommendations on, e.g., social distancing and mask use, and possibly to a degree of herd immunity.

Next, we highlighted the importance of a timely introduction of restrictions. Given the poor quality of data on cases, we created a variable, , that quantifies the escalation of the epidemic prior to the introduction of restrictions based on mortality data alone. Using a number of regression techniques we found to be a very strong predictor of mortality in both waves – in fact, a stronger and more consistent predictor than the proxies for socio-demographic, infrastructural, and environmental factors at our disposal. Potentially suboptimal proxies, collinearity, and possible confounding effects require that we interpret these results with care. But they do suggest that, by “hitting the brakes” on the exponential dynamics of mortality, restrictions play a pivotal role in curbing the spread of the epidemic and mitigating its impact.

Finally, we explored the relationships between mortality and mobility patterns. During the first wave lockdown, mobility exhibited very pronounced and similar contractions across the country. Following the introduction of the second wave color-coded restriction system, mobility contractions were milder and more geographically differentiated. Despite these differences, through additional functional regression exercises, we were able to identify significant and robust positive associations between mobility and mortality in both waves. We note that, through all our regression exercises, we adopted a minimalistic approach in terms of control covariates – using the first principal component of the variables listed in Table 1 as a single “summary” control. In future work, other, more exhaustive strategies should be explored. For instance, even though publicly available data is often not readily available to create appropriate proxies at province-level resolution, efforts to retrieve and analyze data on additional socio-demographic, infrastructural and environmental factors could provide further insights on the unfolding of the epidemic.

This study, alongside our previous work³, demonstrates the potential of FDA for analyzing epidemiological data. Specifically, FDA tools for smoothing and shifting (more generally, aligning) curves provide an effective way to handle outlying observations and compare phenomena that may occur asynchronously or in different time frames. Moreover, functional regression models can naturally incorporate a time dynamic along with predictors of different types. Also importantly, FDA-based analyses do not rely on the strict assumptions often required by standard epidemiological models, and thus allow a different perspective – and more flexibility – in representing the temporal evolution of the epidemic. FDA tools as a means to handle longitudinal data, or other data collected on structured domains in biomedical and public health research, are gaining in popularity⁴² but are still under-utilized. Showcasing well-established and recent FDA methods (including those developed by our group⁴³) is one of the contributions of this study. We point out that smoothing, as well as FDA tools applied downstream from it, can be used to analyze spatial or even spatio-temporal data (2 or 3D domains) – provided the measurements are numerous enough^21,44–46. Our study treated the temporal, but not the spatial, component functionally; extending functional treatment to the spatial component represents a promising avenue for future work in COVID-19 as well as other epidemiological research.

Methods

Data retrieval and pre-processing

Deaths and cases

Differential mortality was computed using the daily all-cause death counts provided by the Italian National Statistical Institute (ISTAT) for 7270 Italian municipalities (comprising about 93.5% of the Italian population). In particular, we take the difference between daily deaths and the average daily deaths in the five years prior to the pandemic (2015–2019), and divide by the total population as of January 1, 2019. This is done for each province and for each day in the first and the second waves of the epidemic; that is, for the two 150-day intervals from February 25, 2020, to July 23, 2020, and from October 1, 2020, to February 27, 2021, respectively.

Daily province-level differential mortality values are then smoothed into curves using cubic smoothing B-splines² with knots approximately at each week (21 knots equally spaced over 150 days – the time domain of each wave) and a roughness penalty on the curve second derivative. Employing one knot per week allows us to focus on general trends and smooth out daily fluctuations. Separately for each wave, the smoothing parameter is selected as to minimize the average generalized cross-validation (GCV) error⁴⁷ across all 107 province-level curves (the same criterion is employed to smooth mobility data; see below). GCV is a well-established, widely used selection criterion in the literature^3,48; in Fig. S12 we compare it to alternative criteria and provide a quantitative justification for our choice. Next, we align the differential mortality curves using landmark registration². Specifically, for each wave, we shift curves horizontally in order to align their peaks. For curves with multiple peaks, we consider the highest peak occurring between day 10 and day 100. The new common peak time is set to be equal to the time of the earliest peak, specifically day 20 (peak of the province of Lodi) for the first wave, and day 33 (peak of the province of Cagliari) for the second wave. Finally, we add (remove) days at the end (beginning) of the domain for the shifted curves, to guarantee the same 150-day length for all curves (curves are smoothed one more time after this “domain integration”, again choosing the smoothing parameter by minimizing the GCV criterion). As a result, the aligned, common time in each wave is the one of the province with the earliest peak. All computations are performed using the R package fda⁴⁹.

We repeat mortality calculations at the level of regions, in order to compare results with the data provided by the Italian Civil Protection agency (Dipartimento della Protezione Civile; DPC), which releases regional daily counts of recorded COVID-19 deaths since February 2020. We divide both ISTAT differential death counts and DPC death counts by the total population as of January 1, 2019, to obtain ISTAT differential mortality and DPC mortality, respectively. We do not perform any further preprocessing, since we employ these data only for validation purposes.

We also consider two additional variables calculated from our differential mortality curves, which we call area before, , and area after, . For each province, these measure the areas under the curve to the left and the right, respectively, of the date when restrictions were introduced (March 9, 2020, for the first wave, and November 4, 2020, for the second). Note that and are computed before shifting the mortality curves, and therefore the number of days used to calculate each of them is the same for all provinces. Moreover, since both and are highly right-skewed, we take their logarithms to normalize them before using them in our regression models (see below).

Local mobility

Daily mobility indexes are provided by Google at the resolution of Italian provinces⁵⁰. These measure the fractional reduction in mobility with respect to levels registered in the first five weeks of 2020 (January 3 to February 6), and are organized in categories based on mobility aims. We focus on two categories that capture the local, short-range mobility of individuals, which was allowed in all provinces also during mobility restrictions: the “Grocery & Pharmacy” index, which accounts for local trips to grocery and pharmacy stores, and the “Workplace” index, which accounts for trips to and from workplaces. We produce mobility curves by smoothing these daily data with the same procedure used for mortality curves (see above).

Socio-demographic, infrastructural, and environmental factors

We focus on six scalar covariates, listed in Table 1. These capture aging of the population ( Over 65); quality of distributed primary health care (Adults per family doctors); the potential of hospitals (Ave. beds per hospital), schools (Ave. students per classroom) and workplaces (Ave. employees per firm) to act as contagion hubs; and pollution levels (PM10).

The variable % Over 65 is retrieved from ISTAT⁵¹ at the province-level. To compute Adults per family doctor, we divide the adult population of the province as of January 2019 (from ISTAT) by the number of family doctors in the province⁵². The latter is obtained by summing the number of family doctors at the level of “Aziende Sanitarie Locali” (Local Health Agencies, ASL) provided by the Ministry of Health for the year 2018. We manually retrieved data missing for some provinces in Lombardia, Molise, Sardegna, and Toscana from the corresponding ASL websites. To compute Ave. beds per hospital we use data from the Ministry of Health⁵³, which provides the number of beds per ward in each hospital in 2019. We first aggregate them over wards belonging to the same hospital, and then average over hospitals in each province. To compute Ave. students per classroom we use data from the Ministry of Education⁵⁴, which provides the number of students in each classroom of each school in the country (public or private, at every level of education), for the academic year 2019/2020. We average them over schools in each province. Missing data for Trento, Bolzano, and Aosta are filled through random forests imputation⁵⁵, with default parameters maxiter=10 (maximum number of iterations performed if the stopping criterion is not met) and ntree=100 (number of trees grown in each forest). To compute Ave. employees per firm we use data from ISTAT⁵², which provides the number of employees per firm at the level of municipalities. We average them over firms in each province. To compute PM10 we use data from ISTAT⁵⁶, which provides the average annual concentrations of PM10 (in ) detected by air quality meters distributed over the Italian territory, and we average them over meters located in each province.

Statistical analyses

Clustering and rank comparisons

We perform functional clustering of the shifted province-level mortality curves, separately for each wave. This is done with an agglomerative hierarchical algorithm employing a Euclidean functional distance () and complete linkage. The (squared) distance between two generic curves and on a generic domain is simply defined as . Starting from the matrix of pair-wise distances , the agglomerative hierarchical algorithm merges singletons based on their distance, and clusters based on the maximal distance between their members (complete linkage).

To further compare province-level mortality between the two waves, we rank mortality curves based on their peak values (the curve with the lowest peak is ranked first and the one with the highest peak is ranked 107th), separately for each wave. Next, for each province, we compute the difference between its rankings in the second and first wave.

Feature selection for scalar and functional regression

To evaluate the role of potential scalar predictors of mortality, we perform feature selection in two separate regression models. The first is a standard scalar regression, where the scalar response is modeled as a function of the scalar predictors ( and the six covariates in Table 1). Here we use an elastic net penalized fit⁵⁷ as implemented in the R package glmnet⁵⁸. In symbols, the penalized fit corresponds to the minimization problem

1

where is the response vector, is the (standardized) design matrix, is the coefficient vector, and n and p represent the number of units and features, respectively. We set , and we run the fit for different values of , starting from , which selects 0 features, and gradually decreasing it. To capture the relevance of each feature, we track its -ratio, defined as , where is the largest penalty weight such that feature j is included in the model. A higher ratio (earlier inclusion in the model) corresponds to higher relevance. Finally, to assess stability of the results, we repeat the procedure 500 times, each time using a random subset of provinces, with itself randomly selected between 90 and 107. -ratios are averaged over the 500 runs. By removing a random number of units randomly selected without replacement, we aim to mitigate the effect of potential outliers on our findings.

The second model is a function-on-scalar regression, where the full mortality curves (functional response) are modeled as a function of the same scalar predictors. Here we use fgen⁴³, which extends the elastic net to this type of regression. In this case, the minimization problem is

2

where is the collection of n response curves, X is again the (standardized) scalar design matrix, is the collection of p coefficient curves, and is the -norm for a generic n-dimensional vector of functions f(t). We handle and as in the scalar elastic net, track again the -ratio for each feature, and perform the same stability assessment repeating the procedure 500 times on random subsets of provinces.

Concurrent function-on-function regression modeling

We formulate both models A and B described in the Results section as “lagged” concurrent function-on-function regressions^2,34 where the functional response (mortality) at time t is assumed to depend only on the functional predictor(s) (one or both mobility indexes) at time ; in this formulation the lag is an additional model parameter. We shift the mobility curves of each province by the same shift used to align its mortality curves (Fig. S10 shows the mobility curves after this alignment), and we fit the models with 10 different values of ranging from 15 to 24 days. This range is consistent with the observed lag distributions (Fig. 4B), previous findings³⁶, and established epidemiological frameworks^37–39. Fitting our models with different lags addresses potential stability pitfalls³⁵, and provides a robust characterization of the relationships under investigation. In symbols, the general formulation we utilize is

3

where, as before, is the collection of n shifted mortality curves, , and are i.i.d. Gaussian errors. M is the number of functional predictors ( in Model A and in Model B), all introduced with the same lag , and J is the number of scalar covariates ( in Model A and in Model B). Finally, , , and are functional intercept and coefficient curves for provinces where the epidemic was mild () while , , and are the corresponding coefficient curves for provinces where the epidemic was hard/intermediate ().

All concurrent functional regressions for Models A and B are fit in R using the pffr function in the refund package⁵⁹. Curves are evaluated on a grid of 100 equispaced points (the model takes evaluated curves, not an explicit basis expansion). Other arguments are left at defaults, as follows. We use tensor-product terms of type ti to construct the smooth, t-varying coefficient functions. Along the response index , the global functional intercept uses cubic B-splines with and a first-order difference penalty; all -varying coefficient functions (including concurrent effects and varying scalar effects) use the response-index basis with cubic B-splines, , and the same penalty. Smoothing parameters are estimated by REML, and uncertainty quantification uses the default mgcv Bayesian covariance of the penalized spline coefficients. Finally, for each functional model, we also compute the and the partial of each covariate.

Author contributions

All authors conceived ideas and analysis approaches. T.B., J.D.I., and L.T. retrieved and processed data from multiple public sources, implemented pipelines and performed statistical analyses. All authors interpreted findings and participated to the writing of the manuscript. M.A.C. and F.C. supervised the research.

Data availability

The data used for all the analyses are available at https://github.com/tobiaboschi/fdaCOVID2.

Declarations

Competing interests

The authors declare no competing interests. 

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-29316-4.