## 2 Materials and Methods
### Dataset Description  
The data at the base of our study was essentially corresponding to two types of time series. The former was relative to the new daily COVID-19 infections registered in all the counties of interest in the period 4-22 March 2020, while the latter was concerned with the air pollution, in particular, the particulate matter PM\({}_{2.5}\) registered on a daily basis, in the period 19 February-8 March 2020, in all the counties of interest. The information relative to the daily COVID-19 infections was retrieved from the website of the New York State Department of Health--COVID-19 Tracker [25], while the daily pollution PM\({}_{2.5}\) levels were collected from the website of the United States Environmental Protection Agency, under the Outdoor Air Quality Data section [26]. Since, in each county, there were different sensor stations returning pollution values, at various times during the same day, the correspondent data was aggregated using an average daily value for each county. The first issue to explain regarding these two time series of data is concerned with the different periods that were analyzed, that is, 19 February-8 March (PM\({}_{2.5}\)) vs. 4-22 March (COVID-19 infections). In regard of this, it is well known that a delay can occur between the day a person comes in contact with the virus (with pollution circulating in the air that might favor an airborne virus transmission) and the day when this person manifests the first symptoms of COVID-19 (being consequently recorded as infected). Unfortunately, such a delay can be influenced by two different factors. First of all, there is an incubation period of this virus that, according to the medical literature, may range from a couple of days to almost fourteen. More precisely, as reported in Reference [27], the mean incubation period for COVID-19 is estimated to be 5.2 days, while 12.5 days are needed to reach the 95 th percentile of the distribution of all the infections. Moreover, other studies have highlighted how an additional delay can be suffered from the time when a person is tested for positivity and the moment in time when this infection is registered by the authorities [28]. For these reasons, the two time series of data we took into consideration, say \(X\) (the daily average PM\({}_{2.5}\) level) and \(Y\) (the number of the new daily infections), were staggered by 14 days, yielding: \(X\) ranging from 19 February-8 March, and \(Y\) ranging from 4-22 March 2020. It is also important to note that our investigation period ends on 22 March, in correspondence with the announcement of the Governor of New York State, [PERSON], who placed the statewide stay-at-home order, starting from 8 p.m. on 22 March [29]. Hence, the reason to limit our study to the pre-lockdown period is that the lockdown measures might have significantly altered the general situation, with a slowdown of human activities and a consequent change of the pollution levels. The two corresponding curves of our interest (pollutant and infections), staggered by 14 days, are shown in the plots of Figure 1, where those curves are reported for all the counties we have examined (precisely: New York, Kings, Bronx, Richmond, Queens, Nassau, Suffolk, Rockland, Westchester, Onondaga, Oneida, and Monroe). In Figure 1, blue lines are the PM\({}_{2.5}\) pollutant values and the corresponding time period, while red lines are the new daily infections and the relative time period of interest. For each plot, one can see (leftmost) the measurement unit of the PM\({}_{2.5}\) pollutant (measured in micrograms/m\({}^{3}\)) and (rightmost) the number of the new daily infected people. The motivations behind our choice of investigating those precise 12 counties in the New York state, represented in grey in the map of Figure 2, was to evaluate whether the (potential) relationship between the PM\({}_{2.5}\) pollutant and COVID-19 diffusion remained valid through very different scenarios. However, scenarios that were all geographically relative to NYC.