# Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case
## 3 Results and Discussion  
### Growth Rate Analysis  
To extract significant information from the epidemics data, we should perform a careful analysis of growth patterns in daily and total numbers of confirmed cases and deceases, looking for details about growth maxima positions and widths for both infected and deceased, and how they correlate with each other. For this purpose, we have fitted growth rate \(r(t)\) data (Figure 2) of infected (panel (a)), deceased (panel (b)) and recovered (panel (c)) people in Italy since the beginning of the pandemic (26 February 2020 for first official case in Italy) up to 24 May 2021, by using seven Gaussian bands (dashed blue lines) plus a constant background (Equation (18)), according to the GGR model. The constant background assumption means that random contagion events are uniformly distributed in time regardless of the general epidemics data trend, in a consistent way with respect to the uncertainties and incompleteness of available data. This leaves a certain margin of error to undetected cases and the resulting, seemingly random, contagion events [11; 43]. The lag in the growth rate of deceased and recovered with respect to infected people is intrinsically included in the theory by means of the Gaussians centroid. In fact, we considered that only the first Gaussian band for infected people is centered at zero (days). We will enter more in details in the next section. For the chosen modelling, we have selected 7 Gaussian components after an optimization procedure that took into account the use of the same standard deviation for all Gaussian bands (within an interval of 15-25 days in order to maintain a comparable spread for the considered events) and the lowest \(\chi^{2}\) corresponding to a statistical significance of the fitting parameters (\(p<0.05\)). Then we have focused our attention on the time and intensity of the events (respectively, the centroid and amplitude of the Gaussian functions). The width of the last Gaussian band reaches the highest bound value because the end of data is still not definitive. Our results indicate that deceased growth rate peaks are shifted from 3 (only the first peak) up to 21 days ahead of corresponding infected peaks. This is overall consistent with the hospitalization time intervals for severe Covid-19 cases [44; 45] and also holds for the time shifts between recovered and infected contributions, ranging  
Figure 2: (**a**) Infected, (**b**) deceased and (**c**) recovered growth rate fitted by a Gaussian mixture (Equation (18)) with seven components (blue dashed lines) plus a constant background (black dotted line). The red solid line corresponds to the total fit and the vertical green dashed lines mark important dates as described in the text. from less than 1 day for the first peak up to 25 for the fourth one. The uncertainties obtained after the fitting procedures are less than 10 days for all centroids, less than 5 days for all standard deviations and of few percent for all Gaussian amplitudes. The vertical green dashed lines reported in Figure 2 remark the date of important events in the timeline of Covid-19 pandemic evolution in Italy [46], useful to understand the implication of relevant episodes (e.g., the application of tight measures for anti-contagion control and the discovery of new dangerous virus variants) [47]. First of all, the national-scale lockdown (\(NL1\)) declared on 9 March 2020 to contain the early spread of Covid-19 and its official ending (_End NL1_) on 10 July 2020. In these days, the infected growth rate reached a minimum in Italy; however, we would like to point out that, in the same period, pandemic was far from slowing down on a global scale, as the number of daily cases was growing up to a plateau of \(\sim\)250,000 daily cases in August 2020 [1]. On 25 August 2020 a local, yet significant, outbreak started in a few touristic sites in Sardinia island, in particular at Billionaire Club in Porto Cervo (\(BO\)).