# Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case
## 3 Results and Discussion
### Statistical Analysis--Cumulative and Density Functions  
In order to numerically compute the number of total cases, deceased and recovered as a function of time, we substituted the obtained analytical fit curves \(r(t)\) (red lines plotted in Figure 2) in Equation (16) for each data:  
\[I_{tot}(t) =\frac{I_{\infty}}{1+\left(\frac{I_{\infty}-I_{tot}(0)}{I_{tot}(0)} \right)e^{-\int_{0}^{t}r(t^{\prime})dt^{\prime}}} \tag{19}\] \[E(t) =\frac{E_{\infty}}{1+\left(\frac{E_{\infty}-E(0)}{E(0)}\right)e^{ -\int_{0}^{t}r(t^{\prime})dt^{\prime}}}\] (20) \[R(t) =\frac{R_{\infty}}{1+\left(\frac{R_{\infty}-R(0)}{R(0)}\right)e^{ -\int_{0}^{t}r(t^{\prime})dt^{\prime}}} \tag{21}\]  
Herein \(I_{\infty}\), \(E_{\infty}\) and \(R_{\infty}\) represent the asymptotic limits of total cases, deceased and recovered for very large times. According to our model, these values are not independent. Instead, we have assumed a simple proportionality relation between deceased, recovered and infected (see Equations (10), (12) and (14)). We noticed that the normalized curves \(I_{tot}(t)/I_{\infty}\), \(E(t)/E_{\infty}\) and \(R(t)/R_{\infty}\) can be considered, mathematically speaking, as cumulative distribution functions (CDFs) \(F(t)\), since they both are monotonic increasing and their output value for large times equals 1. Therefore, we interpret such functions as describing the probability of an entire subpopulation \(I_{\infty}(E_{\infty},R_{\infty})\) having been infected (being deceased or recovered) by day \(t\) so they can be directly compared with the total cases (total infected, deceased and recovered people), normalized at the value of the last considered day (24 May 2021 in our case). \[F_{I}(t) \equiv\frac{I_{tot}(t)}{I_{\infty}}=\frac{1}{1+\left(\frac{I_{ \infty}-I_{tot}(0)}{I_{tot}(0)}\right)e^{-\int_{0}^{t}r(t^{\prime})dt^{\prime}}} \tag{22}\] \[F_{E}(t) \equiv\frac{E(t)}{E_{\infty}}=\frac{1}{1+\left(\frac{E_{\infty}-E (0)}{E(0)}\right)e^{-\int_{0}^{t}r(t^{\prime})dt^{\prime}}}\] (23) \[F_{R}(t) \equiv\frac{R(t)}{R_{\infty}}=\frac{1}{1+\left(\frac{R_{\infty}-R (0)}{R(0)}\right)e^{-\int_{0}^{t}r(t^{\prime})dt^{\prime}}} \tag{24}\]  
where:  
\[F_{I,E,R}(t)\equiv P\big{[}\ y(t^{\prime})=y_{\infty}|\ t^{\prime}\leq t\big{]} \tag{25}\]  
and for each case we can set \(y(t)\equiv I_{tot}(t),E(t),R(t)\). The corresponding probability density functions (PDFs) are then defined:  
\[p_{I}(t) \equiv\frac{dF_{I}}{dt}(t)=\frac{1}{I_{\infty}}\frac{dI_{tot}}{dt }(t) \tag{26}\] \[p_{E}(t) \equiv\frac{dF_{E}}{dt}(t)=\frac{1}{E_{\infty}}\frac{dE}{dt}(t)\] (27) \[p_{R}(t) \equiv\frac{dF_{R}}{dt}(t)=\frac{1}{R_{\infty}}\frac{dR}{dt}(t) \tag{28}\]  
and can be directly compared with the new daily counts provided for these variables. Having interpreted \(F(t)\) as the probability of total cases, deceased and recovered number reaching their maxima by day \(t\), the corresponding PDFs \(p(t)\) are the probabilities of reaching these maxima on a day \(t\). PDFs could provide useful information about future massive outbreaks in the short-term, since they are usually anticipated by a significant increase of growth rate \(r(t)\) and of PDF \(p(t)\). After computing \(F_{I,E,R}(t)\) and \(p_{I,E,R}(t)\) by fitting \(r_{Est}(t)\) with our GGR model, we compared raw and modeled data. The corresponding behaviors are shown in Figures 5 and 6. We underline that raw data series have been estimated by normalizing cumulative infected, deceased and recovered by their latest values reported in Italian Ministry of Health data, while daily variations of the same quantities have been normalized by the sum of all daily increments. It is noteworthy that our GGR model is able to perfectly describe  
\[E(t) =\frac{E_{\infty}}{1+\left(\frac{E_{\infty}-E(0)}{E(0)}\right)e^{ -\int_{0}^{t}r(t^{\prime})dt^{\prime}}}\]0 (**a**) Infected, (**b**) deceased and (**c**) recovered cumulative distribution functions. Blue open circles correspond to raw data, while the red solid lines represent model curves (Equations (22)–(24)).