# Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case
## 2 The Gaussian Growth Rate Model
In fact, \(f\) corresponds to the portion of total population which must be immunized for the epidemic to die out. As reported by WHO, both a good estimate of the numerical value of \(f\) and the achievability of herd immunity has been questioned [35]. So, in Figure 1 the time trends of infected, deceased and recovered growth rate are shown for different values of the ratio \(f\). A clear divergence emerges for \(f<0.1\) while trends remain almost unchanged for \(f>0.1\). Moreover, the obtained result is related with the considered fraction of the population being infected over the total (around 7% [36]), which further justifies the no relevant changes found for \(f>0.1\). In the following, we are going to set \(f=0.7\) (70% herd immunity), a proportion which has been frequently cited as a likely immunity threshold [37; 38], especially in the early days of vaccination campaign. We should keep in mind, however, that there are several variables to be considered, such as social interactions in reopening scenarios and variations of transmission rate for different virus strains, which may alter this threshold value [39]. As for infected people, we may set their number \(I(t)\) as equal to a time-dependent fraction of total cases:  
\[I(t)=\eta(t)I_{tot}(t) \tag{7}\]  
where \(0\leq\eta(t)\leq 1\) for each \(t\). Therefore, we obtain from previous equations:  
\[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t)=-\alpha(t)\eta(t)S(t)I_{tot}(t) \tag{8}\] \[\frac{dI_{tot}}{dt}(t) =N_{H}\alpha(t)\eta(t)I_{tot}(t)\bigg{[}1-\frac{I_{tot}(t)}{N_{H} }\bigg{]} \tag{9}\]  
Furthermore, we use the Italy case-fatality ratio (CFR), which is the ratio (that we call \(m\)) between the number of deceased \(E(t)\) with respect to the total cases \(I_{tot}\):  
\[E(t) =mI_{tot}(t) \tag{10}\] \[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\]0  
Note that, the CFR value is estimated to be of 3.0% by the Johns Hopkins University of Medicine [36] from available data on COVID-19 death toll. We have chosen to use CFR and not IFR value (COVID-19 infection fatality ratio) since estimating IFR is challenging. In fact, it would require to know the real number of total cases and deceased, which may have been underestimated in times when Italy's testing capacity was limited and the pressure on the healthcare system was high [40]. Data currently available provide information only about confirmed cases, possibly underestimating real data, as [PERSON] et al. pointed out already in May 2020 [41]. Indeed, given the relatively high number of tests performed so far, the use of CFR instead of IFR for \(m\) seems a correct assumption. \[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\]1 **(a)** Infected, **(b)** deceased and **(c)** recovered growth rate (see Equation (5) for definition) \(r(t)\) vs. \(t\) (days) in a log-lin plot for different values of herd immunity ratio: \(f=0.01\), 0.02, 0.05, 0.1, 0.2, 0.5, 0.7, 0.9. While \(r(t)\) shows a clear divergence for \(f<0.1\), it does not change significantly for \(f>0.1\). Likewise, we can set the recovered number \(R(t)\) as proportional to \(I_{tot}(t)\) by a ratio \(h\):  
\[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\]2 \[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\]3  
The ratio \(h\) is in general a time-dependent value. However, for our analysis it is sufficient to consider its limit for large times, that is:  
\[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\]4  
because for \(t\rightarrow+\infty\) all infected people will have either died or recovered, whereas the active cases will vanish. So, the problem can be described by a logistic ODE.