# Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case
## 2 The Gaussian Growth Rate Model  
The classical logistic model corresponds to the second-order approximation of the epidemic curve growth described by the original Kermack-McKendrick SIR model [28; 29; 30], which is indeed a mechanistic compartmental model aimed at describing the dynamics of an epidemics in a population. In classical SIR model, the population is assumed to be constant over time, so that the following equality holds for each time \(t\):  
\[S(t)+I(t)+R(t)=N \tag{1}\]  
although deaths caused by the disease itself should be considered as well. However, we note that in the Italian case we are interested in, there have been 125,335 deaths as of 24 May 2021 [31] over an initial population of 59,641,488 reported in 2019, just before the pandemic emergency was declared. Therefore, we will assume the Italian population to be approximately constant, as the decrease caused by COVID-19 pandemic was less than 1% (0.2%) of the initial population. We can translate these words into a system of non-linear ordinary differential equations (ODEs) [32]:  
\[\frac{dS}{dt}(t) =-\alpha(t)S(t)I(t) \tag{2}\] \[\frac{dI}{dt}(t) =\alpha(t)S(t)I(t)-\beta(t)I(t)\] (3) \[\frac{dR}{dt}(t) =\beta(t)I(t) \tag{4}\]  
where \(\alpha(t)\) and \(\beta(t)\) are defined as transmission (\(\alpha(t)\)) and removing (i.e., death or recovery, \(\beta(t)\)) rates [33]. This system should be accompanied by suitable initial conditions \(S(0)>0\)\(I(0)>0\) and \(R(0)\geq 0\)[32]. The first condition requires the susceptible number \(S\) to be non-zero at initial time \(t=0\), because if it were, the disease could not find a vulnerable population to spread through. The second condition states that the initial number of infected should be non-zero too, or else there would not be any epidemics outbreaks. Despite its shortcomings in terms of data availability, SIR model provides a simple, intuitive and effective description of epidemics in a population. Herein we propose a model based only on one equation of the SIR model (Equation (2)). We propose a Gaussian mixture able to describe the growth rate of infected, deceased and recovered people in Italy. We correlate the Gaussian parameters with the number of people affected by COVID-19 as a function of the large-scale anti-contagion control measures strength, and also of vaccines effects adopted to reach herd immunity. In detail, we consider a Gaussian mixture to model the growth rate \(r_{Est}(t)\) of the total cases \(I_{tot}(t)\), deceased \(E(t)\) and recovered \(R(t)\) (estimated from National Health Institute of Italy (it. _Istituto Superiore di Sanita_, or ISS)) daily data reports in json format [31]. Data processing and analysis have been performed by using a custom build algorithm with MATLAB (2018a, MathWorks, Natick, MA, USA). As we will detail in the next, the growth rate \(r_{Est}(t)\) can be expressed by the following relation, according to a generalization of classical logistic models [28; 34]  
\[r_{Est}(t)\equiv\frac{dy(t)/dt}{y(t)\left[1-\frac{y(t)}{N}\right]} \tag{5}\]  
where \(y\) in our case indicates infected, deceased or recovered people, \(N\) is the number of considered people and \(t\) is time. We will show that the estimated growth rate \(r_{Est}(t)\) can be described, and fitted to, by a Gaussian mixture, from which the name of our model: Gaussian Growth Rate (GGR) model. First, we assume that only a sub-population \(N_{H}\) is considered as susceptible rather than the total population \(N\). It is worth to mention that \(N_{H}\) in SIR models coincides with the so-called attack size, i.e., the total number of infected individuals. Since the logistic model is governed by two parameters, the growth rate and the saturation (asymptote), \(N_{H}\) is related with the asymptotic value through:  
\[S(t)+I_{tot}(t)=N_{H} \tag{6}\]  
where \(I_{tot}\) corresponds to the cumulative total cases (i.e., the total number of infected people). Then we assume that the ratio \(f\) between \(N\) and \(N_{H}\) may be connected with the long-debated herd immunity (\(N_{H}=fN\)).