# Gaussian Parameters Correlate with the Spread of COVID-19 Pandemic: The Italian Case
## 1 Introduction
Mechanistic models based on differential equations, such as the well-known SIR model (Susceptible-Infectious-Recovered) and its numerous modifications, are the most used to mimic and predict the spread of COVID-19 by including specific assumptions on the chosen parameters which control transmission, disease, testing capacity and immunity [17; 18; 19; 20]. These models, on the contrary of crude statistical approaches, are able to consider nonlinear dependences such as the increasing diffusion speed with the number of infected people. On the other hand, our poor knowledge of this novel virus, as well as the different variants met in different countries, limits model accuracy. Note that the highest source of uncertainty influencing all models lies in the number of really infected people. In fact, available data depend on the number of performed virologic tests; so a certain number of infected (but not tested) people is not considered (essentially asymptomatic people). This issue is called _under-reporting_ and it has been proposed to model its process in order to avoid an underestimation of the infection rate [21; 22] and therefore to reduce the model uncertainty also considering the heterogeneous (in space and time) distribution of the acquired data [11]. Hence, hospitalizations and deaths could be considered more reliable data although, from one side, they may underestimate disease burden and, from the other side, correlations with the cases in the community should be unveiled [10]. Thus, until today, COVID-19 pandemic evolution is very difficult to predict because many factors (i.e., social behaviors) are involved as well as the occurrence of multiple contributions corresponding to multiple \"waves\". Summarizing, two main modelling approaches are being used today: (i) time series regression analysis furnishes reliable estimate only on a short time window. (ii) SIR model or its modifications that, by using coupled differential equations, justify inter-conversions between the different fraction of populations considered within the model. One of the most extended SIR models is called SIDARTHE and includes inter-conversions between susceptible (S), infected (I), diagnosed (D), ailing (A), recognized (R), threatened (T), healed (H) and extinct (E) [18]. However, for a reliable evaluation of the conversion rates, data need to be continuously analyzed over varying time windows for improved predictive power during virus spreading and evolution [23]. Unfortunately, predicting the next wave is really a hard task, even for mechanistic models, that by employing constant \"conversion rates\" are indeed unimodal. This would require identifying a change (likely increase) in the relevant rate. In this work, we propose a model using a Gaussian mixture to describe the growth rate of infected, deceased and recovered people. Generally, even if it is a simple approach, Gaussian fitting procedure has been largely debated [24; 25; 26; 27] to analyze epidemic trend given the low accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. Nevertheless, in light of some measures currently in use to judge the lockdowns or to prevent spreading of the virus, Gaussian modelling provides exact and simple approximate relationships between the two relevant parameters of the function (peak time and width). Specifically, this manuscript aims to discuss in detail the mutual relations between the growth rate of infected, deceased and recovered people as a function of the large-scale anti-contagion control measures strength, and also of vaccines effects adopted to reach herd immunity. We take into account some relevant aspects such as the slowdown and trend inversion in infected growth for days close to the observed infection peaks, and/or societal system resilience. Indeed, it is found that the infection rate data show a characteristic rapid growth to a peak value followed by a slower decline if countermeasures are employed/deployed effectively.