COVID-19 infections can spread silently, due to the simultaneous presence of significant numbers of both critical and asymptomatic to mild cases. While for the former reliable data are available (in the form of number of hospitalization and/or beds in intensive care units), this is not the case of the latter. Hence, analytical tools designed to generate reliable forecast and future scenarios, should be implemented to help decision makers planning ahead (e.g. medical structures and equipment). Previous work of one of the authors shows that an alternative formulation of the Test Positivity Rate (TPR), i.e. the proportion of the number of persons tested positive in a given day, exhibits a strong correlation with the number of patients admitted in hospital and intensive care units. In this paper, we investigate the lagged correlation structure between the newly defined TPR and the hospitalized people time series, exploiting a rigorous statistical model, the Seasonal Auto Regressive Moving Average (SARIM A). The rigorous analytical framework chosen, i.e. the stochastic processes theory, allowed for a reliable forecasting about 12 days ahead, of those quantities. The proposed approach would also allow decision makers to forecast the number of beds in hospitals and intensive care units needed 12 days ahead. The obtained results show that a standardized TPR index is a valuable metric to monitor the growth of the COVID-19 epidemic. The index can be computed on daily basis and it is probably one of the best forecasting tools available today for predicting hospital and intensive care units overload, being an optimal compromise between simplicity of calculation and accuracy.