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Simple Summary: In the absence of vaccines and antiviral therapies against COVID-19, non-pharmaceutical interventions represent the only weapon to fight this new coronavirus. In this context, mathematical models are an important tool for supporting the study of spread and transmission of this disease and to understand how restrictive measures could control the epidemic. In the current study, we adopt a new mathematical model representing the dynamics of COVID-19. We estimate model parameters with a new Bayesian method while using the public Italian data, in order to reproduce the pandemic evolution for Italy and one of its regions, Umbria. Once the model is calibrated, we also apply an algorithm, called Conditional Robustness Analysis. This algorithm allows for us to understand the influence of epidemiological parameters and non-pharmaceutical interventions on the number of hospitalized patients. This pipeline of analysis provides a quantitative explanation of the number of underestimated positive cases during the first wave of the epidemic and of the impact of lock-down measures on the hospitalization capacity. Moreover, the calibrated model is quite accurate for making updated estimations of the epidemic evolution. This study started from the request of providing predictions on hospitalization and Intensive Care Unit (ICU) rates that are caused by COVID-19 for the Umbria region in Italy. To this purpose, we propose the application of a computational framework to a SEIR-type (Susceptible, Exposed, Infected, Removed) epidemiological model describing the different stages of COVID-19 infection. The model discriminates between asymptomatic and symptomatic cases and it takes into account possible intervention measures in order to reduce the probability of transmission. As case studies, we analyze not only the epidemic situation in Umbria but also in Italy, in order to capture the evolution of the pandemic at a national level. First of all, we estimate model parameters through a Bayesian calibration method, called Conditional Robust Calibration (CRC), while using the official COVID-19 data of the Italian Civil Protection. Subsequently, Conditional Robustness Analysis (CRA) on the calibrated model is carried out in order to quantify the influence of epidemiological and intervention parameters on the hospitalization rates. The proposed pipeline properly describes the COVID-19 spread during the lock-down phase. It also reveals the underestimation of new positive cases and the need of promptly isolating asymptomatic and presymptomatic cases. The results emphasize the importance of the lock-down timeliness and provide accurate predictions on the current evolution of the pandemic. [ABSTRACT FROM AUTHOR]