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Solid estimates describing the clinical course of SARS-CoV-2 infections are still lacking due to under-ascertainment of asymptomatic and mild-disease cases. In this work, we quantify age-specific probabilities of transitions between stages defining the natural history of SARS-CoV-2 infection from 1,965 SARS-CoV-2 positive individuals identified in Italy between March and April 2020 among contacts of confirmed cases. Infected contacts of cases were confirmed via RT-PCR tests as part of contact tracing activities or retrospectively via IgG serological tests and followed-up for symptoms and clinical outcomes. In addition, we provide estimates of time intervals between key events defining the clinical progression of cases as obtained from a larger sample, consisting of 95,371 infections ascertained between February and July 2020. We found that being older than 60 years of age was associated with a 39.9% (95%CI: 36.2-43.6%) likelihood of developing respiratory symptoms or fever >= 37.5 {\deg}C after SARS-CoV-2 infection; the 22.3% (95%CI: 19.3-25.6%) of the infections in this age group required hospital care and the 1% (95%CI: 0.4-2.1%) were admitted to an intensive care unit (ICU). The corresponding proportions in individuals younger than 60 years were estimated at 27.9% (95%CI: 25.4-30.4%), 8.8% (95%CI: 7.3-10.5%) and 0.4% (95%CI: 0.1-0.9%), respectively. The infection fatality ratio (IFR) ranged from 0.2% (95%CI: 0.0-0.6%) in individuals younger than 60 years to 12.3% (95%CI: 6.9-19.7%) for those aged 80 years or more; the case fatality ratio (CFR) in these two age classes was 0.6% (95%CI: 0.1-2%) and 19.2% (95% CI: 10.9-30.1%), respectively. The median length of stay in hospital was 10 (IQR 3-21) days; the length of stay in ICU was 11 (IQR 6-19) days. The obtained estimates could be instrumental to refine mathematical modeling work supporting public health decisions., Comment: 20 pages, 4 figures