Italy suffered heavily with the new pandemic crisis caused by the novel coronavirus SARS-CoV-2. Given the low number of tests performed on the early stages of the outbreak, Italy lost track of most of infections. We use a modified SEIR model to reconstruct the most realistic infection curve using the hospitalization curve of the registered data. Using this method we estimated that, by the end of the first infection wave, about 3-4% of the population will have been infected by the virus. Following the same process, the number of deaths is projected to be between 100000 to 115000. We also find a significant correlation between the number of tests performed, the fraction of undocumented infections and the rate of change dI/dt of the real infection curve. We conclude that herd immunity is not enough to contain further spread of the disease inside the country.