For the exponential phase of the epidemic to kick in, you need a cluster that goes undetected for a week or so. For example it can be a young local person that just shrugs it off as a bad cold and meets a lot of people while he's sick---which is exactly what happened in Italy. Italy found out after someone that "shouldn't have gotten sick" ended up in the ICU with pneumonia and a clever doctor decided to test him for COVID-19. That was in mid February, one week before 10 municipalities were the first to be locked down.

Looking back at the January data, however, there were already more cases of pneumonia than usual in that same area and many of those that survived now have antibodies. And in fact, the strain that is hitting Italy was actually already present in Germany in January based on genome analysis; for whatever reason the real outbreak started a bit later in Germany than in Italy, but that's expected because the initial phases of the epidemic are random.