Each hospital can only treat 30 patients at once. You'll need more than one hospital and physicians to treat larger outbreaks.
When a citizen becomes sick they'll stop working at their job. If no hospital is available, they will stay near their homes, but will still travel around town to collect food and supplies. Moving among the population when sick will cause the disease to spread.
If there is room in a hospital, the sick citizen will stay there for the length of the disease, and they will have a reduced chance of dying.