Greek and Roman cities were generally much cleaner and healthier than earlier cities, so their rates of disease were generally much lower. In fact, the average lifespan of a Roman citizen was not much less than what it is today, somewhere around 60-70 years, I believe. Romans had an understanding that disease was somehow linked with dirt, though they didn't know why.
Many Greeks believed that disease was sent by the gods and only gods could cure them. However, medicine was held in high regard. Hippocrates felt that disease was caused by an imbalance in the humors, and that this imbalance was from natural causes. He was probably the first "modern" physician, in that he believed that a well-trained doctor could cure illness through a combination of personal experience and the study of medical writings.
I'm not sure whether the diseased were excluded or isolated from others. Certainly this is the case with lepers, but I really don't know how much that carried over to entire cities. I do know that at several times in history, diseased corpses were used as primitive biological warfare. Simply catapult a few bodies over the walls of the city you're trying to siege, wait a couple weeks, and voila, free city.
You should try looking up some of the works of Hippocrates, Aristotle, or Galen at the library or online, I'm sure a reading of primary sources would be helpful. The link below has a lot of interesting background material as well.
http://www.umanitoba.ca/faculties/medicine/history/histories/briefhis.html