None of my villagers have gotten sick.... How do they get sick anyways? I'm the one sick though (in real life, not my character) lol.
They get sick randomly. If you talk to your villagers outside often one may bring up so-and-so is sick and someone should bring them medicine. And if you go to that villager's home they will have a depressing black/purple cloud over their head and if you talk to them they say they need medicine. You purchase medicine from the store and give it to them. Make sure you check up on them the next day since they can be sick for three days (from my experience, not sure if they can be longer than that).