To follow up on that, Simon's illness was most likely epilepsy. If you're looking for any one single boy with a physical illness, there's your answer. Piggy's asthma is up to question because he is an overweight, out of shape boy. He believes he has asthma because that's what his aunt keeps telling him; unfortunately, this is the same aunt that keeps feeding him candy. I'm not saying he doesn't have asthma, just that it isn't certain that he does.
In a literal sense, there are a couple boys with diseases. Piggy has asthma and Simon has some illness that causes him to have small seizures and to faint. We learn about both of these conditions in the first chapter almost as soon as we meet each of those characters. On a more figurative level, every boy on the island is infected with a disease and that disease is the one the "Lord of the Flies", the head of the pig on the stick, tells Simon about - the fact that the real evil, or beast, on the island is the evil that lives in each person. Golding's contention was that everyone was essentially evil but most people manage to control that evil because we are civilized people. He felt that if civilization were removed, many people would revert to that savage nature inside of each of us. In that sense, each boy on the island was infected with the disease of savagery.