A couple additional answers here, one in-canon and one meta...
1) As we probably all know, anime features a very high number of kid/teenage characters, and very few adults. There's a fair number of diseases that affect adults more than kids/teenagers, and some of them are very rarely found in kids/teenagers (arthritis, for example). Kids/teenagers in general tend to be pretty healthy relative to the adult population, except when it comes to more minor things, like the flu (I had the flu much more often as a kid than I do as an adult). So to a certain extent, it actually makes some in-canon sense that there's not that much specific disease going around in anime.
2) At a meta-level, writers know that if you give viewers something very specific and based in reality, they're going to expect you to "do the research" and present it with real world accuracy. Some writers just don't have the time to "do the research", and hence go with generic "sickness" so they can just bed-ridden the character, maybe give that character a high temperature or heart rate, and call it a day.
