I agree with keeping a child home when they are sick with a fever, etc.
I do want to comment on a child ending up at school, sick, after feeling that way at home before school. Of course it depends on the child, but if I kept one of my kids home every time he complained of not feeling well he'd be home as often as he'd be at school. And yes, there were times that a nurse would call and say, "Ben said that he wasn't feeling well this morning before coming to school" and I'd have to explain that he was always complaining. Mainly because he didn't want to wake up and go to school. Sometimes he really would be sick--the boy who cried wolf.
There's also been times when the nurse will call and say that he's in the health room. I don't run in right away to pick him up (I do if there's a fever or the nurse things that he's really sick) and will suggest that he go back to class. I do talk to him on the phone and can usually tell if he's really sick. I might sound heartless, but I've learned when to react and when not to with him.
With my 12yo, if he complains about being sick I believe him right away and he stays home if he says he needs to. He's rarely sick and doesn't want to miss school and have work to make up. So if he says he needs to stay home or that I need to pick him up from school I do that.