Kids must be kept out a full 24 hours free of fever or vomit. Strep... anything that requires antibiotics... kids must be on meds a full 24 hours (which is when the child is no longer contagious.)
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Colds, well, you play it by ear. My son (and myself) hold onto coughs for weeks, even months after a bad flu/cold virus and they DO sound terrible... a whole lot worse than they are. Currently, my 10-year-old's cough is JUST starting to taper off and it's been a good 8 weeks since he was actually sick. He was checked out by a doctor and even put on a full course of anti-biotics "for good measure." Of course, it made no difference. He had no fever. No runny nose. No lethargy. He just had a nasty cough. I kept him home about 3 days for the inital cold/flu but he's been in school ever since.
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It's hard to say if this kid is actually the one that passed on the germs to yours. Germs are everywhere and people are most contagious before they really know that they are sick. This means anyone in that class could have brought it in before they knew they were sick. Or, your child could have picked it up at the grocery store that morning.
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Personally, it drives me crazy when people send their kids to school miserable and sick but I've learned that coughs aren't neccesarily a good measure of who has is actively fighting a virus and who is no longer contagious but holding on to that cough.
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