Agree with Liz if its clear there is no infection it's just a winter time thing in some kids but if its green/yellow/orange then it's a sign of infection and the centre should be asking them to stay away.

FWIW my kids get a cough from now until spring that is Asthma related, it sounds awful but not contagious and doesn't bother them but I am forever explaining about it and it drives me nuts.