It is difficult to say if your son does have asthma despite his strong family history of asthma. He has only been ill for 2 weeks this time. His illness appears to have started off with an infection as he was put onto antibiotics. He was diagnosed as having a post nasal drip which may have come from sinus infection. Following this kind of mainly upper respiratory infection, many children develop a cough like you describe here due to a condition called 'tracheitis'. This is a dry cough which is irritating and can last for up to 6 weelks. The main thing you need to do now is to keep your son's nose clear. Use a Sterimar nose spray twice a day. It will help him to give him a safe cortisone syrup such as Prelone Syrup 5ml daily for 5 days to help to heal his tracheitis.
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