Is My Son's Cough an Asthma or Bronchial Spasm?

Question

We had been to my sister’s house for a shot vacation in June and while we were there, my son had an attack of allergy and Asthma brought on by the presence of the cats in her house. He experienced breathlessness and wheezing and we had to get him nebulized and he was given oral Prednisone. It is five years now since he has had any episodes of asthma. He is better now and has no breathlessness or wheezing but there is still some mucous which causes the terrible coughs that he is going through. The doctors do not feel that there is anything very much the matter with him now but I am worried as I am the one who hears the terrible coughing so many times through the day. Do you think that this is asthma? Do we still need to make him use the inhaler and the steroids? I have read that bronchio spasms are generally in the list of symptoms along with all other symptoms. Please, I solicit any suggestions from any of you.