reduced steroids

My husband has been on monthly steroid injections in an attempt to reduce oral steroids. First 3 months at 80mg, 4th month down to 60mg, and 5th month down to 40mg. As dosage is reduced so does his peak flow (down from 660 to 480). Now unwell with constant cough and production of much mucus. Can you get bronchitis as a result of asthma? Any one else on steroid injections? Thanks.