I Had A Cold That Turned Into A Tracheal Infection

I had a cold that turned into a tracheal infection and have since been diagnosed with hyperactive airways. I was given prednisone 5mg. starting with 7 one day then decrease by 1 each day. it didn't help he then gave me 5, 50 mg. prednisone to take 1 each day for 5 days and I still have the cough that feels like I swallowed something the wrong way. still taking puffer and cough syrop but it doesn't seem to make any difference. I finished prednisone last Friday. How long does it take for something like this to clear up?