I don't think i'm sick, i've out grown asthma years ago, and i'm coughing persistently!?

[Where Angel's Dwell]

I grew out of my asthma 10 years ago when i was 8. i'm 18 now, so it's been a long time. My doctor said that i had out grown it, but he told me that whenever i got a cold, it might trigger some coughing. I got sick, and i never really coughed again. I'm not exaggerating i hardly ever cough. But recently i've been coughing every night and during the day too! i went to the docotr, and he prescribed me antibiotics, my mother is a docotr, and she didn't alow me to take it because i didn't have a temperature, the 'mucus' from my cough wasn't coloured, and so i wasn't that type of sick, she also insisted that it was a way of building up my immune system (she doesn't want me to become immune to antiboitcs, which is apparently happening at the hospitals lately). My throat started hurting a little (for one day) but i didn't believe i was sick at all, yet the docotr insisted for me to take some antibotics, and my mother wouldn't allow it, instead she gave me cough syrup, which is what the doctor also prescribed. I don't feel sick, but this coughing appears to be lingering, and i don't know why? the weather is currently freezing, we had hail, storm and rain recently, and a few days prior to that we had scorching hot weather (Melbourne weather, what's new?) so i don't know what this could be?! My mother (and the docotor as well, i think) insist that i'm recovering from some viral infection, and that has to pass on it's own ( the docotr want's me to take antiboitcis though) and although i feel no other symptoms besides a cough, what could it be?! i certainly don't want to get "onset asthma" but that really only occurs much later in life, during your 50's, i'm freaking 18. Could it be the cold, which has made me sick and shown no other symptoms besides coughing, allergies, it's currently summer? And i recently when running in the rain (I had to run home for like half an hour, drenched which wind and cold water) Not fun.

Thank you for this
when i breathe deeply, there is no wheezing. hardly any mucus. and i don't find that i'm panting, because i'm short of breath.