Well, I'm by no means a medical expert, but perhaps if Biggie's cough doesn't go away after another 2-3 weeks, it would seem like he has a tracheal collapse. If however, the vet prescribes some antibiotics and his cough does go away, then I would think that he doesn't have any problem with his trachea. Hopefully that is the case, as a tracheal collapse is definitely a bigger problem. No problem, I'm glad to help, I just did a quick search and was able to find some good info on both problems.
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Kennel Cough Info:
If your dog is diagnosed with kennel cough, your veterinarian will likely prescribe an antibiotic to help prevent any secondary bacterial infection and a cough suppressant. We have found in those persistent cases of kennel cough, the use of a relatively new antibiotic, azithromycin, to be effective. This medication is very effective in the treatment of the mycoplasmal forms of tracheobronchitis. Again, before any treatment regimen administered, is it is imperative that a proper veterinary examination and appropriate diagnostics be performed.
Links:
http://www.cah.com/library/kennekcough.html http://www.marvistavet.com/html/body_kennel_cough.html