Welcome! I am Dr. Altman, a licensed veterinarian and happy to answer your questions today.
I am so sorry no one has assisted you yet today. It sounds like he might have been exposed to a dog with kennel cough
of another bug that he is more sensitive to then your other dog. Just as when children go to daycare one can come home ill while another has no issues whatsoever.
Even dogs with current vaccinations
can get this condition, just a more mild form as with people getting the flu vaccination can still get the flu virus but a more mild form.
There are some home remedies that can help reduce the cough and give relief
1.) The steam of the shower when the cough is severe or constant
2.) A little honey three times daily can help soothe the lungs
3.) If you have a humidifier or nebulizer that would also be wonderful to use in this situation
Unfortunately I prefer to avoid cough suppressants because when suppressing a cough if there is phlegm or buildup and lead to bronchitis or even pneumonia if the material is not coughed up so this is not something I tend to administer.
I hope this information makes sense! If the cough is productive, there is lethargic or a reduced appetite then it would be best to have a veterinary exam to ensure antibiotics are not warranted. Especially with the uptake in the canine influenza in the country the past few months...
Please let me know if I can be of any further assistance to you both!
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