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Runny nose is mostly due to viral infection.
Viral infection usually improves with supportive measures.
He might have bronchospasm.
So this inhaler has been prescribed.
As it is still continuing for a week, I would advise to take him to his pediatrician to have an repeat examination.
If it is needed, doctor can do blood work, chest X-ray.
I hope this information will be helpful.
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Wish his healthy life.