Brief Answer:
Viral infection unlikely

Detailed Answer:
Hi

Thanks for asking on this platform

I have gone through the query and do understand your concern.
Cough without fever can be due to allergy, reflux or atypical bacterial infection.
Cough due to cold allergy or reflux of stomach content into upper airway is common in night.
I won't get overly worried if there is no breathing difficulty.
In such condition giving antibiotic like azithromycin which covers atypical bacteria with cough syrup containing sputum thinning agent like Ambroxol with levosalbutamol which prevents airway spasm (like Ascoril LS or Ambrodil LS) and antiallergic like cetrizine syrup will help.
For this age the dose of azithromycin syrup is 5 ml(each 5 ml contains 100 mg) once a day for 3 days. This is an antibiotic and should be taken after consultation with a doctor to confirm infection.
Cough syrup 5 ml two to three times a day and cetrizine 5 ml once at night time for 5 days will be OK.
Please keep a watch on any breathing difficulty and if you notice any please consult your pediatrician.

I hope this helps.
Please let me know if you have any more queries.
Regards

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