His exposure to the girl made him infected. Most cough and cattarh are usually due to viral infection which will normally run their course with or without any form of treatment

Do not self medicate. Please take the baby to a hospital immediately. TREAT AS URGENT. Babies from 2months below that are sick for whatever reasons should be attended to immediately at the hospital.

Is your child not over dressed?

Is the room temperature cool enough?

Is child sweating in an environment where others are not?

If yes to the above questions you will have to take your child to see a paediatric endocrinologist for evaluation and management.