All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is not obligatory to purify those places in the room unless you want to perform the prayer on them. If they are impure and you put a rug on them, and you prayed on the rug, then this is permissible.
It is not ruled that the impurity is transferred just because of doubt, so if you want to purify this place which is impure, then you do not have to purify other places just because you doubt that the impurity might have transferred to them.
However, if you are absolutely certain that the impurity was transferred, then you are allowed to purify the places you know have become impure. But you must avoid whispers because being carried away with them leads to a great evil.
Allah knows best.