Keep an open mind.
Your family member may not be able to prove she/he has mites, and you cannot prove she/he does not have mites.
If your loved one does have an infestation, saying the person has a psychological problem will add to the burden of getting rid of these vermin. Imagine being in their shoes.

Your family member may get lucky and capture a specimen tomorrow, or get a positive result for a mite DNA test.
Or they may never have that good fortune.
Either way, this is a person you love and respect.
Trust them to let you know the kind of help they need.