If you’re feeling those bites, you’re likely dealing with either fleas or lice, or something along those lines.
Do you have any pets at home? If so, it’s probably fleas; if not, lice seems more likely.
Dust mites—which typically inhabit things like bedding, carpets, or upholstery—aren't actually visible and they don't bite. They mostly just trigger allergies in sensitive people.
Generally speaking, when it comes to killing off bacteria, mites, or bugs, I know steam disinfection is quite effective, and it's a safe way to go about it.
I also managed to find a useful link here in case it turns out to be lice.
http://www.publichealth.gov/lice/ddd/lice.htmI found something else as well; it could be bed bugs. I read somewhere that they tend to lie dormant during the winter, which might actually align best with how you described your situation.
There are some photos of the insect there, so you can compare them to these little monsters.
http://www.epa.gov/pests/bedbugsAnd here is a more detailed description in English:
http://www.hsph.harvard.edu/bedbugs/From what I understand, the real issue with them is that they hide in cracks, walls, floors—everywhere. That would explain why simply replacing your mattress didn't solve the problem.
It might be worth looking for a professional to help you get rid of these pests once and for all.