This does sound like a flea infestation, are most of the bites below the knee, especially around the ankle?
You very often find with a flea infestation that only certain members of the family are being bitten, in fact you are all probably being bitten but only certain people are allergic to the bite.
It is possible to get a flea infestation without pets, especially if you have recently moved into the property, or if you have visited people with pets, allowed pets into your home etc!!
As a pest controller for 13 years, I treated homes on a daily bases for flea infestation, in all that time I think only about half a dozen of those homes were actually dirty, fleas don't like dirt!!!
Check you bed and pillows for blood spots, have you been on holiday recently or stayed in a hotel or had visitors stay in your bed especially foreign visitors.
I am away until late Tuesday night but will come back to check out your response and give some advice!!