If you are looking for hope here, I sincerely apologize. My cousin passed on his own this way, 27 yr old, 5 days after vent placemant and 17 mins without breathing. My own family looked for hope and asked my opinion. I am a hospice nurse. The brain begins to die after 6 mins at a minimum. previous post is correct about what is involved in keeping the body breathing and heart beating. Those are primal brain areas and all the areas that make us human have already gone. As a hospice nurse, my opinion is that this person you ask about is only technically clinging to the shell we call our human body. The spirit is making or has already made that journey we all will some day take. I know it sounds religous. I have seen it happen alot though, this beatiful transition as we leave our shell. Keep praying miracles absolutely do happen but also keep listening and try to hear what the Doctors and nurses are saying to you. God Bless.
The brain stem would account for the breathing etc. - as that controls, to the best of my knowledge, regulates cardiac and respiratory function, hence why the person is breathing. However, it would appear that absent the rest of the brain functions that would include memories, sight, speech, walking, etc... are not functioning.
It is better to consult with the doctors as they would know more. I did some research and they did say that after day 3, the prognosis got poorer, but it is hard to tell without all the details and there is a score that doctors give them.
I am sorry about your situation.