Phyllis Petruzzelli spent the week before Christmas struggling to breathe. When she went to the emergency department on Dec. 26, the doctor at Brigham and Women’s Faulkner Hospital near her home in Boston’s Jamaica Plain neighborhood said she had pneumonia and needed hospitalization. Then the doctor proposed something that made Petruzzelli nervous. Instead of being admitted to the hospital, she could go back home and let the hospital come to her.