406's real name is unknown. She used to live in the same apartment building as the film's protagonist Columbus, her apartment number being 406 and so Columbus referred to her by that number since he didn't know her name.
Columbus became aware of the zombie plague after 406 knocked on his apartment door seeking shelter and companionship. The frightened girl explains to Columbus that she was walking home from work when a sick homeless man rushed at her and tried to bite her. Columbus lets 406 rest on his couch for a while, but later he finds that she has turned into a zombie.
It seems that 406 was indeed bitten by the homeless man she encountered and she turns into a zombie while staying in Columbus' apartment. She attacks him and chases him through the apartment while he frantically tries to reason with her. In the end, Columbus is forced to put 406 down by bludgeoning her over the head with the lid of a toilet tank, snapping her neck in the process.