The rules in the comics are that the zombie bites' kill you because of infection, whether or not you're killed by a normal person or natural causes you will come back as a zombie in this universe. I'm not sure how long it takes for someone to turn into a zombie but it's not instantaneous.
Originally Posted by pruppert
SPOILER: when that lady's sister got bit in the swarm and died. There wasn't too much so far to determine the time line between getting bit and being a zombie (I think the black guy with his son said the fever took his wife, and may have mentioned it, but i didn't remember). Anyways, she bleeds out, and her head turns towards her sisters hand, which is right next to her head/mouth. Thought there was a chance she was going to bite her sister up after a quick transition to a zombie...