That's right if I recall correctly, both in the comics and the show; it's as if once a person is bitten by someone infected, being that one is already infected, it facilitates the conditions for the virus to kick into overdrive and cause death, which is needed for the virus to take over, thereby restarting part of the brain stem and completing the virus' life-cycle. As Jenner described it..if it even is a virus, that all could be wrong, but story-wise it's all we know (a 'normal death', unless decapitation, also results in reanimation; but the bite is what puts it into overdrive; having the body reanimate in as little as 2 minutes).
I always thought of it like the flu, which is why Morgan, the black guy in the beginning, thought his wife was just severely ill instead of dying from infection; except it has the exact opposite purpose, in which the body usually raises its temperature to kill any invading virus, in this case it causes a swift death so that the virus can, for lack of a better word, 'activate' and reanimate the brain stem.
Sadly, Kirkman has said that he doesn't plan to ever reveal the true cause of whatever this is, and we may never even know what it actually is at all. His reasoning (if I'm remembering the interview correctly) is that any reason for the infection/virus would diminish the mystery and ultimately the fear behind the walkers and much of the feel to the series would be lost. I tend to agree, but I'm also kind of bummed if he actually sticks to that statement, even for the TV series. I can see that happening in the comics, that fits the completely hopeless and macabre tone; but the TV series isn't at that point yet and I'm not sure if it will get there, so a bit of hope and explanation in that area might be cool, and would also help to end the series' inevitable end in a more...well, TV-ish way. A kind of nice little bow on the wrapping, tying up everything.
But hopefully that isn't too far off. I actually just read a poll yesterday asking viewers if they'd like to see 7 seasons of TWD. I voted 'hell yes!'. In fact if they are to even remotely follow the comics story and include many of the story arcs and pivotal story moments, I can see it going for much longer than that even by the time the comic series ends (I'm thinking 10-12, but that'd be so awesomely nuts that I won't even hope for that).