This has long been a problem in zombie fiction. As far as I know Mira Grant was the first zombie author to handle this, by segregating the virus into two categories: dormant and active. We all carry dormant virus, but actual shambling or running zombies carry live/activated virus. A critical mass of such causes the zombie change.
Therefore, when you get bit by something with a sufficient active virus load it turns you into a zombie without dying. It's a very clean way of handling the problem. Her series is worth a look. The first book in her series is Feed: http://www.booksofdoom.com/w/Feed
It's surprising that it took so long to address this obvious plot hole.