Zombieism isn't a disease.
People don't turn into zombies because a zombie bit them. People turn into zombies because they die from the zombie bite. In Night of the Living Dead, the recently deceased began to re-animate. It didn't matter how they died, just that they were dead. Car wreck, heart attack, stroke, gunshot, drowning, blood poisoning from being bit by a dead thing, it didn't matter. That's what made it all so terrifying. No matter what, eventually you were going to die and become one of them...the undead.