In the comics, if you die of natural causes you still become a zombie -- bite or no bite. Everyone on Earth is already infected and destined to become a zombie. I'm not sure if this applies to the TV series, who knows -- they might go a different direction.
About, what, 200,000 people died daily? So in one day and without warning, 200,000 people resurrect in their deathbeds as zombies, infecting confused coroners, loved ones, and doctors. The next day, 200,000 people again die of mostly natural causes + newly infected humans. Not sure if it happened worldwide all at once, but if it did, it'd be hard to contain something that was literally everywhere and appearing exponentially.