Let's pretend for a moment that zombies are real (as if half of you weren't already daydreaming about that very thing). Have you noticed how most zombie movies take place only after the apocalypse is in full-swing? By the time we join our survivors, the military and government are already wiped out, and none of the streets are safe.
There's a reason the movie starts there, and not earlier. It's because the early part, where we go from one zombie to millions, doesn't make any sense. If you let the creeping buzzkill of logic . . .