Read World War Z guys!
In this book, the Zombies start in china. As the Chinies authorities want to hide it, they give it enough time to grow almost unoticed so it can spread.
Thin about it, a zombie who wander around into a sub-urban place, or even a city can do some damage befor being terminated.
Ill make a little 'scene' -
lets imagaine a tourist who come to New-york. Back at his homeland, or even at the airport, a crazy Hobbo attacked him a bit, jsut a little scratch. He fly to New york. as he went into Coma, die and rise again, He will in best case scenario will bite a room keeper or a cleaner befor a police man will come and shot him. In worst case, he will bite enough peoples who try to struggle with him, when trying to move hi away from them. For an exmple, he will walk to the lobby, and start attacking the first man he see, on the staires. teh man will run away, after getting bite as it was a surprise attack, The other dude didnt expected that. Then, In the lobby, the bited man will yell "Here he is! he bite me!". A worker will walk up to him to see what is going on, try to talk to him. The ex-tourist will charge at him. 2 injured peopels so far. Some other workers, maybe even some other peoples at the hotler will try to struggel him down, so he wont run away befor the Polcie arrive. He will scratch or bite some other peopels. We have a minimum or 4 Injured peopels befor the autorities can even kick in. And by the time the Doctors will be able to see that something is wrong with there patients, they will attack other peoples too. In a hospital, they can Attack many harmless peopels befor security can arrive, and in some cases, as the security man will propebly want to capture the crazy man and not shot him, maybe even kill the security man and keep on the rampage.
And this is how an outbreak start. From there, As Infected peopels will run away, or even rise as zombies at sub urban places, they can travel to other places. Who will suspect that something is wierd for a man who just came from an Hospital being tired and wanting to sleep? A good night sleep of lets go wild 16 hours is so crazy? espeicely if it is a old guy. Then he go on.
The authorities will propebly deny it at the beggining, claim it is some other Diseases. maybe Rabies.
And which country will dare to say to the peopels "be carefull from crazy cannibals peopels"? which country will dare to move all the peoples to fortify places, wher an outbreak wont start? No one can control what come in and out of a city. There are so many peopels with scartches and bites who are not caused by zombies that you cant stop them all fro a "minor" thing like that. Its take shotung down all the cities and villages. Even years after, there can always be a Hilly billy zombey coming down from the mountains to flame it again.