They actually do. It's been years since I last read the book so the details are escaping me, but the general idea is this:

-The virus is really infectious but slow to spread in the body, depending on a lot of factors (eg where you are bitten). So people moving around while being unknowingly infected becomes a problem.

-The virus can survive on bodyparts outside of the body. So one of the first ways it spreads globally is via the organ black market. It originates in China where someone dies, their organs are harvested and sold to the highest bidder, some rich asshole gets their black market heart transplant in Brazil, and suddenly the zombies are spread in a different continent across the Ocean. Someone else gets some infected skin grafts in Turkey, may fly back home to the US with a layover in London (spreading the infection there), and only collapse and turn into a zombie days later while in their small rural community in Texas where no one has even heard the rumours about the zombie disease.

-Given how relatively easy to hide it is (infection due to slow spread, or even zombies themselves due to being classical slow zombies) a lot of refugees and people moving around have both the classical tropes "Im bitten but I'll hide it, I will probably get cured" and "my family member is infected or has turned into a zombie, I'll just lock them in the back of the car and pretend they're just sick or sleeping". The people smuggling them around (coyotes, human traffickers, etc) turn a blind eye even when they start to realize the truth because they are still getting paid and aren't known for their ethical backbone.

-The largest powers fumble the bag in every way possible. China is where the disease originates, and they try to keep it under wraps until the infection has spread too far and the country is on the verge of collapse. Russia keeps a tight information lockdown. The US first is in disbelief, then thinks they can contain the problem secretly via spec-ops and such because they don't want to spread panic and because, coming out of the WoT, they are worried about lacking the political capital to start another massive military undertaking, and only realize they need to take it more seriously once it's too late and you're having outbreaks everywhere. A lot of other powers/nations just are in denial, we hear a NATO supreme commander confess about how they just didn't believe the rumours at first because sure, there may be a viral outbreak or civil unrest or whatever, but the dead coming back to life to eat the living? That's ridiculous. Which is also why no one hears Israel's warnings (a big point being made about how uber-paranoid the Israeli intelligence community is, which is why they are the ones to take it seriously).

-Shady billionaire tries to make a quick buck promoting fake cures, which increases the problem of people hiding or underreporting infections, especially combined with the lack of proper information given by the governments until it's too late.

Combine all of this, and the book paints a believable version of a slow burning societal collapse where, by the time everyone realizes "this is the real deal, let's get our asses into gear", the zombies are basically popping up everywhere and no place is safe.