The problem with it is that it's stupid. It's not stupid because it's a stupid idea, or a waste of time, or the wrong approach, or anything like that, it's just done really badly.
Basically, the authors make some simple assumptions about the rate at which the dead rise from graves, the rate at which they turn humans into zombies, and the rate at which humans kill them, and show that in any outbreak the zombies will kill everyone. They add a few slightly more subtle tweaks, and show that the zombies will still kill everyone.
Their conclusions rest entirely on one assumption that they make at the beginning and never defend, which is that dead people turn into zombies without any provocation, at a rate proportional to the number of dead people. That is, the number of new zombies rising in a given night is proportional to the number of people who have ever died (and not already risen). Even if you kill a zombie, it just goes back to being a dead person and will rise again in due course (proportional to the model parameter ζ).
Well, duh. Obviously in those circumstances the human race will be replaced by zombies. It really doesn't take a lot of mathematics to work that out.
The problem is that this falsifies their claim to a serious conclusion:
... While the scenarios considered are obviously not realistic, it is nevertheless instructive to develop mathematical models for an unusual outbreak. This demonstrates the flexibility of mathematical modelling and shows how modelling can respond to a wide variety of challenges in 'biology'.
I could believe that if they had shown that the modelling showed how different outcomes related to different assumptions. But that they published this without identifying the key assumption that produced their "zombie takeover" conclusion — the assumption that there is no way, natural or technological, from preventing any corpse from eventually becoming a zombie, contrary to pretty much all authorities as well as common sense — the only conclusion is that the mathematical model distracted them from thinking properly about the scenarios.
There's just no point doing this sort of thing unless you take it seriously.
Update: Wrote my own improved model