Unfortunately, the model appears to be flawed. In the zombie literature, opinions differ as to whether a zombie epidemic can affect those who are already dead. Although many zombie works do have zombies arising from graveyards and such, the most (faux) serious work on zombies, Max Brooks, asserts that the zombie virus affects only the living. But, in either case, the literature is unanimous that destroying the brain puts down a zombie permanently.
In contrast, the paper’s model posits that zombies can arise from the dead population, and that dead population includes not only dead from natural (non-zombie) causes, but zombies that have been put down. This is clearly wrong.
Fortunately, Brooks’s scenario can be recovered by setting to zero the parameter that dictates how quickly the dead become undead. With that parameter set to zero, it doesn’t matter that the dead population is too large. However, the more typical scenario cannot be recovered without a new model, necessitating a new solution.
Alas, the world may have to wait a little longer for a serious mathematical treatment of the zombie problem.