I have in the past seen claims that human activity, specifically overgrazing by domestic goats and salinification of the water table by repeated irrigation, has been responsible for the advance of the desert into previously habitable areas of North Africa. That's a far cry from your book's reported claim, but its at least a nod in that direction.
It appears that a temporary change in the monsoon rain pattern at that latitude caused the Sahara to temporarily become habitable from about 10,000 to 5,000 BP. Scientists have given it the rather uninspiring name Neolithic Subpluvial. Supposedly some inscriptions in Egypt and the Sudan depict this. Here's an article in Live Science from 2006 that discusses the matter. However, before that time the Sahara was actually larger than it is now. So we are still better off that where we started before the monsoon aberration, and can probably expect it to go back toward that larger size, human activity or no.
There are theories that these changes just happen periodically.
I found an article in The Globalist from 2008 that makes the slightly less extravagant claim that human activity accelerated the re-desertification process when the rains went away. So this is an idea that is floating around. I don't know how widely-accepted it is though. Coincidentally, I also heard a story on NPR this morning about "carnivorists", or scientists who study carnivores. They do make the claim that taking carnivores out of an ecosystem can lead to large changes in the vegetation.
Still, we find deserts naturally worldwide in regions where rainfall is less than about 250mm a year. At its current northern limit, the Sahara receives about 100mm a year, and 150 at it southern. So I think it is fair to say that it would be a desert, regardless of what the animals around it are doing.