Best evidence to date suggests that the Sahara dried up about 5,000 years ago, possibly in as little as 300 years, due to climate changes resulting from the precession and rotation of the Earth's rotation axis.
... around 8,000 years ago, the Earth's orbit was slightly different to how it is today. The tilt changed from around 24.1 degrees to the present-day 23.5 degrees.
Given the very strong dependence of vegetation on water availability, the end of the 'Green Sahara' came about quite suddenly around 5,500 years ago," Schmidt said. "Thus, a very slow change in the orbit (led) to an abrupt collapse in that ecosystem.
As regards the claim that human hunting of large carnivores resulted in over-grazing by large herbivores:
I find it an incredulous suggestion that stone-age humans were preferentially hunting large carnivores in an eco-system in which large herbivores were numerous.
The desertification of the Sahara, by all accounts, certainly occurred at least 2,000 years prior to the Founding of Rome in 753 BC.
However, the previous update should not be interpreted as meaning that all of North Africa was exactly as arid 2,000 years ago as it is now. There is considerable evidence that the northern extent of the Sahara is somewhat more arid, and of modestly greater extent, than it was 2,000 years ago. The relationship of these changes to other climate changes, both global and regional, over the past 11,000 years ago remains undetermined.