Modern scientics stated a bit different. Stefan Kröpelin showed in his study of groundtypes in lakes in the tschad area, that the desertification started about 8000 years ago very slowly and finished about 2700 years ago.
after that time, human activities like deforrestation and overusing with goats and sheep herds corrupted more fertility than climate change did after this time - you can see this actually in northern china.
this process was forced more active under the islamic occupation (they didn´t have a tradition of farming).
it´s also lately shown that because of climate change over the last 200 years the sahara is regreening in many parts (expescially in the sahel-zone). many modern day conflicts in this area like in mauretania, dafur are related between the old conflict of herding and farming.