I doubt there were any rats who spread the plague in the desert

While the range of Rattus Rattus excludes the deep desert it nonetheless covers virtually all of the habitable areas of the Middle East.

That map would also seem to be less than entirely accurate as cities can provide a viable habitat for rats even in climates where they might not otherwise survive. So for instance there are certainly rats where I live, in Dubai, despite the UAE not being included in that habitat range map.

So that being said, the Black Plague devastated the Islamic world just as it did Europe and China. Spreading out of the steppe lands where the plague was (and still is) endemic, it spread across Persia to Constantinople and from there to Egypt and beyond. The total population loss was about 1/3rd, but in major cities like Cairo or Baghdad where the disease best spreads the proportion would likely be even higher. All this close on the heels of the widespread destruction wrought by the mongols a century before.

The result was a demographic depopulation which was arguably less easily overcome in the Middle East than it was in Europe, though I'm less familiar with these ideas.

It also had similar ramifications on cultural thinking. Just as Europeans would come out of the Black Death with new concepts about the changing nature of the world, so Islamic thinkers like Ibn Khaldun (who lost both of his parents to plague) would conceive of systems in which violent regime change and demographic shifts were practically a built-in feature of human civilization.