For a variety of reasons the people working the caravans from Petra to Mecca to Medina had remained relatively isolated from these other devastating events, or, most likely, they were affected equally but didn't realize it due to their normally deep level of poverty and their physical isolation from the Mediterranean world.
There are interesting debates regarding the degree of European depopulation. First of all no one doubts that millions died from the plague and whatever the weather anomaly was that gave Western and Northern Europeans an incentive to burn almost every piece of furniture, parchment or other burnable in a short period from 535 to 541. Secondly, back in the days before unemployment insurance, modern machinery, food surpluses, trucks, trains, cars etc. it didn't take much of a disaster for darned near everybody near it to die if only from simple starvation.
Thirdly, there are "secondary diseases" that take the opportunity to kill under these conditions.
Whatever, the death rates were different in different places. The Eastern Empire lost just under half its people. The Western Empire probably lost over half along the Mediterranean and more like 90% in the NW.
The Arabs in the peninsula didn't know anything of this.
BTW, the invasion of the Mediterranean civilization didn't begin until AFTER Mohammad was dead and AFTER Petra was converted. You should note the name of that place ~ it's in Jordan. They appear to have gone cash at the beginning of the disaster and could pay hard money for whatever the Moslem commanders needed.
If the Arabs had been Roman Catholic Monks the outcome would have been the same!