Have you heard about demographic transition?
If Muslims do more children than Christians or post-Christians it's not because of their religion but become they come from poor countries. A lot African migrants are Christians, and their fertility rate is high.
You probably gave it a better perspective than mine - I admit your explanation is better. The outcome still remains the same though. The demographic transition is happening as we speak. Funnily enough though; in Greece the people who migrated here a couple of decades ago (Nigeria, Kenya, Albania) blended in with the rest of us. They only got about 2 children tops. So there you have it... The "virus" got them too.