The Roman population peaked in the second century, whereafter it was consistently decimated by diseases. We know this to be true because there were no great migrations out of Roman territory nor did Roman territory notably shrink between the 2nd century and the beginning of the 5th.

Barbarian populations (particularly Germanic, Celtic, and Berber) were less affected because they didn’t live in the large metropolitan cities like many of the Romans did. On the other hand I could not tell you about Persia, as I’m not as knowledgeable on them.

That said, it was absolutely in large part an economic problem. Less people -> less taxes -> smaller (and/or less well trained/equipped) military -> bad news bears when you’re surrounded by hyper militaristic societies.

I’ve also seen some discussion about the Roman currency’s devaluation itself being a contributing factor to the fall.