I certainly agree that Rome eventually suffered catastrophic population decline, but in the 1st century or earlier? Come on. And a total population of 5 million would be dramatically too low to support the number of legions they maintained in the field, especially in a pre-modern economy that was still agragarian in nature, no matter how refined.
Am I misunderstanding this somehow? The city of Rome itself had at minimum a population of several hundred thousand, with serious estimates ranging upwards to a million or more (and historical claims up to 7 million, I believe, though I think those can be discounted). That would only be a small portion of the total, given the presence of other large cities like Ephesus, Antioch and Alexandria with 6-figure populations each. And yet most people would still have lived in villages or farms, not in the larger towns and cities that were so plentiful at this time.
I'm taking this as an additional and not exclusive approach to estimating the population. And I'm taking it to mean the population of little more than the Italian peninsula. Big assumptions on my part in the lack of a more complete account.