Is there a way to (at least) estimate the population of "SPQR" for the following years?
Do you know any reliable sources?
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Picture a graph starting at 50AD around 41 million, peaking around 200 AD at 45 million and then dropping slowly and steadily to 36 million around 500 AD, and you won't be too far off. At least not in trend, but magnitude is debatable. That's a 10% increase from 50 to 200AD, and then a 20% decrease to 500AD.
The ultimate source I use for these kinds of numbers is McEvedy & Jones' Atlas of World Population History (out of print, but available at libraries and online if you know where to look). However, its primarily geographic-based, and the boundaries of that particular political unit changed a bit over the eras, spanning portions of 3 continents. This means retro-fitting it to a polyglot empire like Rome takes some hoof-work. Honestly, I'm reluctant to do that kind of hoof-work myself (counts 1, 2, 3...) 8 times just for the purpose of this one question.
Fortunately, McEvedy was kind enough to do that for his 362AD map in his New Penguin Atlas of Ancient History, which had a number for TRE at that date of 39 million. His Atlas of Medieval History has a similar map for 737 AD, with an entry that reads:
In its heyday, the second century AD, the Roman Empire had a population of about 45 million. Thereafter, a slow decline set in, and on the eve of the Western Empire's collapse the figure was more like 36 million, a drop of a fifth.
By "eve of the ... collapse" he of course means the rough vicinity of 500AD. So this covers the entire year range you were interested in (minus the first date), and even verbally plots a curve for you.
The "41 million" at the top is my own quick-and-dirty personal estimate, by fitting the curve M&J showed for Europe from 50AD to 200 AD (its population peak) to McEvedy's Roman Empire numbers. A better figure could probably be arrived at by compiling the various appropriate M&J regional numbers for 50AD, but I doubt this is far off enough to be significant. The trend should be right, and the absolute numbers are debatable. The important thing is consistency of sources, if you want to look at trends.
I will note that McEvedy/Jones numbers tend to be on the low side compared to others. I like to use them because nobody else has data that competes in scope, detail, and consistency with theirs. I also think that historians (and amateurs) have a natural bias favoring larger numbers, so theirs feel safer to me. However, it should be pointed out that estimates vary greatly, and this one will be on the lower end.
Wikipedia has an article, Population of the Byzantine Empire, giving the population and area of the Easter Roman or "Byzantine" Empire at 19 dates from AD 300 to 1320.
Only a few of those dates are the ones you are interested in, but that is a start.
Wikipedia also has an article, Demography of the Roman Empire, giving various population estimates.
Here is links to articles about the population of the Roman Empire: