Did plagues reduce the roman population to the point it could not sustain itself?

Yes, there were huge plagues in Ancient Rome that caused all kind of devastation

One of the FIRST of the BIG plagues was the Antonine Plague, 165-180 AD, also known as the Plague of Galen, an ancient pandemic, whether of smallpox or measles, they are not sure, claimed the lives of TWO Roman emperors. 

The disease broke out again 9 years later and caused up to 2,000 deaths a DAY at Rome, one quarter of those infected. 

Total deaths have been estimated at five million. 

Disease killed as much as one-third of the population in some areas, and decimated the Roman army. 

This thing traveled far too, up into Gaul, all over Roman Europe. 

The Plague of Justinian may have been the first instance of bubonic plague and was one of the causes of the Fall of the Roman Empire. 

Smaller but no less deadlier plagues played havoc throughout the Roman Empire over many years. 
Diseases from unkept Roman plumbing with the ground water mixing in with rain water.