Many sources cite that the population declined by 50% during this period. Although this was during the crisis of the 3rd century, I had always assumed that the city of Rome itself was relatively secure during this period?
On whether or not Rome was secured, Italy was invaded multiple times by the Alamanni and Juthungi.
The issue of Rome's population has many layered. Like many ancient capitals Rome depends on supplies from many provinces, notably from African provinces and Egypt. The grain supplies were disrupted multiple times during the 3rd century.
Also imperial government were on constant move, often settling in Milan or Revenna, and the emperor's attention were often directed at either defending the empire, maintaining the fractured empire, or reuniting the empire, their energy was simply too spread out except for Gallienus who had some time on his hand to work on Rome's culture and population needs.