Hi Louis, great question! Here are my thoughts.
How many rats are there in cities
I made some back-of-the-envelope calculations of this a year ago, the goal was to compare the cost of contraceptive treatment per rodent with the cost of contraceptive treatment per pigeon.
Not much has been published on this question, but there seem to be about 220 rats per in Baltimore, if one extrapolates from the numbers presented in this paper.
Here are some other reports to cross-check against:
According to this report from a US agency, "the total rat population of New York City is estimated to be not more than 250, 000 or one rat for 36 persons". A similar claim is made here. Going by the ratio of rats to humans, both of these seem to be at least roughly in accordance with the Baltimore data, which is 47 000 rats and 619 000 humans. According to this source, where they try to estimate the density of rats in urban areas of the UK, "each rat has a rather spacious 5,000 square metres to roam around in." That means that there are 1 rat for every 0.005 , which is equivalent to 200 rats per . Also very close to the numbers for Baltimore.
could this be a potential area to improve animal welfare?
I think fertility control for rodents is promising, here are some somewhat related thoughts I have on contraceptives for pigeons. Although the only commercially available product for rodents that I'm aware of (ContraPest) is quite expensive as far as I can tell; see my back-of-the-envelope calculation.