Diseases generally don't come from domesticated rats or other captive bred rodents and small pets
It is a bit silly to worry about though,considering wild animals and captive animals are extremely different in regards to illness and exposure and even parasitic organisms that they may carry.Of course,there will always be extremists wanting to completely ban small pets(and in some cases pets in general with certain organizations that fool massive amounts of people daily)who will twist things and make it seem like something horrible will definitely happen to those who ever have certain pets.
For example,people in general would panic over the thought of salmonella being transmitted by reptiles so many people refused to own reptiles(ignoring that proper hygiene essentially eliminates the chance!).Or,in the case of rat bite fever people once again jumped to conclusions(most likely with help from the stereotypical filthy sewer rat and such)without understanding the truth about rats.
Regardless,in almost any case you're actually safer handling and keeping the more "exotic" pets(reptiles,small animals including rodents and rabbits,birds,etc)as long as you take responsibility and practice good hygiene compared to owning dogs which are constantly being exposed to various diseases and parasites and other nasty things without anyone thinking about it.Of course,there's also a lot of misunderstanding about raw feeding for carnivorous pets being just as unsafe,but that's a separate issue although yet again lots of assumptions do affect it with little focus on the reality
In the end,people often overthink and oversterilize.This is horrible for the immune system though,but as long as you take standard handwashing and hygiene practices and stick to them you'll be fine