The other day I was at a center for hospice care with my son, bringing cookies to the residents. This is a place I have volunteered at from time to time, but we haven't visited very often lately due to a hectic schedule (I always feel like my son and I get so much from the visit, though we give so little and we both leave the place with a smile on our face). They used to have a pet bird, but I didn't always understand that, from the veterinarian's perspective, birds can transmit zoonosis (meaning diseases that can go from animals to people) especially in immunocompromised people. I don't know the whole story, but the bird was short lived (being a previous bird-owner, it may have been because of the mess they make).
I found out they recently got a cat. I was excited! That is a great pet for the residents to interact with and though some people might think they transfer disease, if they are well-cared for, they can actually prevent it.
The Egyptians even recognized this fact thousands of years ago. They knew then that rats were dirty animals, and I think they knew there was some correlation with rats and disease. If they had a few more cats during the Black Plague, maybe it wouldn't have spread as much (especially if the cats back then had good flea control like we do today). I digress. Cats are good at keeping rodents away. Not all cats are equally good at this (my parents' cat doesn't have the attention span to chase a mouse, I don't think, but she makes a wonderful cuddle companion). My cat actually used to catch mice at my parents' house and would put them in a waste basket. She learned if she caught them alive, she would put them in the basket and then notify us- then she would get her favorite treat. It was a great exchange and I liked that she left them alive and then I would just take the basket and let the mouse free outside. I'm pretty sure those mice would never come back again.
Regarding parasites; flea control is a good idea. Fleas are disgusting and spread disease. Cats are better equipped than dogs at removing fleas on their own (a healthy cat is a meticulous groomer, so a lot of the time people will never even notice fleas on their cat- but they will on their dog). They are also good at removing ticks for the same reason. It only takes one flea or tick to transmit disease or drop off your pet and attach to you, so proper parasite control is strongly recommended. A product such as selamectin also kills parasites that your cat can ingest if it does decide to eat a rodent (Eww!). I joke with my husband that my cat is cleaner than some people and in some ways this statement could be true. She's indoor only and to my knowledge has not been catching things around our house... She's on medication to prevent her from getting parasites, eats a cooked food (raw food can transmit food borne disease) and bathes herself multiple times a day. As long as humans are careful around fecal matter (who wouldn't be?) there is very little chance of disease transmission.