Where I am, strays are usually catch, sterilize, release. But if you caught a cat who was sick they'd treat it, and lots of cats end up at the shelter because people don't fix their pet cat in time, and they get pregnant (even if you are responsible - we had a brother sister pair and I asked the vet what she would he fix them, he said "6 months", I asked but when can they get pregnant? "4 months". Thankfully it didn't happen, they were fixed and lived long kitten free lives).

And pets, if someone gets sick and can't care for them they get surrendered.