In 1665 Europe was engulfed in a deep mess due to the fact they slaughtered many cats believed to be "familiar" of witches. The death toll from the Black (Bubonic) Plague was estimated to be between 25% to 50% because rats and mice carried the fleas that carried the plague. We live on an Island, illnesses come here from all over the world due to our destination as a tourist heaven, do we really want to chance it or will we learn from history?
I heard from someone who read an article in the Garden Island of someone wanting to introduce mongooses to our Island to eradicate rats. Interesting concept except mongoose hunt during the day and rats hunt at night. What will these mongooses eat? Birds nest on the ground, have nests with eggs, easy dinner?
About rats and mice: an average rodent is of breeding age at 6 months. Each female can have 1 to 32 to a litter. If half are female and breed at 6 months of age in a years time you can have - you do the math. As you can see cats are an important part of our ecosystem and even if you are an ailurophobe, you can see it is time to leave our feral's alone Spay, neuter, care for the colonies is the best thing for our Island.