Rats are the ones who spread disease.

Europeans hated cats, so they killed many of them. This allowed the rats to spread and they brought the Plague.

https://owlcation.com/humanities/Cats-and-the-Black-Plague

Because of their secretive nature and their ability to survive extraordinary circumstances, the general population came to fear cats as consorts of Satan. The innocent cats began to be killed by the thousands.

The cats ultimately got their revenge, of course. Since there were few felines left, the rat populations increased unchecked, and the plague grew even more widespread. You’d think that the humans would make the connection by this point, but instead, they made things even worse. They began to associate the plague’s new vigor with the cats and even with dogs. They believed that since both of these animals typically harbored fleas, they must be the cause of the plague. Subsequently, cats were outlawed in many parts of Europe, and huge numbers of cats and dogs were killed. In fact, at one point in the middle ages, there were barely any cats left in England at all.

Even though cat ownership was illegal in some regions, a few people kept their felines. Other people finally noticed that these cat owners often seemed to be immune to the black plague. Word spread quickly, and more observations of this phenomenon were noticed. This resulted in research, crude as it was during the time.