I think one of the reasons there were few cats around is because they used to eat them. And once in a while the restaurant feline catches would pick up an pet cat they could find and take them back again to the restaurant. I remember seeing pet cats in glass shows out the front of the restaurants. But somewhere between then and today they banned eating pet cats. So cats are now not on the menu and today there are cats everywhere. Many of them began in someone's house and acquired thrown away, as most are extremely friendly. Also they may have this thing about tummy worms. And in those days China had not been as modern as now so stomach worms would have been a problem. Fleas aren't a really problem because they may have tile flooring and real wood furniture.
Then the nine lives of the humble pet cat who only wished to help mankind by killing rodents that taken disease took a significant flip for the even worse during the middle ages. Where witch hunts and using at the stake were the norm. A person who didn't like pet cats somehow connected these to witches plus they were burnt at the stake with there owners. The way they made that interconnection I'll never know. It started off with Pope Gregory IX in the 1232 who considered the normal house kitten as diabolical. Then to make matters more serious in1484 along came the greatest feline killer ever, Pope Innocent VIII, who decreed something such as all cats originated from hell and should came back to the fires of hell by fireplace. Felines were mercilessly prosecuted and were burnt so they would return to the fires of hell. It really is amazing any survived because if you were captured with a feline you were burnt mutually. I still go to cathedral and sometimes I just wonder why. And undoubtedly the Black Loss of life was in the background surviving and thriving because there were no pet cats to kill the rats that transported the fleas that triggered the condition. Then along came up the cat's savior Ruler Louis XIII of France, who in the 1630s repealed the law and cats once again began to live. However they were still hated and killed for many ages from then on. I don't know if the Dark Death subsided but cats once again guarded mankind against the diseases taken by rodents. As well as the famines due to mice plagues.