I think one of the reason why there were not many cats around is basically because they used to eat them. And every now and then the restaurant cat catches would pick up an cat they could find and take them back again to the restaurant. I recall seeing felines in glass displays out leading of the restaurants. But somewhere between then and now they banned eating pet cats. So cats are actually not on the menu and today there are felines everywhere. Most of them began in someone's house and got thrown away, as most are incredibly friendly. Also they have this thing about abdomen worms. And in those days China had not been as modern as now so abdominal worms would have been a problem. Fleas are not really a problem because they may have tile floor surfaces and wooden furniture.
Then your nine lives of the humble feline who only wished to help mankind by killing rodents that transported disease took a serious switch for the even worse during the middle ages. Where witch hunts and losing at the stake were the norm. Somebody who didn't like felines somehow connected those to witches and they were burnt at the stake with there owners. The way they made that connection I'll never know. It started off with Pope Gregory IX in the 1232 who regarded the normal house kitten as diabolical. Then to make concerns more serious in1484 along arrived the greatest cat killer ever, Pope Innocent VIII, who decreed something along the lines of all cats came from hell and should delivered to the fires of hell by open fire. Felines were mercilessly prosecuted and were burnt so they would return to the fires of hell. It is amazing any survived because if you were trapped with a feline you were burnt along. I still go to church and sometimes I just wonder why. And of course the Black Death was in the background surviving and thriving because there were no cats to destroy the rats that carried the fleas that brought on the disease. Then along emerged the cat's savior King Louis XIII of France, who in the 1630s repealed the law and cats once more started to live. However they were still hated and wiped out for many hundreds of years from then on. I don't know if the Black color Loss of life subsided but pet cats once again secured mankind from the diseases carried by rodents. Plus the famines brought on by mice plagues.