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 grab an pet cat they may find and take them back again to the restaurant. I recall seeing pet cats in glass displays out leading of the restaurants. But somewhere within then and now they forbidden eating pet cats. So cats are actually not on the menu and now there are cats everywhere. Many of them started off in someone's house and got thrown away, since many are incredibly friendly. Also they may have this thing about abdominal worms. And in those times China was not as modern as now so belly worms would have been a problem. Fleas aren't a really problem because they have got tile flooring and wooden furniture.
Cats and mankind have been residing in harmony for quite some time but that hasn't always been the situation. There have been times when felines have been practically wiped out in the name of God. And what took place the Black Death reigned supreme for considerably longer than it will. So we will get back to the origins of recorded background to enough time of the pharaohs of Egypt. Here cats were revered as Gods and many were mummified to previous for eternity. As the pet cats guarded the grain warehouses and wiped out the rats and mice that given on the grain. Therefore reducing famines, which were the curse of Egypt. It was illegal to get rid of cats in those times, like cows in India today.
Then the nine lives of the humble kitty who only wished to help mankind by eradicating rodents that carried disease took a significant change for the even worse during the dark ages. Where witch hunts and using up at the stake were the norm. Somebody who didn't like cats somehow connected these to witches plus they were burnt at the stake with there owners. How they made that interconnection I'll never know. It started off with Pope Gregory IX in the 1232 who considered the normal house cat as diabolical. Then to make things even worse in1484 along arrived the greatest pet cat killer ever, Pope Innocent VIII, who decreed something along the lines of all cats originated from hell and should returned to the fires of hell by hearth. 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 found with a kitten you were burnt together. I still go to church and sometimes I simply wonder why. And of course the Black Fatality was in the backdrop surviving and growing because there were no pet cats to kill the rats that taken the fleas that induced the condition. Then along came up the cat's savior King Louis XIII of France, who in the 1630s repealed the law and cats once again began to live. But they were still hated and wiped out for many centuries after that. I have no idea if the Black color Fatality subsided but cats once again covered mankind resistant to the diseases transported by rodents. Along with the famines due to mice plagues.