I think it's all okay. China has the biggest cities of the world and they censored it for 2 months, while the virus was spreading.
This isn't happening in the rest of the world. ( Except some dictatorships)
Some minor outliers like you mentioned, eg. Italy. But these are getting handled. It will not spread anymore like it did in China.
It's possible that some parts will need extra handling, but sharing information is the key for handling this.