Actually, public healthcare means people will visit the doctor sooner, and can get diagnosed sooner (or quarantined or whatever needs to happen). Means it should help over having a large part of your population unable to pay their medical bills and just wait till they really need that doctor or die.
Borderwise: You think there's an open border between China and Italy? Virusses don't stop at borders. You can close every border, and someone coming to the border and caughing at someone on the other side will get you sick just as well. People will go on vacation and come back sick. The only way it'd work is in a completely totalitarian regime which enforces a quarantine at penalty of death, for everyone who enters the country, no matter which nationality (well, it might work against direct infections, plenty of other ways to get infected depending on which virus).