Canada, which has a public health care 'system,' has taken the tact of testing as many people as possible. Both to determine the extent of the spread of the infection and also to be able to identify individuals affected. Then action can be taken to contain the virus spread, self-quarantine, testing people that the infected person has been in contact with, determining where the possible source of the infection. Anyone showing up to hospital with flu symptoms, across the country, is tested.
It's all covered under that national public health insurance program (no wasted time on determining coverage through a myriad differing private insurance companies).
I don't know if you consider that 'surveillance state' - but it's organized, thought through, and so far seems to be effective. One death reported so far, a man in his 80's with numerous underlying conditions beyond COVID19. One community contact. All the rest, so far, are from overseas travel.