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I propose a less effective, but easier to implement, way to optimise who to test. It relies upon a simple result from network/graph theory that I will outline here.

Assume, if you will, that typical social networks contain a small number of people who are highly connected (hubs) and a large number of people who are much less connected (spokes). The hubs could be e.g. your GP/physician, your teacher, or even just your popular friend. It would not be surprising if these people have a much high transmission rate for a virus than people who have a much smaller social circle - all other things being equal, and also crucially, they often connect mutliple sub-networks (GP is a good example here).

We would of course like to ensure that we test (perhaps regularly) the people who are at the center of these networks - the hubs. How do we find out who these people are? It turns out that you can do this probabilistically. First, I pick someone at random from the population. I then instruct them to pick a contact/friend at random. The person they pick is statistically much more likely to be a 'hub' than a 'spoke', even though we have no explicit knowledge of the network and carried out this process randomly. A good way to visualise this is to imagine a toy network of say one person in the middle, connected to everyone else, and 10 other people, all connected to the person in the middle but nobody else. You can see that in 10/11 cases, a 'spoke' is selected who then goes on to select the 'hub' (what we want) whereas in only 1/11 cases the hub is initially chosen, who then chooses one of the other 10 contacts at random.

A practical implementation of this could be to choose N people at random and send them a letter or text message instructing them to pick e.g. the (modulo) 5th person from their contacts list whose name begins with R. They would then contact this person and inform them that they should present themselves at a doctor and be tested should they develop any symptoms. There are obvious optimisations to be made here in terms of name distributions and other subtleties of course, and certainly providing a list of fallback randomiser instructions, if they can't find someone fitting that criteria.

Bottom line is, even if many people don't comply, you can increase the probability that you end up prioritising testing of people who are very connected, and thus likely to spread disease, which can be invaluable when your testing capacity is constrained.

A practical implementation of this could be to choose N people at random and send them a letter or text message instructing them to pick e.g. the (modulo) 5th person from their contacts list whose name begins with R.

I'm sorry but in modern America, you can count on the average person screaming bloody murder when confronted with a scheme like that. No one likes to be part of someone else's mathematical model but a random electrician or hairdresser is toss that away or maybe put out an angry tweet that will get more leverage than this idea.

2 problems:

1. The bottleneck in testing capacity in the US will likely be labor.

2. Testing asymptomatic people is probably not very effective because they are unlikely to have high viral loads.

If the US cannot mobilize sufficient labor to provide testing, this is a heinous failure of American leadership. And indeed, maybe the US cannot do this but if a country with a huge state and military sector cannot effectively "draft" people to engage in the fairly simple processes that would be involved in just poking a needles or whatever into people, our inability to survive whatever the next crisis is seems pretty guaranteed.
This is a thought experiment at best. If we ignore the privacy concerns, there is the basic practical issue that location tracking is often wildly bad and would lead to tons of false positive and negative selection of people to test. Over the years I've used a variety of Garmin watches and iPhones to track runs and walks. In areas with tall buildings or steep hills and trees all of these systems will jump around and track me as if I were superman bounding over/through buildings and city blocks from one second to the next.
The idea that one should limit testing seems extremely misguided.

I would note: South Korea and Italy had equivalent infection figures today. South Korea's mortality rate has held steady at .6% while Italy's has been around 4% - quite a bit more death. Basically, strong evidence South Korea is good model for controlling Covid. [1]

Korea has had a policy of aggressive testing, aggressive tracking infections and publishing infectees previous locations online (but not names).

There's literally drive-through testing. Anyone who wants to get tested can be tested. [2]

[1] https://covid19info.live/

[2] https://www.cbsnews.com/news/coronavirus-south-korea-drive-t...

Edit: Article from the Atlantic: "Exclusive: The Strongest Evidence Yet That America Is Botching Coronavirus Testing" [3]

[3] https://www.theatlantic.com/health/archive/2020/03/how-many-...

Ideally, everybody would be tested! But the tests are expensive and in limited numbers (even if free in some countries for people, health authorities still pay for them). And people also won't volunteer to get tested unless knowing they are at high risk.
Apparently there are two strains. One is lower transmission but higher mortality and is prevalent in Iran. If Italy has more of the one with higher morality, that might make their numbers higher.
I spoke to someone working in an ER in New York City. People are walking in, no symptoms, no travel history, no contact with any known cases - demanding to be tested for coronavirus. They'll say ridiculous things like "my neighbor (who they've had no contact with) went to Thailand last month. I want to be tested.". People are panicking.

If tests were infinite, then fine - but as long as we have a finite number of test kits, there have to be limits.

...but I think we fundamentally agree - we need MANY more test kits. ...and it would be great to have so many that we could operate at-home or drive-thru testing. I think we'll get there.

Can we use homomorphic encryption to deal with the privacy concerns?