The currently deployed tests are based on detecting viral nucleic acid. I haven't been able to find sensitivity or specificity info on line, but I have seen gripes about the lack of that information in the open literature, so maybe it isn't just me. The FDA literature says that the tests have been designed to minimize the false positive rate, suggesting the cutoff values favor specificity (i.e. true positives).
The currently available test doesn't help much with defining the epidemiology, since it is truly a snapshot. A negative test cannot discriminate between someone who is unexposed and uninfected, an asymptomatic infected person early or late in the infection, whose viral load is below the cutoff for a positive test, and a person who has been infected and recovered. Tests that measure antibody formation can give you information regarding that group of asymptomatic but infected patients. I haven't seen anything currently available that uses serology for Coronavirus testing.
There are a couple glaring problems with mass testing as implemented. As the test seems to be biased toward true positives, it isn't a great screening test. Another big problem is there isn't a clearly defined effective treatment, so diagnosing the disease doesn't give you all that much actionable intelligence. Once a treatment becomes available, mass testing with an adjustment of the test characteristics or methodology has a place. Finally, a negative test may lend a false sense of security, again as the available test appears to be weighted away from sensitivity, and that viral RNA may be present, but below the test cutoff, and could be either rising or falling depending on where in the course of the illness the patient may be. Oh, and the test takes 48-72 hours to yield results.
V/R
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