But nearly three weeks into the in-person school year, early data from the city’s first effort at targeted testing has shown the opposite: a surprisingly small number of positive cases.

Out of 15,111 staff members and students tested randomly by the school system in the first week of its testing regimen, the city has gotten back results for 10,676. There were only 18 positives: 13 staff members and five students.

And when officials put mobile testing units at schools near Brooklyn and Queens neighborhoods that have had new outbreaks, only four positive cases turned up — out of more than 3,300 tests conducted since the last week of September.