The plight of the department has reached Washington. The White House, responding to what an official described on Thursday as an “urgent S.O.S.” from the Police Department, arranged shipments of 4,000 Tyvek suits for homicide detectives processing suspected Covid-19 deaths and 6,000 gallons of hand sanitizer.

In New York, security officers have been posted outside precincts to screen people entering in an effort to minimize foot traffic, and face masks are being used inside some precinct station houses. Some commanders have moved their daily roll calls outdoors so that officers can spread out. Officers on patrol are carrying N95 masks to pop on if they feel endangered.

Gov. Andrew M. Cuomo has promised to send state police, if necessary, to fill in for sick officers, but Commissioner Shea has declined the offer for now. He has also said he does not see a need, yet, to ask officers to work 12-hour tours to cover staff shortages.

While paramedics were responding to a record surge in emergency calls, Commissioner Shea said the calls funneled to police were down. There have also been no parades or large gatherings.

Since the city imposed emergency rules, reports of the seven most serious crimes — including rape, robbery and assault — have dropped to about 187 a day, compared with about 267 a day in the 11 days before the rules went into effect. “Nobody’s on street and that’s really helping us,” Mr. Shea has said.

But at the same time, nearly 700 officers have been assigned to enforce Mayor Bill de Blasio’s new social-distancing rules. That has pulled officers from specialized assignments, like film and narcotics, onto patrol.

The nature of police work and the officers’ lack of training in public health puts them at a greater risk of contracting the virus, researchers say. So far, almost 4 percent of the force has tested positive, and the number increased sevenfold last week.