Three Houston police officers were recently in what their chief called a “tussle with a suspect” who was displaying fever-like symptoms: a hallmark of the novel coronavirus.

“We immediately sent them home; they didn’t even go back to the station,” Chief Art Acevedo said in an interview. All three officers tested positive for the coronavirus.

The officers in Houston are part of a mounting toll across the country as police officers and other first responders are falling ill with the coronavirus. Departments are now grappling with mounting infections in their own ranks as they navigate an extraordinary public health emergency.

In New York, hundreds of uniformed officers have tested positive for the coronavirus. Infections have also been confirmed in departments in Boston, Chicago, Cincinnati, Dallas, Detroit, Los Angeles, Houston, Miami, Miami Beach, Milwaukee, Nashville, New Orleans and Philadelphia, among others, along with sheriff’s offices across the country.

“You have to remember, as police officers . . . we can’t walk around in biohazard suits,” Tampa Police Chief Brian Dugan said in an interview. One of his officers has tested positive for the coronavirus. “We have to be able to get to our guns, our radios, our handcuffs, that type of stuff. There’s no way for a police officer to do their job and not potentially be exposed to the virus.”

Officers are still contracting the virus even as police in some major cities have had fewer calls to answer.

A Washington Post analysis of reported crimes from more than a dozen cities — including the District, Los Angeles, Chicago, Dallas and Atlanta — found a notable decline in crime that started when people started being urged in mid-March to work from home and stay away from others.

Most cities saw a 10 to 20 percent decrease in property crimes, the analysis found. Violent crimes have remained relatively flat overall, the data show, though police in some cities reported drops in some of those categories.

But officers continue to confront the virus, with departments nationwide quarantining waves of officers for possible exposure. Three Tampa officers were potentially exposed after giving CPR to a person they later learned was infected. Officials in New Jersey said Saturday that 700 officers statewide have tested positive for the coronavirus. In New Orleans, a police spokesman said about 5 percent of the force is sick or in quarantine.

Some agencies are mourning officers and personnel who have died of the virus. Detroit police said a 38-year-old 911 operator died Monday, and Capt. Jonathan Parnell, a three-decade veteran of the department, died Tuesday. The Wayne County, Mich., sheriff’s office said Donafay Collins, a 63-year-old commander, died Wednesday.

The New York Police Department said Thursday that Dennis Dickson, a custodial staffer at police headquarters, was the department’s first death from the virus. The toll only grew after that. On Friday evening, the NYPD said another civilian employee — Giacomina Barr-Brown, who worked in a precinct roll call office — had become the second. Then, on Saturday, the department said Det. Cedric Dixon had died.

“We have lost three members of our family in a little over 48 hours,” Dermot F. Shea, the New York police commissioner, said Saturday. “As I stand here, I cannot begin to describe what we are feeling. . . . We are hurting. We are crying. And we continue to fight.”

New York City has become the epicenter of the American coronavirus outbreak, with more than 29,000 people confirmed to be infected with the virus and 517 dead from it, according to the New York City Department of Health and Mental Hygiene. Some of the infected work for the NYPD.

More than 550 New York Police Department employees had tested positive for the coronavirus, more than double the number reported just two days earlier, Shea said, and more than 4,100 people from the uniformed part of the department were out sick. Police officials have said many of those testing positive were uniformed officers. Shea called the situation facing the department “deeply troubling.”

“You are on the front lines of this,” Shea said in a recorded message for officers.

Firefighters and paramedics nationwide have also been diagnosed with the coronavirus. The New York City Fire Department said that 235 of its people — including firefighters and civilians — had tested positive as of Saturday. In Kirkland, Wash., firefighters were quarantined after responding to calls at the Life Care Center nursing home, where there was a deadly coronavirus cluster.

While many professions have adapted to stay-at-home orders and official advice to practice social distancing by having employees work from home, core parts of police work can only be done by heading outside and interacting with others. Police can’t scour a crime scene for shell casings over Zoom, nor can they provide first aid to someone in dire medical straits from an office. So departments have responded to the virus by rethinking key aspects of how they police their communities.

“It’s a unique thing to try to police in a social distancing world,” said Paul Pazen, the Denver police chief.

People who call 911 now get asked if they have a fever. Officers are cutting down on arrests and not responding in person to some calls as they try to avoid possible exposure to the deadly virus.

Acevedo said people who call his department now are now asked whether anyone is showing any symptoms of illness, including a cough or fever. Officers going to a scene where a person might have those symptoms will go in protective gear, Acevedo said, though the department is “running out of everything.”

Police in Oklahoma City are asking 911 callers about their health before responding and requesting that, when possible, people step outside when officers show up. In Philadelphia, police are detaining people charged with some nonviolent offenses only for paperwork and planning to arrest them later. Chicago police said they are seeing less activity overall, with car and pedestrian stops down 40 to 60 percent, and are encouraging officers to minimize public contact.

Dallas police said they will keep responding in person to crimes including assaults, sexual assaults and robberies but will not physically show up for graffiti, shoplifting or lost property, among other things. Police in Tampa, Nashville, Oklahoma City and Charleston, S.C., say they are taking more reports over the phone rather than in person.

Other departments have responded by shutting their doors when possible. Memphis police closed their stations to the public, while Charleston police did the same with the lobby of their headquarters.

Police also say they are trying to avoid bringing additional people into jails — which experts warn are dangerously susceptible to disease outbreaks — if possible.

Police officials stress that they are still patrolling communities and responding to calls and serious crimes. Pazen said his officers have “been able to address some pretty high level crimes but do it in a safe way,” including keeping distances when possible and wearing protective equipment.

But they are worried about what happens the longer people stay at home, uncharted territory in its scope and severity.

“What’s kept me up at night as chief for many years is the possibility of a pandemic,” said Acevedo, the Houston police chief. “Because in law enforcement, we’re used to dealing with human behavior, we’re not used to dealing with Mother Nature and a virus and medical issues.”

The pandemic is also raising questions about how departments can operate longer-term if officers keep contracting the virus and calling in sick, said Eugene O’Donnell, a former New York police officer who teaches at the John Jay College of Criminal Justice.

“The whole notion of policing in a crisis is that you can turn to them and hand them the ball,” O’Donnell said. “But they are human institutions — people worry about their families, worry about their own health. If this is a prolonged issue and we’re just at the bottom of the curve that’s going to go up, we’ll have to see whether it can be sustained over the long haul.”

Officials said it is too soon to know if the drops in crime will continue. In New York, major felony crimes declined by 17 percent for the week ending Sunday, police said. In Seattle, where the virus first took root in the United States, officials say the number of 911 calls has dropped in recent weeks. Chicago police had reported several days of decreased activity before 14 people were injured in shootings Wednesday night, a department spokesman said.

In the week after California residents were ordered to shelter in place March 20, San Jose saw violent crime drop nearly 44 percent from a year earlier and property crime fall 36 percent, said Eddie Garcia, the city’s police chief.

“But we’re in the first quarter of the football game,” Garcia said. “I’m not doing a touchdown dance just because our first quarter went well.”

Police and experts fear certain crimes — including domestic violence and child abuse — will surge during this period of economic and existential uncertainty.

The Rape, Abuse and Incest National Network (RAINN) is experiencing a “noticeable increase in visitors voicing fears about their safety during this pandemic,” Heather Drevna, the group’s vice president of communications, said in a statement.

Pazen, the Denver police chief, said authorities were similarly worried about this because of the sheer number of people.

“Now we’re putting young people, multiple generations in the same household, in proximity with a lot of stress,” Pazen said. “We’re concerned about that.”

After the first Tampa officer tested positive this week, Dugan, the police chief, told reporters at a briefing he is worried about the health and safety of his officers and the general public.

“My biggest concern is, it’s gotten real,” he said. “And it’s only going to get worse.”

Devlin Barrett, John D. Harden and Griff Witte in Washington and Michael Majchrowicz in Fort Lauderdale, Fla., contributed to this report.