But falling case numbers may soon lead these men to believe that the threat has passed. “We can’t ask people to change their behavior forever,” Dr. Daskalakis said. “That didn’t really work with H.I.V., so it’s not going to work here, either.”

Vaccination is likely to be a more effective containment strategy in the long term, he added.

As of Sept. 20, health officials had administered nearly 700,000 doses of Jynneos in the 48 jurisdictions for which data were available. While that is a substantial improvement over the early weeks of the outbreak, it accounts for only 22 percent of the doses needed to protect the 1.6 million Americans estimated to be at high risk.

Even as infections decline, the proportion of cases among Black and Hispanic men has grown to 70 percent in mid-September from 37 percent in late May. Yet Black men have received less than 9 percent of the doses administered so far, and Hispanic men about 16 percent.

Federal health officials are intensifying efforts to reach high-risk groups and have vaccinated at least 11,000 attendees at large gatherings where Black and Hispanic men congregate, such as Atlanta Black Pride.

The C.D.C. has announced a new program that would make up to 10,000 vials of vaccine — or 50,000 doses, under the new dose-sparing strategy — available to communities where hesitancy, language barriers, immigration status or other obstacles prevent widespread vaccination.

Eligibility for the vaccine is scattershot by location, and the criteria often opaque, according to an analysis by the Kaiser Family Foundation. Some states, like Indiana and New Mexico, offer no information online about who qualifies. Laboratory and health care workers who may be exposed to the virus are eligible in only 18 states and cities.

Many men at high risk have opted for a single dose, which may not be sufficiently protective. Although the proportion of second doses has increased, so far 77 percent of administered doses are first doses.

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