The state agency was slated to provide an updated weekly tally for Massachusetts cities and towns Wednesday afternoon.

Roberto Santamaria, Nantucket’s health and human services director, said in a telephone interview that the Board of Health will discuss the possible responses to the recent spike at its meeting Thursday. Measures, he said, could include closing restaurants at 11 p.m. or midnight.

Massachusetts in late June permitted restaurants to resume indoor seating with restrictions for the first time since mid-March.

Santamaria said earlier closures would prevent “a lot of these large gatherings we’re seeing after 11, when people have been drinking for a significant amount of time” and less likely to wear face coverings, which are required in the downtown area.

“It’s not quite a curfew,” he said of the possible earlier closing time. “More so, a [message that] we would just like you to disperse.”

One large party on the Fourth of July at a rental property resulted in the owner receiving a $1,000 fine, he said. Attendees were initially told they had to either wear face coverings or break up the party.

“After they had become pretty belligerent and insulting, there was no other course” of action to take, Santamaria said. “We issued a fine per person” not wearing a mask.

He said the town hasn’t had to fine anyone else, though multiple verbal warnings have been issued and beach parties and bonfires have been broken up.

“It got their backs up,” Santamaria said of the public reaction to the $1,000 fine. “It got people’s attention.” In addition, Sanatmaria said, masks are being handed out to people on the street, and there’s signage “all over the place” about the face covering requirement.

The island, he said, hopes to prevent larger spikes in August.

“That’s why we’re being a little militant about the masks,” Santamaria said.

The Cape Cod Times first reported on the spike in cases on the island.

“We have seen an increase in the number of confirmed cases of COVID-19 on the island since late June, which we expected given the increase in the island’s population and expanded testing capacity,” said Jason Graziadei, a Nantucket Cottage Hospital spokesman, in an e-mail message Wednesday.

As a hospital, he wrote, “we have prepared for this, in collaboration with the town, with greater testing, contact tracing, as well as outreach and education to the public.”

The hospital on Tuesday had a succinct message on Twitter for its roughly 2,600 followers.

“Wear a mask,” the facility tweeted.

That missive was quickly seconded by state Senator Julian Cyr, a Democrat who represents the Cape and Islands.

“And wash your hands,” Cyr tweeted in response. “And physically distance. And isolate if symptomatic. And don’t give up.”

The Town of Nantucket sounded a similar refrain in a recent message posted to its official Facebook page.

“While temperature goes [up], we want to get out and enjoy the [sun],” the town posted last week. “There are so many activities we can enjoy while keeping physical distancing and avoiding crowds. The risk of spreading covid-19 is still high. In order to stay safe and healthy, we must enjoy summer responsibly.”

The posting directed residents and visitors to avoid crowds, maintain six feet of distance from other people, wear a mask when distancing isn’t possible, wear a mask in public in the downtown and Sconset areas, and frequently wash hands and use sanitizer.

Also, the posting said, “be responsible. If you feel sick or you suspect you have been in contact with a person that has been confirmed with covid-19, stay home” and call your health care provider.

Santamaria said Wednesday that Nantucket is prepared to take more restrictive measures if necessary to contain the virus.

“We don’t want to roll back to some of the governor’s previous orders” put in place before reopening began, he said. “But we would if we needed to.”

Travis Andersen can be reached at travis.andersen@globe.com. Follow him on Twitter @TAGlobe.