The Steamship Authority on Sunday released statistics showing a modest increase in out-of-state vehicles coming to the Vineyard in the first half of March, as Massachusetts Gov. Charlie Baker urged residents with second homes on Nantucket and Martha’s Vineyard to stay home.

The preliminary figures for March 1 through March 15 show that 102 more vehicles with a New York or New Jersey address in their vehicle profile arrived on the Island than during the same period in 2019. There was a net increase of 264 more vehicles with a Massachusetts address and 21 fewer vehicles from New England states other than Massachusetts compared with last year.

The figures were released by boat line communications director Sean Driscoll Sunday afternoon, following persistent concern that the Islands have become a magnet for out-of-staters seeking refuge from the coronavirus pandemic.

Speaking to reporters in Boston, Governor Baker addressed the issue of second-home owners opting to come to both Nantucket and Martha’s Vineyard. Both Island hospitals have urged people to stay away, saying the health care system is ill-equipped to deal with a large influx of people in the off-season.

“We would prefer they not do that and that they stay on the mainland and don’t create additional issues for both of those Islands at a point in time when they don’t have the level of service capacity in place they typically would have in the summer,” Governor Baker told reporters, according to Boston news station NECN.

The statement came as Nantucket announced its first confirmed case of Covid-19. So far, only one case has been confirmed on the Vineyard, a 50-year-old Vineyard Haven homeowner. Nantucket Cottage hospital said Sunday the patient was quarantined at home, but gave no other details. A stay at home/shelter in place order has been issued for all of Nantucket, to take effect at 5 p.m. Monday and run for two weeks, through April 6.

Statewide, cases of the coronavirus continued to climb. The Massachusetts Department of Public Health reported 646 cases on Sunday, up 121 from the day before. Four more people in Massachusetts have died from the pandemic, bringing the toll to five.

On Saturday, West Tisbury and Chilmark became the first town Island towns to adopt emergency construction bans, out of concern for public health and safety. On Sunday, the Aquinnah Board of Health announced a more modest restraint on construction.

Adopted by a unanimous vote of the two town boards of selectmen and health at meetings Saturday, the emergency orders are nearly identical and effectively ban all construction activity beginning Monday and running until April 6.

Effective Monday at 9 a.m., the towns will begin the process of halting all regular activity at construction sites and all work authorized by building permits.

Construction sites must be secured and any work wrapped up by 5 p.m. Wednesday, according to the emergency orders. Only skeleton crews will be allowed after that. Emergency work will be allowed on a case-by-case basis.

The Aquinnah board of health, by contrast, said that construction sites employing five or fewer workers could proceed with no restrictions provided reasonable precautions are taken to guard against transmission of disease. Any site employing more than five workers must submit a written mitigation plan to the board of health. The policy was approved unanimously by the three members of the board of health, James Glavin, Sarah Saltonstall and Gerald Green.

The Chilmark and West Tisbury votes followed a series of meetings over the hectic past two days among town officials during the rapidly escalating emergency around the pandemic. In both towns there was broad discussion about the scope of the ban, including whether it should include landscape workers, painters and other non-permitted work. In the end it was decided that most large landscape projects are on work sites and would be shut down anyway under the construction ban.

“I’m sure the document is not perfect . . . but it shows that the town is trying to stop social interaction,” West Tisbury selectman Kent Healy said at that town's meeting, the second of the day on Saturday. “And that is the primary goal.”

Stemming disease transmission is the reason for the ban.

Signed by the selectmen and board of health members, the full emergency order is posted on the town website.

At a meeting Friday, West Tisbury health agent Omar Johnson called the decision fraught but vital.

“We are all united in this effort,” Mr. Johnson said.

Chilmark selectmen and board of health members, who also had been debating the issue for days, also took action Saturday to enact a ban in their town.

Excavating contractor John Keene pushed for including landscapers, said he had already begun layoffs at his company and suppported the ban despite the ramifications. “It is just time to stop and I say that knowing that I just shot myself in the foot,” Mr. Keene said.

Longtime Chilmark real estate broker Deborah Hancock concurred.

“I think it is awfully important that we do this,” she said. “I think it is going to hurt an awful lot of people, but it is going to hurt an awful lot more people if we don't.”

Selectman Warren Doty, who had been reluctant to support the ban out of fear for the economic repercussions, joined the vote of support in the end.

“Here we go, we're going to destroy our economy and move into a recession,” Mr. Doty said. “But what the heck, let's go.”

Enforcement of the orders will fall under a combination of the boards of health, the town building inspectors and police, town counsel Ron Rappaport said.

There has been one confirmed case of Covid-19 on the Vineyard.

The Tisbury board of health confirmed the case in a press release Friday.

Health agent Maura Valley said the patient, a 50-year-old male, is a Tisbury homeowner who is under quarantine at home and appears to be recovering. Ms. Valley said in a follow-up email that the circumstances of the virus’s transmission were unknown.

“The patient’s family and close contacts have been identified and are in self-quarantine and taking all recommended precautions,” the statement said.

Due to privacy concerns, no other identifying information will be released, the statement said.

Hours after the first confirmed case was reported on the Vineyard Friday, the screws began to tighten on the Islandwide shutdown. Using the Steamship Authority website, the Vineyard and Nantucket hospitals issued a joint statement strongly advising people — on both sides of Vineyard Sound — to stay at home and avoid travel unless absolutely necessary.

“If you come to the Island — you will only strain our limited resources, putting your life and others at risk," the statement says in part.

Signed by Denise Schepici, president and CEO of the Martha’s Vineyard Hospital, and Gary Shaw, president and CEO of the Nantucket Cottage Hospital, the statement went up on the SSA website early Friday evening.

Also Friday, the SSA announced that it would eliminate trips in response to a drop in ridership over the past two days. Schedule reductions will begin Sunday.

The hospital said it has no patients hospitalized with Covid-19.

The hospital also said it currently has adequate stock of protective masks, suits, gloves and testing equipment, but that they do not have unlimited bandwidth for virus testing or supplies.

“We have appropriate personal protection equipment (PPE) but as other hospitals are experiencing, our supplies are limited,” a spokesman said.

The hospital also noted that coronavirus testing does not require documentation of immigration status.

Around the Vineyard Saturday, an emerging new way of life was apparent. On quiet streets in downtown Edgartown walkers were out enjoying the abundnant sunshine, albeit in small groups of three or less. At Ghost Island Farm in West Tisbury, where a new outdoor ordering and pickup system had begun, cars lined up and people bought fresh greens, frozen summer tomatoes, Island meat and other provisions. Elsewhere home gardeners were outdoors puttering, and indoors sharing posts on Facebook with fellow gardeners about seed and flower swaps.

Susie Middleton and Will Sennott contributed reporting.