Even as Martha’s Vineyard vaccination rates continue to climb, Island Covid-19 case rates refuse to relent, pushing hospital and public health officials to urge vaccinations with the state’s full reopening date approaching in just over one week.

On Wednesday, 17 residents tested positive for the virus, 13 at Martha’s Vineyard Hospital and four at TestMV. There have now been 28 residents to test positive since Monday — nearly topping the prior week’s total in three days.

Tisbury health agent Maura Valley expressed surprise at the large positive case numbers, saying that they were mainly close contacts of Islanders who had previously tested positive for the virus. Dukes and Nantucket counties have the highest Covid-19 positivity rates in the state, health agents said.

Ms. Valley said preliminary contact tracing had shown three cases traced back to employees at the Cardboard Box restaurant in Oak Bluffs. The restaurant was closed on Tuesday, according to a Facebook post.

Three customers who dined together at the restaurant on May 7 have also tested positive for the virus, Ms. Valley said. It was not immediately clear from contact tracing whether the customer and employee positive tests were connected.

“That’s going to need a little more digging,” Ms. Valley said.

Other recent positive tests were largely among close family or social contacts, Ms. Valley said, with people in their 20s continuing to test positive in large numbers.

“It looks like several of the people testing positive were close contacts of previous positives,” Ms. Valley said. “We’re seeing recently, that it just seems more infectious than before.”

At a press briefing earlier Wednesday morning, hospital officials said the Island vaccination process was rolling along smoothly, with more than 12,500 residents receiving first shots and 9,996 receiving both doses, representing about 80 and 60 per cent of the Island population respectively.

But as the state plans to fully reopen May 29, officials said receiving a vaccine was the best way to ensure a return to normalcy, especially as case rates remain alarmingly steady.

The incongruity between Island vaccinations and cases proved difficult to square Wednesday, prompting hospital officials to note that the virus remains a threat. “The virus is still here,” hospital CEO Denise Schepici said. “Everybody should just get vaccinated.”

Vaccine appointments at the hospital are readily available, and a mobile vaccine bus made rounds on the Vineyard over the past five days offering free shots with no appointment needed.

The hospital has now vaccinated more than 100 children ages 12 to 15, out of an estimated 600 on-Island, and plans to hold a Johnson & Johnson vaccine clinic Friday. Residents interested in the clinic can make an appointment for Friday, officials said, and added that appointments were widely available.

Although Ms. Schepici expressed concern about hospital capacity as summer arrives, she expressed support for the governor’s fast-tracked reopening plan so long as Islanders continued to exercise common sense. Hospital chief nurse and head of operations Claire Seguin agreed.

“They are basing these decisions on vaccine data, and science,” Ms. Seguin said. “Although we are a little out of sync with what the rest of the state is seeing.”

In an interview later in the day Wednesday, Ms. Valley said she thought the governor’s decision might have been slightly premature considering the Island’s case situation. But she agreed that vaccines were widely available on-Island, offering residents a good opportunity to protect themselves.

“I keep hoping that the numbers start going down,” Ms. Valley said. “But I think that there’s plenty of vaccine available. So people do need to take personal responsibility to be vaccinated, to protect themselves and their families.”