State and federal supply issues continue to stymie the Covid-19 vaccine rollout on Island, as recent expansions in eligibility have not been met with a matching increase in doses.

The Martha’s Vineyard Hospital, which serves as the state vaccination site on-Island, has received a steady supply of approximately 1,500 vaccine doses per week, officials said at a press briefing Wednesday morning. But for the time being most of those shots are going to individuals who require a second dose, leaving few available appointments for newly-eligible residents.

On Monday, individuals 60 and older, as well as workers in the grocery store, restaurant, transportation and funeral sectors, along with others, became eligible for the vaccine. But the hospital only had 100 available first-dose appointments, leaving many Islanders shut out of registration.

The 100 appointments filled in approximately 10 minutes, a hospital spokesman said. A similar process occurred on Saturday, and is expected to occur again for signups this weekend.

“We certainly asked for a lot more than we get allocated,” hospital president and CEO Denise Schepici said at the briefing. “It’s all supply dependent . . . and that’s a week-by-week guessing game.”

“I understand the frustration. I really, really do.” she added.

State population estimates show that there are at least 1,335 newly eligible Dukes County residents between the age of 60 and 64. Hundreds more who work in the food services and transportation industries are also now eligible for shots.

On Wednesday, hospital officials tempered their excitement about the eligibility expansion with the need to set realistic expectations for the broader rollout, saying they had no timeline for when the state or federal government would increase vaccine supply.

“The focus of this call is to set expectations,” Ms. Schepici said. “Even though by mid-April, everyone will be eligible, it unfortunately does not mean everyone will get an appointment for a vaccine by then. The challenge, as always, will be the supply we receive from the state, and in turn the supply the state receives from the federal government.”

Individuals age 55 and older, as well as those with one pre-existing condition, will be eligible to sign up for vaccines on April 3, according to state guidelines. All state residents will be eligible to sign up April 17, with appointments beginning on April 19.

To date the Island hospital has administered more than 10,000 vaccinations, broken down to about 6,500 first doses and 3,500 second doses.

It remains unclear when the national vaccine supply will increase, leaving questions about the reality of the Massachusetts rollout timeline. Hospital officials said they had only been able to schedule 241 first-dose appointments for the current week, along with 1,614 second doses. The numbers mark a reversal from past months, when more shots were going to first-dose patients.

Officials said the hospital has the capacity to administer 1,000 vaccinations per day, with three different stations, more staff and a pop-up tent outside the building, if vaccine supply was not limited.

Instead, Ms. Schepici and hospital head of operations Claire Seguin said they expected a comparably small number of available first-dose appointments for signups this Saturday at 8 a.m. They said they did not yet have word from the state on the amount of vaccine they would receive for next Monday’s signups.

“We don’t have a timeline yet,” Ms. Schepici said.

“It’s really week-to-week conversations,” Ms. Seguin added.

The state allocates vaccines weekly based on a variety of factors, officials said, including regional population estimates and the percentage of vaccinated people in each county. Dukes County currently has the highest first-dose vaccination rate in the state, at 36 per cent. While the allocation process remains mysterious and involves some lobbying on the part of local officials, Ms. Seguin said the state essentially tries to keep the percentage of vaccinated people in each county fair.

Although Dukes County’s allocation has not decreased in recent weeks, it hasn’t increased, either — an issue facing other regional vaccination sites statewide.

The state estimates the Island population at 17,400, based on 2019 census data. Hospital officials said the state Department of Public Health is aware the population has likely grown during the pandemic, although the issue is not unique to the Vineyard.

“We have advocated about that,” Ms. Schepici said. “They are aware that we’ve had Covid migration here on the Island, but there have been other areas impacted by Covid migration too.”

The hospital currently uses a first-come, first-served online vaccine signup process that opens at set times on Mondays and Saturdays. Officials said they thought the system worked well, and that while minor improvements had been made week to week, there are no immediate plans for broader changes, like a pre-registration process.

Officials also said they decided not to release the number of available appointments before signups because the numbers from the state could change, leading to disappointment.

“I think what the frustration is with, is really the supply chain issues,” Ms. Seguin said. “When we were at a point when we had more first doses available, folks were less frustrated with the app, has been my experience. But I get it, and understand, and hope they can be patient.”

Ms. Schepici and Ms. Seguin also said they did not know when the Johnson and Johnson one-shot vaccine would become available, although they were hopeful that manufacturing and downward pressure from the federal government would lead to supply increases for all three vaccines soon.

“I wish I had a crystal ball,” Ms. Schepici said.