Faced with the daunting task of parceling out a limited supply of coronavirus vaccines, Trump administration officials came up with a seemingly simple formula last year to streamline distribution of the shots.

First, federal administrators would run an automated algorithm to divide vaccine doses nationwide, based on the size of each state’s adult population. Then each state would decide how to dole out the shots to local hospitals, nursing homes and clinics.

But rather than streamline vaccine distribution, public health experts say, the algorithm has increased the burden for many states. It requires them to come up with multiple delivery plans for their weekly quotas of Pfizer and Moderna shots, even if the different shipments are destined for the same clinics and hospitals.

“The way it’s set up is so cumbersome,” said Dr. Michelle Fiscus, the medical director of the Tennessee Department of Health’s Vaccine-Preventable Diseases and Immunization Program. Her team initially resorted to planning out the vaccine deliveries with a big paper map of the state, she said, covered with pink and yellow sticky notes. “If these artificial allotments were scrapped,” she said, “it would help us tremendously.”

The federal vaccine allocation formula is among the dozens of algorithms—some automated systems and others simple prioritization lists—being used by government health agencies and hospital systems across the United States to help determine where the vaccines are sent and who can get them.

The algorithms are intended to speed Covid shots from pharmaceutical plants to people’s arms. The formulas generally follow guidelines from the Centers for Disease Control and Prevention recommending that front line health care workers, nursing home residents, senior citizens and those with major health risks be given priority.

Yet federal agencies, states, local health departments and medical centers have each developed different allocation formulas, based on a variety of ethical and political considerations. The result: Americans are experiencing wide disparities in vaccine access.

Oregon, for instance, has prioritized teachers over the elderly for Covid shots, an approach that could help schools and businesses reopen. New Jersey has put smokers before educators, which could save lives.

Some prioritization formulas also conflict with one another or impose such prescriptive rules that they hinder immunizations, public health experts say. Yet many Americans may not be aware of the layers of algorithms influencing their access to vaccines.

Ellen P. Goodman, a professor at Rutgers Law School who studies how governments use automated decision-making systems, said algorithms were needed to efficiently allocate the vaccines. But public agencies and health centers should be transparent about the prioritization formulas, she added.

“We want to know who is using them, what they are trying to do, who owns the proprietary algorithms, whether they are audited,” she said.

The vaccine prioritization formulas fall roughly into three tiers: federal, state and local. At the top level, Operation Warp Speed—a multiagency federal effort, created by the Trump administration—has managed nationwide vaccine distribution through Tiberius, an online portal developed by Palantir, the data-mining giant. The Biden administration, which has retired the program’s name, has taken over and is continuing the effort.

To divvy up doses, federal administrators use a simple algorithm. It automatically divides the total amount of vaccine available each week among the 50 states—as well as U.S. territories and a few big cities like New York—based on the number of people over 18 in each place. Some health officials and researchers, however, described the Tiberius algorithm as a black box.

“Why can’t they make public the methods that they use to make these estimations?” said Dr. Rebecca Weintraub, an assistant professor of medicine at Harvard Medical School who was a co-author of a recent study on state vaccination plans. “Why are the states receiving a different number of doses than they expected per week?”

States began warning about Tiberius’ drawbacks last fall. In interim vaccine plans filed with the CDC, some state health administrators complained that the platform seemed overly cumbersome and that the algorithm’s week-by-week allotments would make it difficult to plan monthslong vaccination campaigns.

Another potential drawback: The Tiberius algorithm calculates state vaccine allotments based on data from the American Community Survey, a household poll from the U.S. Census Bureau that may undercount certain populations—like unauthorized immigrants or tribal communities—at risk for the virus.

Although demographics experts said the survey data was the best available resource, they cautioned that it could have high margins of error at the smallest census tract levels. That could potentially lead to problems in states using Tiberius for local vaccine allocation.

Already, public health officials in Oklahoma have discovered that a federal vaccine allocation formula overestimated the number of doses nursing homes would need and have reallocated the shots to people 65 or older who don’t live in long-term care facilities. And states like Washington have created their own allocation systems—using federal and local data sets—to plot vaccine distribution for the weekly dose shipments.

Some states also say that they have received fewer doses than the Tiberius algorithm allotted or that the federal system canceled their orders without notifying them.

In an emailed statement, the Department of Health and Human Services coronavirus vaccine operation said that the census survey data provided a baseline for the effort and that officials were working to make sure tribal communities had equitable access to the virus shots. The statement added that 99.9 percent of state vaccine orders had been delivered according to schedule and that the agency was working with states to improve their experience and add new features to the system.

The Biden administration plans to soon start sending shipments of the shots directly to thousands of retail pharmacies, an effort that will not cut into the vaccine doses that states are allotted through Tiberius.

Once the Tiberius algorithm has done its work, state health departments use their own approaches to divide the virus shots among authorized vaccination providers and decide who is eligible to receive them.

Florida and Alaska, for instance, have given residents 65 and older priority for the vaccine. Massachusetts began allowing residents 75 and older to get the shots in early February. Oregon is opening the shots up to people 80 and over this week.

Some health administrators are trying to redress the disparities. States like Tennessee have developed prioritization algorithms based on a CDC database called the Social Vulnerability Index. It uses variables like poverty and crowded housing to identify areas that could suffer the most harm from disasters like tornadoes or disease.

As a result, Tennessee has sent extra vaccine doses—beyond its standard, adult population-based allocation—to 35 of its 90 counties hardest hit by the virus. Among them is Haywood, a county whose population is about 51 percent African American and where the coronavirus mortality rate is more than double the state average.

“We want to give more vaccines to that county to be able to vaccinate the population faster because their people are dying,” said Fiscus, the health official leading Tennessee’s vaccination program.

Public health administrators in Alaska said they were taking a similar equity approach to allocation—using spreadsheets to look at community risks and needs line by line.

“We’re making sure we’re identifying these stakeholders and communicating with those who are prioritized but might not know that yet,” said Tessa Walker Linderman, a co-lead of the Alaska Covid Vaccine Task Force.

Once states have allocated the shots to vaccination providers, medical centers use their own formulas to decide which health care workers and patients may receive them. The process can be fraught. In December, administrators at Stanford Medical Center in Palo Alto scrambled to fix a vaccine algorithm that failed to prioritize resident physicians, who were treating coronavirus patients, for the shots.

The process has been smoother for Providence Health, one of the nation’s largest hospital systems. In December, Providence asked its more than 200,000 employees and contractors across seven states about their work roles and locations, generating virus risk scores for each person. Those in the highest-risk group were then invited to make vaccination appointments first.

The scoring algorithm has helped the Seattle-based hospital system to fully vaccinate 75,000 workers, moving from those at highest risk to those at lower risk, including some administrators. “The goal is to get as many shots as possible into arms,” said B.J. Moore, Providence’s chief information officer who collaborated on the algorithm with the health system’s clinical experts.

Now the hospital system is using a similar approach to analyze tens of thousands of patient records, stratify their risks and notify those who are eligible to get the vaccine under state rules. Providence declined to comment on whether the hospital system had notified patients that their records were being data-mined for public health purposes.

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