Colleges Vowed a Safer Spring. Then Students, and Variants, Arrived.

She is known on campus as Patient Zero, the unidentified student who returned to the University of Michigan after winter break carrying an unwanted stowaway from her trip to England — a highly contagious variant of the coronavirus first detected in Britain.

Quickly the case became a cluster, with at least 23 confirmations of the B.1.1.7 variant, concentrated in the Wolverines’ athletic program. Late last month, the university instructed students to stay in their rooms as much as possible and paused campus sports, disrupting a winning basketball season and any hope that the spring semester might be less chaotic than the fall.

“I’ve started wearing two masks,” said Alyssa Frizzo, a junior from Rockford, Mich., who described the variant as a haunting presence on the Ann Arbor campus. “I think a lot of people have.”

With nearly a year of coronavirus experience behind them, leaders at Michigan and other U.S. universities ushered in the new term pledging not to repeat the errors of last year, when infection rates soared on campuses and in the surrounding communities. A New York Times effort has recorded more than 397,000 cases and at least 90 deaths connected to campuses since the pandemic began.

But although most schools have pledged to increase testing as a way of spotting outbreaks early, it is an expensive proposition at a time when many are struggling financially, and not all are testing students as often as recommended by public health experts.

The plans to keep the virus under control at Michigan, which had more than 2,500 confirmed cases by the end of the fall semester, included increased testing, offering more courses online, limiting dorm rooms to one occupant and establishing a policy of no tolerance for rules violations. Yet already more than 1,000 new virus cases have been announced by the school since Jan. 1.

Other universities across the country have also encountered obstacles to a smooth spring, ranging from the unexpected challenge of emerging variants — also detected at the University of Texas at Austin, the University of Miami, Tulane and the University of California, Berkeley, in recent days — to the more common problem of recalcitrant students.

At Vanderbilt University in Nashville, students returning after winter break were required to be tested upon arrival, and then asked to avoid social interactions while awaiting results. But some had other ideas.

“We identified a cluster of positive Covid-19 cases linked to students who did not follow the arrival shelter-in-place rules,” a campuswide email reported on Jan. 23, blaming two student organizations for violating protocols. “More than 100 students are now in quarantine.”

Tulane University in New Orleans, which is testing students at least twice a week, said it had placed 18 students and six Greek organizations on interim suspension after they violated social distancing rules in the first weeks of classes.

The foundation of most university plans for the spring semester is built on ramped-up testing to quickly identify infected students before they display symptoms, then place them in isolation to prevent the virus from spreading. The testing push has grown since July, when a study by researchers including A. David Paltiel, a professor of public health policy and management at the Yale School of Medicine, recommended that college students be tested twice a week to better detect asymptomatic infections.

The American College Health Association later embraced the idea, issuing guidelines in December. “For the spring, we specifically recommend that all students are tested on arrival and twice a week thereafter if possible,” said Gerri Taylor, a student health expert who serves as co-chair of the organization’s Covid-19 task force.

Ms. Taylor said her organization did not know what percentage of schools had adopted the recommendations, and a survey of colleges across the country revealed a variety of requirements, ranging from only voluntary testing to mandatory testing twice a week.

After more than 2,000 students were quarantined in two high-rise dormitories as the virus raged through its campus last fall, the University of Wisconsin-Madison beefed up its testing regimen, with plans to administer up to 80,000 tests a week this spring.

“A high degree of testing — and quarantining those who test positive or were exposed — is key to keeping the university operating and our community safe,” Lori Reesor, vice chancellor for student affairs, said in a conference call with reporters.

The University of Florida, where testing was voluntary for the fall, now requires tests twice a month for students living in campus housing, in fraternity or sorority houses or attending classes in person — a population that accounts for nearly half of the school’s 52,000 students.

Terry W. Hartle, senior vice president of the American Council on Education, an industry group, estimates that testing has cost U.S. colleges and universities an average of $9 million per institution, part of $120 billion in total expenses and lost revenue stemming from the pandemic. Some schools have already had to cut programs, lay off staff and even close for good as costs mount.

Federal stimulus funds have helped offset some of the testing expenses, and President Biden’s proposed stimulus plan includes an additional $35 billion for higher education. But even the least-expensive tests — which have ranged from $6 at the University of California, Davis, to $25 for a consortium of schools partnering with the Broad Institute in New England — can become costly when applied aggressively at a large campus.

The more expensive P.C.R. swab tests used to diagnose the coronavirus typically cost $50 or more apiece from commercial suppliers, though some schools have lowered the price by developing their own tests or partnering with a nonprofit lab.

Michigan requires only one test per week for students, but even with fewer of them on campus this spring, it has tripled its testing since the fall semester to 15,000 tests per week from about 5,000. The school has also increased the use of the more expensive P.C.R. tests because their results can be used for genomic sequencing to identify variants, said Emily Toth Martin, an associate professor of epidemiology who devised the campus testing program.

Dr. Martin said that the first case of the variant at Michigan was identified because both the state health authorities and the university had geared up to increase genomic testing, particularly for people who had traveled to hot spots. By the end of last week, more than 600 genomic sequencing tests had been carried out in an effort to locate variants, she said.

“This is a variant that moves 50 percent faster than anything we had to deal with last semester,” she said.

Dr. Martin supported the decision, recommended last month by the Washtenaw County Health Department, to issue stay-at-home advisories for students, which expired Sunday, and to pause University of Michigan sports practices and games because the variant cluster was tied to its athletics program.

“We were working with a fast-moving problem and a slow technology,” she said. “We had thousands of students arriving in the city. I didn’t have the confidence to say we knew where the spread was going.”

The decision led to the postponement of at least five Wolverines men’s basketball games, which has angered some students and alumni. More than 5,000 people signed a petition calling for a “team by team” shutdown, which would have allowed those with no positive tests — including basketball — to keep playing.

The associate athletic director at Michigan, Kurt Svoboda, said on Friday that athletics would restart for the most part this week except for two teams — women’s water polo and men’s ice hockey.

As of Friday, the university had identified nine new cases of the variant since the previous week, according to Susan Ringler-Cerniglia, a spokeswoman for the Washtenaw County Health Department. “The good news, if there is any, is that it’s still associated with the campus community, and we haven’t had any cases in the broader community.”

Ms. Ringler-Cerniglia said that the first-identified carrier of the variant at Michigan — Patient Zero — “did everything right.” After testing positive in early January, she advised the health authorities that she had been in England during winter break, leading to additional testing that isolated the variant. Another student, who traveled domestically to a state where the U.K. variant had already been identified, appears to have brought it back to campus, as well.

Ryan Glauser, a Ph.D. student in history, is a co-chair of the Covid-19 task force for a union of graduate teaching assistants. His union went on strike last fall to demand improved coronavirus safety measures, and won concessions. He had been optimistic about this semester, until the variant emerged.

“The B.1.1.7, with its arrival, has thrown a lot of question marks in the air,” he said.

Ms. Frizzo, the junior from Rockford, said the university’s testing protocols were more convenient this semester. But the variant has made her more cautious even about everyday activities, including the communal bathrooms in the dorms.

“You take off your mask to brush your teeth,” she said.