At the University of Texas, Austin, the difference between a student vaccination rate of 60% and 80% would cost the school about $4 million to prevent the spread of Covid-19 and treat additional students projected to get infected, the head of the school’s Covid-19 Modeling Consortium said.

If fewer students return to campus vaccinated, the school will spend more on tests, contact tracing, quarantine housing and online classes, said Dr. Lauren Meyers, a professor of integrative biology and data science who directs the consortium. The school recommends vaccines, but state law prevents them from being mandated.

“There are these two different worlds: There’s a world where we do all this mitigation, it’s successful enough that we could actually safely continue school in person and we don’t have to go online,” she said. “Or there’s the world where we don’t do enough to mitigate, or there’s not enough vaccination in the community, and we’re forced to go online, and that incurs additional cost to UT.”

The consortium modeled 35 scenarios with student vaccination coverage ranging from 40% to 80% in a recent report. It estimates that 57% of the school’s 50,000 students would be fully vaccinated when classes start on Aug. 25 and about 209 would arrive on campus for the fall semester infected with Covid-19.

If 60% of students are vaccinated when the semester starts, the consortium recommends proactive testing of unvaccinated students twice weekly to keep Covid-19 spread under control. If 80% are vaccinated, then symptomatic testing alone would suffice.

About 700 schools are mandating vaccines for students, according to a tally by the Chronicle of Higher Education. Indiana University was among the first to require students to be vaccinated unless they are exempt for religious or medical reasons. Last week, the U.S. Supreme Court rejected a request by a group of Indiana students who were seeking to block the school from enforcing the vaccine requirement.

About 85% of students have so far been vaccinated, said school spokesman Chuck Carney. Students at Indiana, and most other schools that have mandates, are asked to submit proof that they have been vaccinated.

Twelve states, all led by Republicans, have prohibited schools from mandating vaccines, according to the National Academy for State Health Care Policy. Some, including South Carolina State University, are delaying the start of the fall semester by several days to give students more time to get vaccinated.

“As the university steps up its response to this pandemic, we will do all we can to encourage students, faculty and staff to get vaccinated if they have not already done so,” said school President Alexander Conyers.

Some schools are passing the approximately $50 cost of a surveillance test onto unvaccinated students. Federal aid that colleges and universities received to cover the cost of preventive measures earlier this year isn’t recurring, and insurers generally don’t cover surveillance testing.

Birmingham-Southern College, a private liberal-arts school in Alabama with 1,100 students, recommends students get vaccinated. The school received about $1.5 million in federal money to cover the cost of testing and other expenses created by the pandemic last academic year, said school President Daniel Coleman.

That money isn’t available this year, and Mr. Coleman said he believes the 70% of students who will return to campus already vaccinated shouldn’t be forced to pick up the tab for weekly surveillance testing for their unvaccinated classmates.

“It’s not going to cover all the costs, but we wanted something there to demonstrate that if they make this decision they need to pay for it,” he said.

Clint Reid, chairman of the College Republican Federation of Alabama, likened the fee to a ransom. “It’s not something you should be forced into,” he said. “But if the school is going to mandate tests, they should probably foot the bill for them.”

Alabama has one of the highest infection rates in the nation and among the lowest vaccination rates. The Republican-led state Legislature has banned schools from mandating Covid-19 vaccines.

The nation is evenly divided over vaccine requirements in higher education: 48% of Americans support colleges and universities mandating vaccination and 49% oppose it, according to a Quinnipiac University poll of 1,300 people this month.

At Penn State, more than 1,200 faculty signed a petition asking for a vaccine mandate, and the faculty senate voted “no confidence” in the university’s public-health plan that recommends—but doesn’t mandate—vaccines.

In an open letter to the university, Penn State President Eric Barron said he was mindful of maintaining support across the political spectrum. Republicans, who tend to oppose rules requiring vaccines, control the state’s Legislature, and the governor is a Democrat.

“Regulations across the country clearly reflect state-level political realities,” Dr. Barron said in the letter. “State funding of our University requires a two-thirds vote of the Pennsylvania Legislature, meaning that our funding relies on strong bipartisan support.”

Penn State Prof. James Tierney is scheduled to teach an economics class this semester to 590 students in a hall that holds about 750, he said. He asked if he could teach this semester’s class online. When that request was denied, he submitted a letter of resignation effective at the end of the year.

“We’re in the middle of a pandemic,” he said. “Why not just allow for the class to be remote?”