Gov. Greg Abbott of Texas may have been overly optimistic on Sunday when he said on Fox News that his state could be “very close” to herd immunity — the point where so much of the population is immune to Covid-19, either from being vaccinated or previously infected, that the virus can no longer spread.

“When you look at the senior population, for example, more than 70 percent of our seniors have received a vaccine shot, more than 50 percent of those who are 50 to 65 have received a vaccine shot,” Mr. Abbott, a Republican, told Chris Wallace. Mr. Wallace had asked why statewide infection, hospitalization and death rates were more under control than in other states, in spite of Texas reopening many activities and eliminating mask mandates.

The governor added, “I don’t know what herd immunity is, but when you add that to the people who have immunity, it looks like it could be very close to herd immunity.”

Michael Osterholm, an epidemiologist and director of the University of Minnesota’s Center for Infectious Disease Research and Policy, said, “There is no way on God’s green earth that Texas is anywhere even close to herd immunity.”

He added: “Look no further than Michigan and Minnesota, which have much higher rates of vaccination than Texas. And we’re already seeing widespread transmission.”

About 19 percent of people in Texas are fully vaccinated, while Michigan has 22 percent and Minnesota has 24 percent.

Estimates of what it would take to reach herd immunity have edged up since the pandemic began, ranging from requiring immunity in 60 percent to more than 90 percent of the population to halt transmission.

No one is sure what the level should be, Dr. Osterholm said. “Anybody who will tell you exactly what the level of herd immunity is, is also likely to want to sell you a bridge.”

He predicted that within a few weeks or a month, Texas and other parts of the southern and western United States would see cases rising the way they are now in the Upper Midwest and Northeast.

A major factor in the relentless spread of the coronavirus is the increasing proportion of cases caused by the virus variant first identified in Britain and known as B.1.1.7, which is more contagious than the form of the virus that first emerged.

That variant “surely resets the meter” and makes herd immunity harder to achieve, Dr. Osterholm said. Additional variants could further complicate the forecast.

“These variants are game changers,” he said. “They really are. It’s really remarkable.”