MIAMI—Florida is recording more Covid-19 cases than any other U.S. state, as hospitalizations in some areas increase at the fastest rate since the start of the pandemic.

The state accounts for one in five new infections in the U.S. and logged 67,413 cases over the past week, according to the Centers for Disease Control and Prevention. Florida had 314 cases per 100,000 people over the past week, second only to Louisiana. The weekly total of new cases reported by Florida jumped more than fourfold between July 1 and July 22, reaching its highest point since mid-January.

Epidemiologists say various factors are at play: large numbers of unvaccinated people, a relaxation of preventive measures like mask-wearing and social distancing, the spread of the highly contagious Delta variant of the coronavirus and the congregation of people indoors during hot summer months.

“All of those things together are a recipe for the data that we’re seeing,” said Jason Salemi, an epidemiologist at the University of South Florida. “It’s a cause for concern...the speed with which the indicators are going up.”

The number of people hospitalized in Florida has climbed steeply over the past month, reaching 3,849 on July 17, the largest tally since late February, according to a data dashboard created by Dr. Salemi. Patients are skewing younger, with 53% under age 60, compared with 30% at the start of the year.

Fifty-five of 67 counties had test positivity rates of 10% or more as of July 9. A Covid-19 testing and vaccination site in Orlando on July 21.

Fifty-five of 67 counties had test positivity rates of 10% or more as of July 9. A Covid-19 testing and vaccination site in Orlando on July 21.

Photo: Paul Hennessy/Zuma Press

Among people age 12 or older in Florida, 55% are fully vaccinated, compared with 57% nationally, according to the CDC.

At University of Florida Health Jacksonville’s two hospitals, 146 patients have been admitted with Covid-19—a record high for the facilities, exceeding the previous peak of 125 in January, said Chad Neilsen, director of accreditation and infection prevention. Covid-19 patients take up three-quarters of the beds at the system’s north campus, which draws people from more-rural areas with low vaccination rates, he said.

“This is a rate of admissions to our hospital that we have not seen before ever,” Dr. Neilsen said. “It really has been unbelievable.”

In response, the medical center is shifting some patients from the north campus, which is over capacity, to a downtown campus. The facilities took measures to prepare, stocking up on personal protective equipment and testing materials.

But they are contending with a staff shortage, a challenge hospitals across the U.S. are facing. Nearly 110 staff members at UF Health Jacksonville are out because they either tested positive for Covid-19 or are awaiting results, Dr. Neilsen said. About 52% of staff are vaccinated, he said, and the hospital system, like many in the U.S., doesn’t mandate vaccines. The hospital system is trying to make up shortfalls by hiring short-term travel nurses, who are expensive, or local student nurses from training programs.

Memorial Healthcare System in the Fort Lauderdale area has 246 Covid-19 patients, 98% of whom are unvaccinated, said Chief Operating Officer Leah Carpenter. That is well below the 674 patients the hospital network had a year ago, but the numbers are increasing far more quickly, she said. Two of its four adult acute-care hospitals are over capacity.

Memorial is converting spaces like a conference center and an auditorium into additional patient-care areas, Ms. Carpenter said. And it is trying to address personnel shortages by offering bonuses and incentive pay, as well as by working with a staffing agency.

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At Jackson Health System in Miami, the number of Covid-19 patients more than doubled to 143 over a two-week period ending Tuesday. The increase prompted the hospital to cancel visitation for most inpatient units, with some exceptions.

“The people that are getting sick and [are] unvaccinated are getting sick very fast,” likely because of the Delta variant, said Dr. Lilian Abbo, chief of infectious diseases, on a conference call Tuesday. “We were not seeing this in any of the other waves of the pandemic.”

Infections are increasing across Florida, with 55 of 67 counties notching test positivity rates of 10% or more as of July 9, compared with 11 such counties two weeks earlier, according to Dr. Salemi’s dashboard. Counties with high positivity rates often have low vaccination rates, data show.

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At John Knox Village, a retirement community in Pompano Beach, near Fort Lauderdale, some residents felt ill earlier this month and tested positive for Covid-19, said Chief Operating Officer Bill Pickhardt. Administrators contacted the county health department, which sent a team to conduct widespread testing at the complex of nearly 1,000 residents. At least 11 residents and one employee tested positive.

Because of high vaccination rates, those who tested positive mostly are experiencing mild symptoms, Mr. Pickhardt said. Administrators have canceled indoor dining, suspended programming that involves gathering crowds, and made mask-wearing mandatory for staff and highly recommended for residents.

Republican Gov. Ron DeSantis said at a news conference Wednesday that he expected an increase in cases during summer, but a spokeswoman ruled out new lockdowns or mask mandates, which he considers ineffective. Instead, he urged people to get vaccinated.

“If you are vaccinated, fully vaccinated, the chance of you getting seriously ill or dying from Covid is effectively zero,” Mr. DeSantis said.

Democratic Mayor Jerry Demings of Orange County—home to Orlando and facing a surge of cases—said he would like to have leeway to impose rules for businesses like requiring mask-wearing in crowded spaces. But a measure Mr. DeSantis signed into law in May limits local governments’ ability to do so.

“Our goal is not to infringe on individual business rights,” Mr. Demings said. But, he added, “I’d like to have options to hold businesses accountable.”

Write to Arian Campo-Flores at arian.campo-flores@wsj.com