According to a small study that assessed the likelihood of re-infection, unvaccinated people who have had Covid-19 are more than twice as likely to be re-infected as those who test positive and maintain their natural immunity with a vaccine have strengthened.

The study, published Friday by the Centers for Disease Control and Prevention, looked at the risk of reinfection in May and June in hundreds of Kentucky residents who tested positive for the virus in 2020.

Those who weren’t vaccinated this year were 2.34 higher risk of reinfection than those who received their vaccinations. The study, published Friday, suggests that adding a vaccine provided better protection to those who survived infection than the natural immunity created by their original battle with the virus alone.

Although the study looked at only a small number of people in Kentucky, it appears to disprove the argument made by one of its US Senators from his home state, Rand Paul, who has repeatedly claimed that vaccination for people like him who had the virus is unnecessary and developed immunity.

Dr. Rochelle P. Walensky, director of the CDC, said the data reinforces the importance of vaccination, including for those who have already had the virus.

“If you have ever had Covid-19, please get vaccinated anyway,” said Dr. Walensky on Friday. “The vaccine is the best way to protect yourself and others around you, especially as the more contagious Delta variant is spreading across the country.”

The study’s authors warned that not much is known about how long natural immunity to the virus lasts and that genome sequencing has not been performed to confirm the reinfections weren’t just flares of the remains of the initial infection of the subjects.

The CDC and the Biden government have been aggressively advocating an increase in vaccinations over the past few weeks as the number of cases and hospitalizations has risen sharply in the last month, largely due to the Delta variant and particularly in regions of the country where vaccination rates are low.

Last week, the number of new virus cases reported daily on Thursday averaged 100,200, and for the first time since mid-February the daily average exceeded 100,000, according to a database from the New York Times. On Friday, the country recorded 106,723 new cases a day.

Another study published on Friday reported that vaccinations drastically reduced hospital admissions for Covid in the elderly in February, March and April. The study looked at data from 7,280 patients from a Covid hospitalization monitoring network and used government records to check their vaccination status. The vast majority of hospital patients were not or only partially vaccinated; only 5 percent were fully vaccinated.

Although vaccination did not completely eliminate infection, the risk of hospitalization was significantly lower for people who were fully vaccinated. Among those 65 to 74, Pfizer-BioNTech and Moderna vaccines reduced the risk of hospitalization related to Covid by 96 percent, and Johnson & Johnson’s single-dose vaccine reduced hospital admissions by 84 percent. In the 75+ age group, Pfizer vaccination reduced hospital admissions by 91 percent; the Moderna vaccine by 96 percent; and the Johnson & Johnson vaccine by 85 percent.