The proportion of COVID-19 infections and hospitalizations among vaccinated Pennsylvania residents rose sharply last month, reflecting the fact that a higher percentage of the population has now gotten the shot but also likely that protection has waned somewhat, state health officials said Friday.

The latest Department of Health data on so-called “breakthrough” infections shows that between Sept. 5 and Oct. 4, vaccinated people represented just over a quarter — 26% — of more than 135,000 new infections and nearly 5,000 hospital admissions across the state. Death statistics for the last 30 days were not available because of lags in reporting and verification.

When the Health Department released its initial set of data on breakthrough cases Sept. 14, just 6% of cases and 5% of hospitalizations since January were among vaccinated residents.

As more people get vaccinated, the number of vaccinated people who contract the virus and wind up in the hospital will inevitably represent a greater share of the overall total.

State health officials cited that mathematical certainty but also the rise of the highly transmissible delta variant of the coronavirus, along with waning immunity among some who have received the vaccine.

“The vaccines were designed to prevent severe illness and hospitalization, but no vaccine is 100%,” the state’s acting physician general, Dr. Denise Johnson, said in a phone interview. “As we have more people vaccinated, there will be more cases in those vaccinated people. It doesn’t mean the vaccines aren’t working. They’re working as designed.”

The situation in Pennsylvania mirrors what is happening nationally. Breakthrough cases accounted for 14% of U.S. hospitalizations and 16% of deaths in June and July, about twice the percentage as earlier in the year, according to a study released last month by the Centers for Disease Control and Prevention.

At a press briefing Friday, Johnson said state officials are seeking to learn more about Pennsylvania’s breakthrough cases, including the length of time that people had been vaccinated before getting infected, whether they had underlying health conditions that increased the risk, and their age.

She said the data reinforced the case for booster shots. Last month, U.S. health officials approved a third dose of the Pfizer vaccine for all Americans 65 and older, along with higher-risk younger people.