Low immunization rates are indeed responsible for the outbreak of highly infectious measles linked to Disneyland in California, a new study suggests.

The outbreak that began in December 2014 spread rapidly, suggesting a significant percentage of the exposed population may have been susceptible because of a lack of vaccination, the researchers say.

They analyzed public health data and case data from case clusters in California, Arizona and Illinois.

In those states, vaccination rates for the measles, mumps and rubella (MMR) vaccine among the exposed population may be as low as 50 percent and probably no higher than 86 percent, they report in the journal JAMA Pediatrics.

Because measles is so contagious, vaccination rates of 96 percent to 99 percent are necessary to preserve what is known as herd immunity and prevent future outbreaks, the study authors explain.

"Clearly, MMR vaccination rates in many of the communities that have been affected by this outbreak fall below the necessary threshold to sustain herd immunity, thus placing the greater population at risk as well," they say.

With vaccines, measles was virtually eliminated in the U.S. in 2000 and cases only occurred when someone arrived with an infection from abroad, where the disease is still common.

In a population fully vaccinated against measles, one such imported case will give rise to just two additional cases over 70 days, experts say, whereas if only 60 percent of a population is vaccinated, an additional 2,800 cases will occur over the same time period.

In a population fully susceptible to measles, one infected person will spread the virus to between 11 and 18 additional people, they say.

"As long as measles continues to circulate in the rest of the world, the U.S. will always be at risk of importation cases, either from Americans [returning from] traveling abroad or visitors coming into the U.S.," says Jessica Atwell, a doctoral candidate in global health and epidemiology at Johns Hopkins Bloomberg School of Public Health.

If we are complacent or if vaccination rates remain low, she cautions, "measles can come back in a heartbeat."

The researchers behind the new study say they agree.

"Our data tell us a very straightforward story - that the way to stop this and future measles outbreaks is through vaccination," says epidemiologist John Brownstein.  "The fundamental reason why we're seeing the number of cases we are is inadequate vaccine coverage among the exposed."