SCITUATE — The first of 15,000 hand sanitizer units were given out to school departments from a state Emergency Management Agency storage facility yesterday afternoon, part of a campaign to ensure cleaner hands in Rhode Island classrooms and prevent the infections that led to the closing of three school systems last week.

The schools in all three communities are reopening this morning, two days after Dr. David R. Gifford, the state health director, announced that test results had shown that a fourth suspected case of mycoplasma bacteria at Hopkins Hill Elementary School in Coventry was a false alarm.

The bacterium, which normally produces mild illness, is believed to have caused three students in Warwick and West Warwick to develop encephalitis, an inflammation of the brain. One, a 7-year-old second grader from Warwick, died. The other two have recovered.

Because it has been so long since the third case surfaced, Gifford said Saturday, the immediate threat appears to have passed.

Distribution of the units and the disinfectant gel began at 1 p.m. yesterday. In the first half-hour, workers from 9 of the state’s 36 school districts had picked up their allotments. All the districts are supposed to pick them up by day’s end tomorrow and install them as soon as possible.

State Department of Education spokesman Elliot Krieger, who briefed the media on yesterday’s distribution, said the goal was to have one dispenser in every classroom where there is no convenient sink to wash up.

Twenty seconds of washing with soap and warm water is the best, health officials have said. The disinfectant gels are a practical alternative.

The Health Department is encouraging teachers and nurses to have children clean their hands before preparing or eating food and after going to the bathroom, handling garbage, blowing their noses, coughing or sneezing. Officials also want children to be taught to cough or sneeze into their arm. Parents are being told to keep their sick children out of school.

Krieger said the dispensers are not new to many schools. “They’ve been put in as part of the pandemic flu program,” he said, referring to the campaign to prevent the spread of bird flu if it, or an equally virulent virus, were to appear. Good hygiene “is something our nurse-teachers work with all the time.”

Putting the hand sanitizers in every classroom, lunchroom, library and anywhere students congregate “brings it to another level,” he said.

The units are also being donated to the state’s 150 nonpublic schools. They are being packaged with “Cover Your Cough” posters.

The cluster of encephalitis cases was unusual enough that the federal Centers for Disease Control and Prevention in Atlanta sent doctors to Rhode Island to search for an underlying cause. That probe will continue, said Dr. Matthew Moore of the CDC.

Concern began with the first case of encephalitis, which caused the death of 7-year-old Dylan Gleavey.

A second-grade classmate at Greenwood Elementary School, Hannah Leahy, was infected, but recovered.

In response, the state closed the school a week ago.

But a third case, involving a middle school student in West Warwick surfaced, followed by a report that the Coventry student had developed a similar illness, leading health officials to close schools in all three communities as a precaution.

The Coventry student was found to have viral meningitis, a disease that appears — on average — once a day in Rhode Island. Unlike encephalitis, it is caused by a virus, so it is unaffected by the antibiotics that were widely distributed in response to the outbreak.

Doctors had originally suspected that the Coventry child had meningitis, but the three encephalitis cases had also initially appeared to be meningitis as well.

Gifford said the Coventry student has recovered and no other children have been found to be suffering from the bacterial infection.

The health director said the state will continue to monitor hospitals and schools, but he added that cases of meningitis and pneumonia are not uncommon this time of year.