Israel’s Ministry of Health on Monday began offering a third dose of the Pfizer vaccine to severely immunocompromised adults in what health experts say could be the first phase of an experiment to provide coronavirus booster shots for older people and the most vulnerable. The recommendation, published Sunday by the ministry, said that the goal of the new program was to raise antibody levels among immunocompromised citizens, including cancer patients, recipients of liver transplants, and others who have recently exhibited weakened vaccine protection, according to data. It said that it had still not made a decision on administering third shots for the general adult population. The decision comes as Israel, which was among the fastest to vaccinate in the winter and then among the first to begin reopening in the spring, is experiencing a surge in new cases, spurred by the prevalence of the highly transmissible delta variant first identified in India. Over the past month, infection rates in Israel have spiked from single digits to more than 400 a day.