Fifteen children in New York City have been hospitalized for what officials called a “multi-system inflammatory syndrome” that the local health department is investigating as a possible consequence of a Covid-19 infection. Doctors in Europe have recently reported similar cases — the latest potential twist in the coronavirus pandemic. The children in New York, ages 2 to 15, had high fevers and elevated levels of inflammatory markers, signs that are common in shock and an acute pediatric heart condition called Kawasaki disease, the city’s health department said Monday. The patients, who were hospitalized from April 17 to May 1, also showed symptoms including rash, abdominal pain, vomiting, and diarrhea. Five of the children had to be placed on ventilators; none have died. Only four of the patients tested positive for an active case of Covid-19, but six of the negative cases showed evidence of a previous infection based on blood tests.