Blood is being collected from dozens of children in New York to determine whether they share any genetic variations that might make them susceptible to the illness.Tissue samples from at least one of the three patients to have died from it — ages 5, 7 and 18 — have been sent to a public health laboratory for intensive testing.A team of more than 30 disease detectives — epidemiologists, clinicians and statisticians — is poring over thousands of pages of medical records.Even as doctors and scientists around the world race to develop treatments and vaccines for Covid-19, New York State has become the center of a parallel effort to investigate an unnerving aspect of the outbreak: a mysterious new syndrome that is sickening a small but growing number of children.The ailment has now been reported in at least 161 children in New York, making the state’s caseload one of the largest publicly reported anywhere. Hundreds of other children across the United States and in Europe have also been sickened with the illness, now called multisystem inflammatory syndrome.The syndrome can be characterized by severe inflammation of the heart, blood vessels, the gastrointestinal tract or other organs, believed to be caused by a reaction to the coronavirus. The inquiries into why it is occurring, and whether a treatment can be found, could have an impact on how the authorities handle the reopening of schools and other activities for children.Doctors in New York involved in the effort say that not since health officials began investigating the connection between Zika in pregnant women and microcephaly in their children in 2015 and 2016 has there been such an urgent search to understand a mysterious threat to children.“We’re looking at everything possible — therapies, diagnostics, clinical picture — and we’re trying to pull this all together,” said Dr. Howard A. Zucker, the state health commissioner.He added: “It does keep you up at night. It’s the worry that is there something we’re going to figure out, in six months from now, and we’re going to say, ‘How did we not think about that possible therapy? How did we not figure out that this would have helped?’”The rush to understand the syndrome has seemed to strike a chord with Dr. Zucker, who is trained in pediatric cardiologist and has reviewed charts of patients with the illness.He described in an interview the frustrations of doctors trying to treat one of the victims, who at the time was battling against a syndrome that did not have a name. The child was in shock; cardiac function had deteriorated, and with it, breathing.Tubes were fed down the patient’s throat, and aggressive, life-sustaining measures — mechanical and pharmacological — were deployed, even as a battery of tests were run to try to determine the cause of the illness.The young patient did not survive.“Everything was done for that child,” Dr. Zucker said. “Everything.”Another doctor spoke of troubling heart ailments in children afflicted with the illness. A third talked about treating a 14-year-old in the Bronx, who was admitted in good condition but then rapidly deteriorated.“Twenty-four hours later, he was aggressively delirious and had to be restrained and sedated,” that doctor, H. Michael Ushay, recalled.The illness was so baffling that Dr. Ushay, the medical director of the pediatric critical care division at Children’s Hospital at Montefiore, said he was not sure whether the child’s downturn was caused by the disease or a reaction to the medicine the boy was given at the hospital, which included steroids and ketamine.But with Kawasaki disease, few children end up in an intensive care unit or requiring life support. Yet a significant number of the patients with the new syndrome did.Dr. Ellen Lee, an epidemiologist at New York City’s Department of Health, was online, reading through her medical discussion groups in late April, when she came across reports of a mysterious illness affecting children in England that seemed to be linked to the coronavirus. The symptoms included abdominal pain and an inflamed, weakened heart.She asked her team to call hospitals to inquire if they were seeing anything similar. Some had not. Others had treated children with matching symptoms, but had not understood they were seeing something new. Within a day or two, Dr. Lee’s team had learned of nine cases.“Those early days, it was just trying to get as much information from the providers, to understand what they were seeing, what was unusual,” Dr. Lee said.Members of her team contacted the Centers for Disease Control and Prevention. They asked the New York City’s medical examiner’s office to see if any children had recently died who showed any of the symptoms. They checked with health departments in other major cities. Nobody seemed to know much.But it would soon become very clear to Dr. Lee, the Health Department and city officials that this new syndrome was rapidly becoming a threat to children in New York.Across the state, about 70 percent of patients with the syndrome have been cared for in intensive care units, according to Dr. Marcus Friedrich, a senior official at the state’s Department of Health. Some required medicine to raise their blood pressure, Dr. Friedrich said, and 19 percent were put on mechanical ventilators.A few have required even more drastic intervention, including extracorporeal membrane oxygenation, in which blood is removed from the body and oxygenated by means of a membrane before being pumped back into the patient, doctors said.Most patients with the new syndrome were previously healthy, without any underlying health problems, doctors said in interviews. And the surge and timing of the cases have led many doctors to conclude the illness is linked to the coronavirus.Many, though not all, of the children ill with the syndrome tested positive for the coronavirus or for antibodies to it, which indicate a prior exposure. But often, based on interviews with the parents, doctors have concluded that the children had been exposed to the virus several weeks earlier.“Is it the virus causing this, or the body’s response to the virus after a few weeks?” asked Dr. Edward Conway Jr., the chief of pediatric critical care at Jacobi Medical Center in the Bronx, saying that he suspected the latter.The time lag and the signs of inflammation have led many doctors to suspect the new syndrome is caused by “an aggressive, abnormal immune reaction to Covid exposure,” said Dr. George Ofori-Amanfo, chief of pediatric critical care for the Mount Sinai Health System.For now, doctors have been administering similar treatments as they would to children with Kawasaki disease or toxic shock syndrome. Patients often receive a combination of steroids; intravenous immunoglobulin, an antibody-rich serum made from donated blood, pooled together from thousands of people; and intravenous aspirin.So far, many patients have responded well, four doctors said in interviews.“Lots of long-term studies are going to emanate from this group of children,” Dr. Michael Gewitz, of Maria Fareri Children’s Hospital in Westchester County, said on last week’s video conference.As more patients with the syndrome emerge, doctors say they have been struck by the variety of ways the new illness seems to manifest.Recently doctors have begun describing a few cases in which patients had brain inflammation or other symptoms involving the central nervous system — a part of the body that did not seem a focal point of the syndrome initially. At least one patient has had Guillain-Barré syndrome, which is marked by temporary paralysis, Dr. Gewitz said in an interview.One boy with the new syndrome who died — a 7-year-old — had a severe headache and grew increasingly unresponsive when doctors tried to rouse him, Dr. Gewitz said. “The child who succumbed had overwhelming brain swelling,” he said.At Cohen Children’s Medical Center on the border between Queens and Nassau County, which has treated more than 40 patients with the new syndrome — the highest number in the state — one doctor expressed some hope that the number of new patients with the syndrome was beginning to slow in recent days.“We’re starting to see a flattening” of the number of cases, the doctor, James Schneider, said.But Dr. Lee, of the city’s Health Department, said it was too soon to know if the number of cases was slowing or still surging.