“The idea was to give him more soldiers in his body to fight this war,” said Dr. Diaz, who is part of the team treating Dr. Laroche at Palm Beach Gardens Medical Center in South Florida.

But there was no donor and no prospects and no time.

So Dr. Laroche’s family in Port-au-Prince, Miami and New York began a desperate search on social media to find someone who beat Covid-19 and was willing to donate blood. Some 80 miles away from Dr. Laroche’s hospital bed, an administrator of a Miami seaport was just returning to work after a mild bout of the disease. He was in his office one morning when the phone rang. He picked up and received an urgent plea: “Will you help save my brother?”

Thus began a frantic journey to deliver hope to a critically ill doctor, with stops at a mobile blood donation bus in Hialeah, a laboratory in Orlando and, finally, the first floor of a Palm Beach Gardens hospital.

In the absence of a vaccine or proven treatment, Covid-19 survivors are being viewed as potential saviors for patients with a disease that has killed more than 52,000 people in the United States. Demand for what is known as “convalescent plasma” has outstripped supply by roughly two to one, setting off a kind of pandemic free-for-all for survivor good will.

Under a national program overseen by the Mayo Clinic and authorized in early April by the Food and Drug Administration, about 2,500 patients at U.S. hospitals have been given the experimental treatment that Dr. Laroche’s family sought. The F.D.A. has also granted “numerous” applications from individual doctors to use convalescent plasma for Covid-19 patients, the agency said in a recent statement.