The company has said that it has a limited supply of remdesivir, but that it is increasing production. It is also drawing on a stockpile Gilead created for use in future pandemics after Ebola outbreaks in West Africa.

Mr. Allard, who lives in Metuchen, N.J. and works as an equity analyst for Bank of America in Manhattan, was admitted to the hospital on the night of March 16 after he had a high fever, back pain and was throwing up, his mother said. She described Mr. Allard, a former All-American lacrosse player at Bates College, as healthy and with no underlying conditions.

In New Jersey, officials announced 935 new positive cases on Monday, bringing the total to 2,844, including 27 deaths.

Mr. Allard was tested Tuesday for the coronavirus, but the sample was sent to Quest Diagnostics, and his mother said the hospital never received the results.

As the week went on, Mr. Allard’s condition rapidly declined, Ms. Allard said. He was placed in a medically induced coma and put on a ventilator. Ms. Allard said his doctors wanted to try remdesivir, but they needed to have a confirmed positive test showing he had the illness caused by the virus before they could seek the drug from Gilead. Finally, after the family made a flurry of calls to elected officials and others, the hospital tested him again on Saturday and received the results from another laboratory in less than six hours.

His results came back positive and that night — around 10:30 — the doctor submitted her request to Gilead. By the next morning, Ms. Allard said, she was hearing that Gilead was shifting to a new system, and news stories later on Sunday confirmed her fears.

His doctor has submitted another request under the new system, but Ms. Allard now fears his treatment will be delayed. Ms. Allard said the doctor had previously treated another patient with remdesivir and the turnaround time for approval had been 48 hours.

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