Included in Mr. Gustafson’s suspicions of an undercount: himself. He is convinced that he had the coronavirus in January, when he was so crushingly tired and feverish, he could scarcely summon the strength to walk to the bathroom from his bed.

“I think it was here long before we knew it,” said Mr. Gustafson, who is also a nurse and said he believes that he contracted the virus from one of the recently deceased people who was brought to the coroner’s office long before anyone in Illinois was looking for positive coronavirus cases. “That’s the only logical thing I can think of.”

Some people have spent part of their days sheltering at home going over the details of their bouts with what could have been the coronavirus. In Rothschild, Wis., Tommie Swenson and his girlfriend, Tammy Swikert, keep thinking of the illness they contracted during the winter that spread widely through their village of 5,000 people.

It was nothing like the flu, said Mr. Swenson, a retired truck driver. Milk and soda tasted funny, or like nothing at all. He could barely sleep at night, he had such a rattling cough and felt a crushing weight on his chest.