BERLIN — By the time German health officials in the state of North Rhine-Westphalia had confirmed the state’s first case of the coronavirus, the patient was already so ill that he had been placed on a respirator. He was in no position to help authorities, who urgently needed to retrace his steps to try to stop, or at least slow, the virus from spreading.

Already, they had lost time. The virus has a 14-day incubation period, so they knew that the man, who was hospitalized on Monday, could have infected many more people before he started receiving care, all of whom were now in a position to pass the virus across Germany’s most populous state and beyond.

“We were faced with a puzzle that we thought had 10,000 pieces, only to realize it has endless pieces,” Ulrich Hollwitz, a spokesman for the Heinsberg district where the man lives with his wife and two children, said on Thursday. “Where was he and when? Who did he have contact with and how close? We had to start working like detectives.”

What the authorities were able to discover was worrisome: The man had not traveled to China, or to other virus hot spots like Italy, nor was he known to have come in contact with someone confirmed by the authorities to have been infected.