In April, with hospitals overwhelmed and much of the United States in lockdown, the Department of Health and Human Services produced a presentation for the White House arguing that rapid development of a coronavirus vaccine was the best hope to control the pandemic.

“DEADLINE: Enable broad access to the public by October 2020,” the first slide read, with the date in bold.

One of the first school districts in the country to reopen its doors during the coronavirus pandemic did not even make it a day before being forced to grapple with the issue facing every system actively trying to get students into classrooms: What happens when someone comes to school infected?

Just hours into the first day of classes on Thursday, a call from the county health department notified Greenfield Central Junior High School in Indiana that a student who had walked the halls and sat in various classrooms had tested positive for the coronavirus.

Contact tracing, a cornerstone of the public health arsenal to tamp down the coronavirus across the world, has largely failed in the United States; the virus’s pervasiveness and major lags in testing have rendered the system almost pointless. In some regions, large swaths of the population have refused to participate or cannot even be located, further hampering health care workers.

As schools, universities and businesses struggle to reopen without the coronavirus testing they need to curb outbreaks, the Trump administration’s testing czar testified to Congress Friday that it was currently impossible to get all tests back within three days.

The testing czar, Adm. Brett P. Giroir, told lawmakers that getting all coronavirus tests back between 48 and 72 hours, which many health officials have said is critical, “is not a possible benchmark we can achieve today, given the demand and the supply.”

ROME — When the coronavirus erupted in the West, Italy was the nightmarish epicenter, a place to avoid at all costs and a shorthand in the United States and much of Europe for uncontrolled contagion.