For Germany, the breaking point could come in December. France and Switzerland might crack by mid-November. Belgium could hit its limit by the end of the week.

Europe, in the throes of a savage second wave of the pandemic, is on the verge of a medical crisis, with intensive care units filling rapidly. Governments are finding that when confronted by the unforgiving reality of an exponentially spreading virus, even vast investments to expand hospital capacity can be washed away in days.

Germany, Europe’s best-
resourced nation, risks being swamped even after increasing its intensive care beds by a quarter over the summer. Belgium, which had doubled its intensive care capacity, is now preparing for decisions about which needy patient should get a bed.

“This huge capacity we’ve built gave a false impression of security. It gave a higher buffer, but ultimately it only represents a week when you’re in an exponential phase,” said Emmanuel André, a leading Belgian virologist who has advised the government on the pandemic — and has bitterly criticized leaders for acting too slowly this fall.

In retrospect, the warning signs could be seen as early as July, when cases in Europe started ticking up again after the relaxation of spring lockdowns. In absolute terms, the numbers were still tiny. Spanish emergency room doctors enjoyed a respite, after being hammered in March and April. Italian nurses headed to the beach. Central European leaders — their countries among the worst hit now, but back then largely untouched — gathered at the end of August for a triumphant conference to discuss the post-pandemic era.

But the math for exponential growth is as simple as it is scary. When two coronavirus cases double to four, and four cases double to eight, it doesn’t take long for the numbers to reach the tens of thousands — and beyond.

“An exponential phenomenon starts with very small numbers, and it is not tangible for weeks and weeks and weeks for people out there,” André said. “If you look at the numbers, you have very strong indicators early on that things are going wrong, but it is only at the very end that things explode.”

Europe is now feeling the explosion.

The European Union reported 1.2 million cases over the past week, the highest yet during the pandemic. Weekly deaths rose by half. Occupancy of intensive care units doubled in 17 days leading up to Oct. 25 in countries tracked by the European Center for Disease Prevention and Control, according to figures released Thursday.

“Europe is at the epicenter of this pandemic once again,” the World Health Organization’s Europe director, Hans Kluge, told an emergency meeting of health ministers Thursday.

A week ago, half of French intensive care beds were occupied. Now, 70 percent are occupied, with more than 3,500 coronavirus patients. When President Emmanuel Macron announced a second national lockdown last week — something he and other European leaders had sought mightily to avoid — he warned that “at this stage, we know that whatever we do, nearly 9,000 patients will be in intensive care by mid-November, which is almost the entirety of French capacities.”

In Germany, Chancellor Angela Merkel — a physicist — has tried to explain the dynamics of viral growth to her citizens in terms that are simultaneously calming and unnerving.

“In summer, at the end of June, beginning of July, we had 300 new infections on most days,” Merkel said at the end of September. “And now we have on most days 2,400 infections. And that means that in July, August, September, in three months, the infection rate has doubled three times.”

If the cases kept increasing at the same rate, she warned, they would be up to 19,200 a day by the end of December, a number that at the time seemed fantastical and chilling. In the end, it took only a month: Germany reported a record 19,059 cases Saturday, a 173 percent rise over the past two weeks.

The full impact of those record infections will be felt in intensive care wards in two weeks, Merkel said Monday, as Germany began its new lockdown.

The numbers are leading “with increasing speed to an acute emergency situation in our hospitals,” she said at a news conference. “We can’t allow this rapid spread of the virus to continue to overwhelm our health system.”

There were 2,061 coronavirus cases being treated in German intensive care beds Sunday, a doubling from 10 days earlier and a more than fivefold increase since Oct. 1. While the country still has more than 20,000 available ICU beds, including a new emergency reserve, it will max them out by early December at the current pace, according to a simulator created by researchers at the University of Saarland.

“We are alarmed,” said Joachim Odenbach, a spokesman for the German Hospital Association. There is a shortage of 4,700 intensive care staff members, he said.

In Britain, coronavirus-related hospital admissions have doubled in two weeks, and top public health advisers have warned that hospitals could be overwhelmed within six weeks, forcing a reluctant Prime Minister Boris Johnson to announce a lockdown Saturday.

Without a lockdown, “doctors and nurses could be forced to choose which patients to treat, who would live and who would die,” Johnson told the House of Commons on Monday. “The virus is doubling faster than we can conceivably add capacity. Even if we double capacity, the gain would be consumed in a single doubling of the virus.”

In Belgium and the Netherlands, staffing shortages are leading nursing homes and hospitals to ask doctors and nurses to keep working if they have tested positive for the virus but are asymptomatic, because so many other medical workers are sick or quarantining.

“Employees who are infected with the coronavirus and have no or very mild complaints are ­allowed to continue working at Argos Care Group in an ­intensive-care corona department if they wish,” wrote one Dutch nursing home chain Thursday. Some Dutch hospitals are full enough that the country has started transferring a handful of patients by helicopter to Germany. Coronavirus patients in Dutch intensive care beds have doubled in three weeks.

“We will support by taking patients into our ICU capacities as long as we can,” German Health Minister Jens Spahn said Friday, declaring that his country was also preparing to take in Belgian and Czech patients.

The situation in Central Europe is especially dire. The medical systems of Poland, the Czech Republic and some of their neighbors have fewer resources than do the richer countries to the west, and leaders did not use the relative calm of the spring and summer to build capacity for the fall. Polish hospital cases have doubled in two weeks, and authorities are just now building a 1,200-bed field hospital in the National Stadium in Warsaw.

Czech authorities are frightened — a sharp change from the spring, when the country was among the least touched in Europe. Prime Minister Andrej Babis predicted that the health system would collapse between Nov. 7 and 11 if he did not impose the lockdown he announced Oct. 21. The Texas and Nebraska National Guards are flying people in to bolster Czech medical staffs. Hospitals around the country are seeking volunteers.

Elements of hospital capacity can be elastic, especially for patients who do not need the most intensive care but simply need someone to keep an eye on them. Nurses can be spread more thinly. Doctors can do fewer rounds. Patients can be discharged earlier. But intensive care units need heavy staffing, and access to them can be a matter of life and death for those who are ailing.

Managing the coronavirus has proved particularly difficult because patients often need long hospital stays and because doctors and nurses need to take so many time-consuming precautions that they can see fewer patients any given hour, said Marc Noppen, chief executive of University Hospital Brussels.

His hospital is not yet at capacity, but it could be within weeks, he said. Hospital leaders have been reviewing an ethics charter they drew up in the spring “in case you have to make a choice, in case you have only one ventilator left and two patients. Who gets it?” Noppen said. “We have to hope for the best and prepare for the worst.”

The risks were already visible midsummer, Noppen said: On July 22, a day when Belgium reported 145 new cases, he circulated a model that predicted that the country’s hospitals could be saturated around Nov. 6 if on the same trajectory.

On Saturday, the country reported 23,921 new cases, a record. And, as predicted in July, hospitals could be saturated by Nov. 6.

Birnbaum reported from Riga, Latvia, and Morris from Berlin. Luisa Beck in Berlin and Quentin Ariès in Brussels contributed to this report.