There are outbreaks. There are epidemics. And there are pandemics, where epidemics become rampant in multiple countries and continents simultaneously. The novel coronavirus that causes the disease named covid-19 is on the verge of reaching that third, globe-shaking stage.

Amid an alarming surge in cases with no clear link to China, infectious disease experts believe the flulike illness may soon be impossible to contain. The World Health Organization has not declared covid-19 a pandemic, and the most devastating effects, including more than 2,400 deaths, are still in China. But the language coming from the organization’s Geneva headquarters has turned more ominous in recent days as the challenge of containment grows more daunting.

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“The window of opportunity is still there, but the window of opportunity is narrowing,” WHO Director General Tedros Adhanom Ghebreyesus said Friday. “We need to act quickly before it closes completely.”

At the beginning of any disease outbreak, public health experts painstakingly trace the contacts of every person who becomes sick. The experts build a family tree of possible illness, with branches that include anyone who might have shaken hands with, or been sneezed on, by an infected person. But with confirmed infections approaching 80,000 people, tracing contacts on a case-by-case basis could soon be impractical.

If the coronavirus becomes a true pandemic, a large proportion of the human population — a third, a half, two-thirds even — could become infected, although that doesn’t necessarily mean that all will become ill. The word “pandemic” evokes fear, but it describes how widespread an outbreak may be, not its deadliness.

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“If we went across the whole world and had a magic ball and were able to detect everyone who’s positive, we’d see it in lots of countries,” said Michael Mina, an infectious disease specialist at the Harvard T.H. Chan School of Public Health. “It’s never clear until it’s happening.

Experts suspect the virus is spreading stealthily.

“I think we should assume that this virus is very soon going to be spreading in communities here, if it isn’t already, and despite aggressive actions, we should be putting more efforts to mitigate impacts,” said Jennifer Nuzzo, an epidemiologist and senior scholar at the Johns Hopkins Center for Health Security. “That means protecting people who are most likely to develop severe illness and die.”

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The count in Italy jumped to 58 cases within days, for instance, concentrated in the prosperous Lombardy region and neighboring Veneto. A 78-year-old man on Friday became the country’s first coronavirus death, and on Saturday, a 77-year-old woman was found dead in her home, after having tested positive for the virus. But authorities said the woman had other health problems, and it was unclear whether she had died of the virus.

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Veneto’s regional president, Luca Zaia, said it is becoming harder to track the virus’s jump from place to place.

“It goes to show you that having other cases of contagion is absolutely possible,” Zaia said.

Containing the virus would be easier if people who are contagious were obviously so, as was the case with severe acute respiratory syndrome, or SARS, which started an outbreak that burned itself out in 2003. But the new virus appears to spread among people who are not always noticeably sick. In fact, most cases of covid-19 have been mild. Taxi drivers and people at business meetings have spread the virus, and among the more than 600 passengers from the Diamond Princess cruise ship who have tested positive for it, about half had no obvious symptoms.

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Reports that some patients are falling ill long after they were exposed are also raising questions about whether the virus’s incubation period is longer than 14 days, potentially casting doubt on current quarantine criteria.

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A new study from an international team of researchers, posted on a medical preprint site Monday, estimates that two-thirds of the coronavirus infections in Wuhan, China, before the imposition of travel restrictions Jan. 23 were transmitted by people who were not documented as infected. A report in the New England Journal of Medicine in the past week suggested that the disease reaches peak infectiousness shortly after people start to feel sick, spreading in the manner of the flu. A study published in JAMA on Friday chronicled the case of a 20-year-old Wuhan woman who infected five relatives, even though she never showed signs of illness.

“What we find is that this virus is going to be very difficult to contain,” said Jeffrey Shaman, an infectious disease researcher at Columbia University and co-author of the study posted Monday. “Personally, I don’t think we can do it.”

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Harvard epidemiologist Marc Lipsitch estimates that 40 to 70 percent of the human population could potentially be infected by the virus if it becomes pandemic. Not all of those people would get sick, he noted. The estimated death rate attributed to covid-19 — roughly 2 in 100 confirmed infections — may also drop over time as researchers get a better understanding of how widely the virus has spread.

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The novel coronavirus may be particularly suited for stealth community transmission since its symptoms can be indistinguishable from those of a cold or the flu, and testing capabilities are still being expanded.

Experts estimate it takes about a week for the number of infections in a given community to double. Based on that, it probably would take several weeks for a new infection cluster to be picked up by a local health department, said Trevor Bedford, a computational biologist at Fred Hutchinson Cancer Research Center in Seattle. By mid-March, he estimated, officials should know whether there is community transmission and a true pandemic.

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The virus has infected people in every province of China and is spreading in communities in Singapore, South Korea, Thailand, Vietnam, Hong Kong and Japan, according to Nancy Messonnier, a top official at the U.S. Centers for Disease Control and Prevention.

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“I want to be clear that we are not seeing community spread here in the United States yet,” she said Friday. “But it’s very possible, even likely, that may eventually happen.”

So far, the United States has 35 cases, about half of them repatriated passengers from the Diamond Princess cruise ship, where the virus spread widely while the ship was under quarantine in Yokohama, Japan. Additional American evacuees were positive in tests conducted by the Japanese, but U.S. authorities won’t count them until they confirm the findings.

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“It is impossible to predict at this point if the current outbreak will progress to a true pandemic,” said Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases and a member of the U.S. coronavirus task force. “If a large number of countries are unsuccessful in preventing sustained multi-generation transmissions, then we could witness the next pandemic.”

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A pandemic is a line in the sand, and every expert has a slightly different definition for when an outbreak crosses it. Generally, it means that there are self-sustaining lines of infection in multiple countries and continents — where the family tree of possible illness begins to encompass the entire population.

Some specialists remain optimistic that this won’t happen, in part because the Chinese government has imposed extreme measures to keep people isolated in their homes. Although the cumulative number of cases keeps going up, the rate of increase has apparently slowed. Changes in how the Chinese are tracking cases has impeded efforts by outside experts to understand the numbers.

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“I don’t want to be complacent. I don’t want to say we’re out of the woods,” said Ian Lipkin, a Columbia University epidemiologist who traveled to China recently to assist with the epidemic response and who isolated himself for two weeks after returning. “But I think we’re not in as dire straits as we might be, and that’s because everyone is pulling together internationally.”

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The WHO may be hesitant to declare a pandemic, as the label comes with significant political and economic consequences. When the WHO declared a pandemic for the H1N1 influenza outbreak in 2009, the decision was later criticized by some countries, which felt the decision caused unnecessary fear and overly aggressive responses. The declaration, for example, prompted many countries to spend large sums on vaccines, even though the H1N1 strain of influenza proved to be relatively mild.

The lethality of the new coronavirus remains difficult to estimate. But across the planet, many health systems are preparing for a pandemic emergency. That includes making plans for treating people who are suspected of having the disease, and for protecting health-care workers.

In China, the death of Li Wenliang, the doctor who was also a whistleblower about the new virus, underscored the risk to those on the front lines. More than 3,000 health-care workers have been infected, according to a report from Chinese public health authorities.

A major Boston health-care system set up emergency operations in late January — treating the threat like a fire in the building or the Boston Marathon bombing. At the University of Minnesota Medical Center, a “scrum” team was activated in early February to focus on the health system’s preparedness for possible patients.

Public health experts are devising strategies on how to conserve N95 respirators, specialized masks that are in limited supply amid surging demand. They’re even thinking about seemingly small details such as how to make sure patients don’t cause new infections if they use a touch screen to check in.

“We have to be ready,” said Paul Biddinger, chief of the division of emergency preparedness for Massachusetts General Hospital in Boston. “Extrapolating from some of the numbers we’ve seen on the impact to the health-care system in China, it means we’ll have to surge fast.”