Models are helping scientists understand Covid-19 infections rates and hospitals plan for surges. This one—a generic version—illustrates the concept of how infection rates can be changed by social distancing measures.
Tuesday, March 31, 2020
The Mathematics of Predicting the Course of the Coronavirus
Epidemiologists are using complex models to help policymakers get ahead of the Covid-19 pandemic. But the leap from equations to decisions is a long one.
People at a supermarket in Barcelona, Spain, practice social distancing
PHOTOGRAPH: DAVID RAMOS/GETTY IMAGES
IN THE PAST few days, New York City’s hospitals have become unrecognizable. Thousands of patients sick with the novel coronavirus have swarmed into emergency rooms and intensive care units. From 3,000 miles away in Seattle, as Lisa Brandenburg watched the scenes unfold—isolation wards cobbled together in lobbies, nurses caring for Covid-19 patients in makeshift trash bag gowns, refrigerated mobile morgues idling on the street outside—she couldn’t stop herself from thinking: “That could be us.”
There’s no way to know those answers for sure. But hospital administrators like Brandenburg have to hazard an educated guess. That’s the only way they can try to buy enough ventilators and hire enough ICU nurses and clear out enough hospital beds to be ready for a wave of hacking, gasping, suffocating Covid-19 patients.
That’s where Chris Murray and his computer simulations come in.
Murray is the director of the Institute for Health Metrics and Evaluation at the University of Washington. With about 500 statisticians, computer scientists, and epidemiologists on staff, IHME is a data-crunching powerhouse. Every year it releases the Global Burden of Disease study—an alarmingly comprehensive report that quantifies the incidence and impact of every conceivable illness and injury in each of the world’s 195 countries and territories.
In February, Murray and a few dozen IHME employees turned their attention full-time to forecasting how Covid-19 will hit the US. Specifically, they were trying to help hospitals—starting with the UW Medicine system—prepare for the coming crisis. Brandenburg says the collaboration could turn out to be, quite literally, life-saving. “It’s one thing to know you may be getting a surge of patients,” she says. “If you can make that more tangible—here’s what it’s actually going to look like—then we’re in a much better place in terms of being able to plan for the worst.”
But it’s a big if. During a pandemic, real data is hard to find. Chinese researchers have only published some of their findings on the spread of Covid-19 in Hubei. The ongoing catastrophe of testing for the virus in the United States means no researcher has even a reliable denominator, an overall number of infections that would be a reasonable starting point for untangling how rapidly the disease spreads. Since the 2009 outbreak of H1N1 influenza, researchers worldwide have increasingly relied on mathematical models, computer simulations informed by what little data they can find, and some reasoned inferences. Federal agencies like the Centers for Disease Control and Prevention and the National Institutes of Health have modeling teams, as do many universities.
As with simulations of Earth’s changing climate or what happens when a nuclear bomb detonates in a city, the goal here is to make an informed prediction—within a range of uncertainty—about the future. When data is sparse, which happens when a virus crosses over into humans for the first time, models can vary widely in terms of assumptions, uncertainties, and conclusions. But governors and task force leads still tout their models from behind podiums, increasingly famous modeling labs release regular reports into the content mills of the press and social media, and policymakers still use models to make decisions. In the case of Covid-19, responding to those models may yet be the difference between global death tolls in the thousands or the millions. Models are imperfect, but they’re better than flying blind—if you use them right.
THE BASIC MATH of a computational model is the kind of thing that seems obvious after someone explains it. Epidemiologists break up a population into “compartments,” a sorting-hat approach to what kind of imaginary people they’re studying. A basic version is an SIR model, with three teams: susceptible to infection, infected, and recovered or removed (which is to say, either alive and immune, or dead). Some models also drop in an E—SEIR—for people who are “exposed” but not yet infected. Then the modelers make decisions about the rules of the game, based on what they think about how the disease spreads. Those are variables like how many people one infected person infects before being taken off the board by recovery or death, how long it takes one infected person to infect another (also known as the interval generation time), which demographic groups recover or die, and at what rate. Assign a best-guess number to those and more, turn a few virtual cranks, and let it run.
“At the beginning, everybody is susceptible and you have a small number of infected people. They infect the susceptible people, and you see an exponential rise in the infected,” says Helen Jenkins, an infectious disease epidemiologist at the Boston University School of Public Health. So far, so terrible.
The assumption for how big any of those fractions of the population are, and how fast they move from one compartment to another, start to matter immediately. “If we discover that only 5 percent of a population have recovered and are immune, that means we’ve still got 95 percent of the population susceptible. And as we move forward, we have much bigger risk of flare-ups,” Jenkins says. “If we discover that 50 percent of the population has been infected—that lots of them were asymptomatic and we didn’t know about them—then we’re in a better position.”
So the next question is: How well do people transmit the disease? That’s called the “reproductive number,” or R0, and it depends on how easily the germ jumps from person to person—whether they’re showing symptoms or not. It also matters how many people one of the infected comes into contact with, and how long they are actually contagious. (That’s why social distancing helps; it cuts the contact rate.) You might also want the “serial interval,” the amount of time it takes for an infected person to infect someone else, or the average time before a susceptible person becomes an infected one, or an infected person becomes a recovered one (or dies). That’s “reporting delay.”
And R0 really only matters at the beginning of an outbreak, when the pathogen is new and most of the population is House Susceptible. As the population fractions change, epidemiologists switch to another number: the Effective Reproductive Number, or Rt, which is still the possible number of people infected, but can flex and change over time.
You can see how fiddling with the numbers could generate some very complicated math very quickly. (A good modeler will also conduct sensitivity analyses, making some numbers a lot bigger and a lot smaller to see how the final result changes.)
Those problems can tend to catastrophize, to present a worst-case scenario. Now, that’s actually good, because apocalyptic prophecies can galvanize people into action. Unfortunately, if that action works, it makes the model look as if it was wrong from the start. The only way these mathematical oracles can be truly valuable is to goose people into doing the work to ensure the predictions don’t come true—at which point it’s awfully difficult to take any credit.
Speaking at a White House briefing on Thursday, Deborah Birx, response coordinator for the Coronavirus Task Force, admonished the press against taking those models too seriously, even as New York governor Andrew Cuomo begged for federal help with acquiring ventilators and protective equipment for health care workers. “The predictions of the models don’t match the reality on the ground,” Birx said.
Responding to Birx in a thread on Twitter, Harvard infectious disease epidemiologist Marc Lipsitch said Birx had been talking about work from his lab, which the federal government had asked for two days prior. In a preprint (so not peer-reviewed), his team had used an SEIR model with numbers tweaked to simulate the tightening or loosening of social distancing measures, as well as a potential flu-like seasonal variation in Covid-19 infections. He was varying R0, essentially. In the model, putting a stop to strict social distancing (without something like a vaccine or a cure coming along) allowed infections to climb right back up to their peak of about two critical cases per 1,000 people—which could be 660,000 Americans getting seriously ill or dying. And even with the strictest lockdown-type measures lasting from April through July, his team’s model finds that the disease surges back in autumn.
Remember, the whole point of social distancing is to slow the epidemic, to keep the numbers of sick people at any one time below the maximum that the health care system can handle and stall so scientists can work on treatments. If Lipsitch’s team is right, the characteristics of Covid-19 might require a cyclical flux between strict social distancing and viral resurgence, on and on, perhaps until 2022. If everything goes right, Lipsitch wrote—massive testing and quarantines of the ill, and aggressive social distancing—it’ll be possible to keep numbers down and maybe shorten the timeline. But, Lipsitch said on Twitter, he didn’t see any of that underway.
So in dismissing all that, was Birx making a policy determination to assume that the model’s most Panglossian all-is-well prediction will turn out to be correct? “That was my impression of her comment,” said Yonatan Grad, an infectious disease epidemiologist at the Harvard School of Public Health and a lead author with Lipsitch on that study, speaking at a press conference on Friday.
Except Ferguson wasn’t really walking back his estimates or his model. As he explained later in his own series of tweets, the new numbers resulted from two things: The UK government’s implementation of social distancing measures, and a slightly higher R0 gleaned from new data from around Europe that suggests the outbreak is moving faster than at first believed—so more people are infected than anyone knows, with milder symptoms. Ferguson said this should lend more evidence, not less, to the importance of social distancing measures.
If it’s the second scenario, says Sunetra Gupta, the theoretical epidemiologist who led the Oxford work, “that would be great news,” because it would mean a substantial chunk of the UK population is already immune. Nevertheless, even though it’s only one of the scenarios that Gupta’s model implies, the fact that it seemed to contradict Ferguson’s work at Imperial College was enough for commentators and some media to tell a story of conflicting (and therefore untrustworthy) models.
To be clear, it does seem like some number of people, possibly a large number, transmit the virus without being diagnosed. They’re House Infected posing as House Susceptible. That seems clear from researchers showing that in January, the strict travel measures that China imposed on people trying to leave Wuhan, the center of the outbreak, slowed the spread of the disease. That shows up in location data drawn from mobile phone apps; the extent of the disease followed people’s travel patterns. When Wuhan locked down, that spread almost completely stopped. It bought time for the world to prepare—which large parts of the world, including the United States, squandered.
One clever study even tried to use a complicated model to estimate exactly how many undiagnosed, or “undocumented,” cases the Chinese population had in January. The researchers—some at Imperial College and others in the US—broke the infected population into two groups, diagnosed and undiagnosed. (Or, in the language of this particular study, “documented” and “undocumented.”) Through surveys and an app-based data collection project, the US team had actual data on how many people were wandering around at any given time with a respiratory virus; their calculations see that number peaking at more than 10 percent.
Using that as a kind of baseline, and combining it with location data for travel among 375 Chinese cities including Wuhan, the researchers tried various models to infer, given the number of overall infections and where they happened, how many undiagnosed infections had to have been out there. Their conclusion was stunning. Just 14 percent of infections were diagnosed, they wrote. Fully 86 percent of infected people were the walking ill, stealth transmitters of the virus. “Those undocumented infected people were about half as infectious. However, because there are many more of them, they are the dominant driver of the outbreak,” says one of the creators of the model, Jeffrey Shaman, director of the Climate and Health Program at the Columbia University School of Public Health. “This virus needs these undocumented cases to successfully move through a society. If you’re identifying cases, you’re going to have a better handle on it.”
ONE THING THAT models will try to predict next, given the tranche of new data that these first 80,000-plus cases in the US will yield, is when social distancing and shelter-in-place measures can end. “You don’t make the timeline. The virus makes the timeline,” Anthony Fauci, head of the National Institute of Allergy and Infectious Diseases told CNN. But how will anyone determine that timeline? The role of children as spreaders of the disease still isn’t clear. Nor is the role of adults with mild symptoms.
“One of the crucial things that has to happen is serology tests, testing large numbers of the population for antibodies,” Jenkins says. This data would show whether a person has been infected—whether or not they had symptoms. Population wide, that can give you more certainty on the numbers of susceptible and recovered people. The tests really only work a week or so after a person contracts the virus, but that’s valuable enough that the UK has purchased 3.5 million blood tests already, and researchers in the Netherlands have begun testing blood bank donations.
“The modeling is going to be absolutely key for how we come out of this and what things look like longer-term. But for places in Western Europe and the United States, the scientific evidence is not particularly complicated. If you act soon, with a firmer response, you limit early deaths and get through the initial phase more quickly,” Jenkins says. That’s based less on models, she says, and more on research like a 2007 article about the “non-pharmaceutical interventions” different US cities used to respond to the 1918 influenza pandemic. Places that locked down more quickly had death rates half as high as ones that waited—and one of the coauthors of that study was the same Marc Lipsitch who tangled with the White House last week.
So Brandenburg’s team got to work. They had already begun ordering more masks, gloves, face shields, and ventilators. They had opened a drive-thru testing station. Now they moved to deploy a surge plan, canceling all elective surgeries and trying to clear out as many beds as possible. They built wedding-party-size triage tents outside hospitals’ emergency room departments to keep potential Covid-19 patients away from others seeking care. They called up ICU nurses who’d retired within the past five years. They began shuffling in nurses, respiratory therapists, and technicians from other departments, training them up on the specifics of critical care.
Adam Rogers writes about science and miscellaneous geekery. Before coming to WIRED, Rogers was a Knight Science Journalism Fellow at MIT and a reporter for Newsweek. He is the author of the New York Times science bestseller Proof: The Science of Booze.
Megan Molteni is a staff writer at WIRED, covering biotechnology, public health, and genetic privacy. Previously, she freelanced as a reporter, audio producer, and fact-checker. Her work has appeared in Popular Science, Discover, Undark, Nautilus, and Aeon. She studied biology and ultimate frisbee at Carleton College and has a graduate...
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