A Mathematician’s Guide to How Contagion Spreads

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A Mathematician’s Guide to How Contagion Spreads

Adam Kucharski didn’t expect to publish a book about contagion in the middle of a global pandemic. But consider him less surprised than the rest of us. “In my field we always have the next pandemic on the radar,” he says.

Kucharski, an epidemiologist at the London School of Hygiene and Tropical Medicine, is a mathematician by training. He uses data and models to predict how disease outbreaks will progress. His new book, The Rules of Contagion: Why Things Spread—and Why They Stop, lays out those tools and how they can be applied to other parts of life. Think methods to predict how panic might course through the global financial system, or how bad information is transmitted on Facebook. But most important, Kucharski says, is what he calls “epidemiological thinking.” That’s a mindset for dealing with incomplete information, as infectious-disease researchers must when they encounter a novel, fast-moving pathogen. Sometimes you might make bad assumptions, and your models might make predictions that never come to pass. But in a crisis, coming up with a hypothesis, even if it’s a rough one, is often the only way to get people to act.

Courtesy of Hachette Book Group

Kucharski spoke to WIRED from London, where he lives. This interview has been edited for length and clarity.

WIRED: I want to ask about something you tweeted just a few minutes ago. It was an excerpt from a book by the British geographer Richard Burton about mosquitoes and malaria. Could you describe what he wrote and why it left an impression?

Adam Kucharski: In the book I dig into the history of many principles of transmission, and one thing that struck me was this quote by a British geographer in the 1850s. Burton noted that in Somalia local people were drawing a link between the rise of mosquitoes in a given season and these fevers, which were probably malarial fevers. And he was very dismissive. He called it superstition. At the time, a lot of the thinking around malaria, certainly in the West, was one of “bad air.” The word malaria comes from mala aria—Italian for bad air. And it was very interesting to see that there were cultures in the world that hit on this other link. But it would take decades for the supposedly “enlightened” Western scientists to reach that same connection. It really does illustrate with some of these ideas that history is more complex than perhaps we think. In many cases, the credit is given very prominently to certain individuals without acknowledging some of the deeper origins.

WIRED: It’s not a perfect analogy, but when I read that, I couldn’t help but think about the relearning of facts and evidence that’s happening right now with Covid-19. Other countries had learned a lot from diseases like SARS. Wuhan had months to learn about this virus before it spread widely in the US and Europe. Now we’re here rediscovering the effectiveness of things like masks.

AK: I think that’s a very important point. Learning from what other countries are doing and not making the same mistakes again is crucial. It was quite striking how a lot of the discourse around the Ebola outbreaks, for example, was at times quite patronizing. People were saying, ‘We know how to control this. We know what measures work, so why aren’t people doing them? Why aren’t people just doing what they should be doing to reduce transmission?’ And of course, we’ve now got major epidemics in a number of countries that supposedly had high levels of preparedness. But transmission is not going down, because people aren’t following those supposedly obvious things that can help reduce transmission. So I think it shows that the challenges are far more fundamental than we think they are, and we have to look widely for solutions to a problem this great. It’s not the case that one group or one country is going to have the answers.

WIRED: What were some of those fundamental challenges you saw with Ebola?

AK: There were a couple of things. In many Ebola outbreaks, people were slow to take the threat seriously because of concerns that certain organizations were trying to advance some kind of agenda by playing up the danger. That’s exactly what we’ve seen with some of the speculation on Covid.

Contact tracing was also a big part of the Ebola response, and more recently, vaccination response has been embedded within contact tracing. Essentially, you establish the contact-tracing network and then vaccinate people rather than just quarantining them. But again, that’s very reliant on trust, because you need people to give that information to contact tracers. You need people, when they’re symptomatic, to come forward.

And of course, if you’ve got diseases that are associated with stigma, then that can reduce the kind of engagement that people have with health systems. And again, it’s the same for Covid—that if we want to use these targeted measures like contact tracing, we need people to report when they’re symptomatic. We need them to get tested. We need them to contact people who might be exposed as well or engage with contact-tracing systems. So it’s those same basic principles of really taking it seriously as a threat and engaging with the measures that we know can help reduce transmission. It’s just a different setting in a slightly different period of time.

WIRED: Let’s step back and talk about the “rules of contagion.” You frequently come back to what you refer to as the DOTS model. What is that?

AK: A very important number we use to measure contagion is R0, the reproduction number. So if you have a case, on average, how many additional cases do they create? There are four things that influence the value of that number, and understanding those four processes can reveal a lot about how different kinds of outbreaks work. I refer to these four things as the DOTS.

So it is, first of all, the duration of infection. How long is someone spreading contagion for? But it’s also the opportunities for spread during that period. How many interactions they have, how many others might they expose? It’s also the transmission probability during one of those interactions. To give an example with Covid, you might have a conversation with someone, but depending on whether you’re wearing a mask or not, that might change the transmission probability. And then the final component is susceptibility—that we might have a situation where people are interacting, that they’re not having any protective measures during this interaction. But if people are vaccinated, then that’s not a problem. It turns out that if we multiply these components together we get the value of the reproduction number. We can target any of those four things to reduce transmission.

WIRED: You’ve studied many epidemics caused by all sorts of pathogens. What’s unique about Covid-19?

AK: One of the really striking things about this infection is that it’s sort of a middle case in terms of our ability to contain it. So it’s not like SARS, which has features that mean contact tracing can bring it under control in many countries. But it’s also not like the flu, where it just spreads so rapidly and so few people have symptoms that there’s very little hope of controlling it. It sits in between the two. That’s why we’ve seen such a wide range of outcomes in different countries, in terms of the success of the response.

WIRED: One thing that really stuck with me, hearing these stories from the history of epidemiology, was that so many of them involve a disbelieving bureaucrat. Sometimes the modelers make bad assumptions and they get laughed off. Or else the predictions are right and they don’t get credit for having prevented disaster. Did that resonate as someone creating mathematical models for Covid-19?

AK: I think so. One of the big challenges with outbreaks is we have to deal with very patchy information, often quite contradictory sources of information, and yet we still need to make decisions. One of the lessons from the history of epidemiology is, first of all, there are things that in hindsight seem completely obvious. For example, in early experiments that people did with mosquitoes, they thought it might be transmitted because people were drinking water that had dead mosquitoes in it. Today you know it’s a bite. It’s totally obvious that’s what’s going on, and people aren’t really drinking dead mosquitoes. But people just hadn’t made that step yet. So I think, first of all, it’s important to realize that some of the insights that are obvious are actually only possible in hindsight.

But I think it’s also accepting that you may not have perfect evidence, but if you have sufficient evidence that there’s a threat, you still have to act upon it. One example would be debates around the actual level of severity of Covid. Early on, we did some analysis with other groups suggesting that somewhere between 0.5 percent and 1 percent of infections are fatal. And there’s obviously been a lot of debate, with many groups speculating that perhaps it’s much lower and perhaps many more people have been exposed. There’s still no conclusive evidence about exactly what the fatality risk is currently. But we have to, I think, make a conclusion or working theory from that. Until we have strong evidence, then we have no other way.

Epidemiology is full of such examples. For instance, the link between smoking and cancer. You can’t run an experiment and get a bunch of people to take up smoking and see what happens. You have to piece together from observation what you think might be going on and ascertain the level of threat you’re dealing with. Ultimately, I think that’s where epidemiological thinking can be very useful.

WIRED: You finished this book before the pandemic began. But what was your mindset when you set out to write? Was the next pandemic on your radar?

AK: In my field we always have the next pandemic on the radar. Flu has obviously been top of the list. But I think a novel coronavirus was always seen as a major potential threat, based on what we knew about SARS and MERS. Revisiting the book, it’s kind of striking how many of the issues are still there. The book opens by talking about a second wave and closes talking about how we’re going to use cell phone data and tracking to help us with public health problems. Those things were perhaps somewhat abstract when I was writing the book. I gave a talk at TED a year ago on a project we’d done that was trying to collect detailed movement and location data, but do it in an ethical, anonymized way, and a lot of the challenges that entails. But actually that debate around privacy versus public health is exactly what we’re having now. So, yeah, it’s been interesting to see how these ideas pre-pandemic are now playing out during this outbreak.

But the book was also an effort to understand emerging threats in different areas. In recent years, a big contagion focus has been on misinformation. I think financial crises as well have shaped, and probably will continue to shape, society. And then, of course, other forms of offline contagion, like violence. The aim of the book was to outline some fundamental ideas that can be useful for helping us tackle these different problems—and also show how we can learn from mistakes and things that have been overlooked in different fields. Often, if you’re facing a particular form of contagion, another field might have already dealt with some of these exact situations. An example being the 2008 financial crisis, which was caused in large part by network structures that researchers on sexually transmitted diseases knew about 30 years earlier.

WIRED: You write about your summer working in finance at Canary Wharf in 2008, just when everything was tumbling down. It sounds like people had thought a lot about financial networks in terms of how you could profit from them, but not so much in terms of how they could fail. You write about how, similar to a person trying to avoid an STD, there were multiple forms of network exposure to worry about.

AK: I don’t think, at that point, there was much thinking about how things were linked together. You could have banks, for example, that individually looked well diversified in their portfolios, but they may have all diversified in the same direction. So when an asset crashes, you’ve implicitly got a contagion effect, because all of them are exposed in the same way.

I think that’s when people really began to understand the role of network effects, particularly some of these features where you had these large banks widely connected to create these hub effects, which meant that contagion spread far wider than it might have otherwise. During that period, you also saw a lot of hidden links being created, which meant there were all of these loops of contagion that no one really knew about. No one was quite sure exactly who owed what to whom. Which is one of the reasons why the system froze during that period, because people didn’t know who to trust. It was, I guess, similar to a pandemic spreading through [an airline] flight network, where you’ve got many different routes through which cases can come. If you don’t know what those links are, it’s very hard to understand where the risk is.

WIRED: You left finance after that, but from what you can tell, has that perception changed? Do people have a better sense of those network structures now, having seen how they fail?

AK: There’s been a lot of effort to understand this. But I think with any outbreak, understanding is the first step and doing something about it is the second. There were a lot of efforts by central banks to try to make changes that can improve resilience. So one is obviously things like capital requirements, particularly on banks that are structurally important to the network. But talking to people who work for central banks, as I wrote in the book, I think it is very clear that once you actually start changing the structure of the network, then that becomes much harder to persuade people to do.

So when you’re talking, for example, within banks about “ring fencing,” which essentially cuts the links between the investment side and the retail side, that’s something that people oppose far more strongly. So I think you can end up with a situation, unfortunately, as we often get with infectious diseases, where you can understand how contagion is happening, but actually implementing things that reduce those risky links may come in for a lot of opposition.

WIRED: One thing that really struck me was in thinking about contagion on social media. Your conclusion was that misinformation isn’t as infectious as people might think, at least compared to very infectious diseases like measles.

AK: This is really striking. So two things influence the dynamics of an outbreak. One is, at the individual level, on average, how much transmission is being created? Say someone shares something: How many additional people shared it on average? The other thing that’s important is the timescale. How long does it take for that transmission to happen?

For a lot of biological infections, this occurs over days, maybe a couple of weeks. But online, it can be a matter of seconds. But one study of the most popular content on Facebook found that on average, each person would lead to only about two additional shares. So it’s the equivalent of a reproduction number of two. Each case causes about two more cases, on average. The timescale is obviously much faster. Things took off and went viral very quickly. But at the individual level, the average extent of transmission, this popular content on Facebook has a transmissibility very similar to Covid.

WIRED: Should that make misinformation any easier to control?

AK: It definitely influences how you go about trying to contain it. If the reason for your outbreak shape is speed, then reactive strategies are going to be very hard to pull off. So actually trying to trace down bits of content and reactively remove things or stop people from sharing them is going to be very difficult on that kind of timescale.

But another feature of online contagion is that a lot of the transmission is concentrated in a fairly small number of individuals. A lot of things that are posted aren’t shared by anyone. And the things that do take off are often amplified by a small number of individuals. So understanding that dynamic of amplification is important, as is finding preemptive ways to try to reduce susceptibility or to change your network structure to perhaps make some of that less likely. For example, WhatsApp in recent years has changed how much things can be shared with large numbers of people to reduce the spread of misinformation. So I think it’s those structural changes, or those preemptive attempts to reduce susceptibility to this kind of messaging, which probably can be more effective than trying to frantically chase down an outbreak that’s occurring on a timescale of seconds.

WIRED: I’m curious what you make of the past few weeks, where it does seem like there’s been a more coordinated—maybe that’s too strong a word—effort on the part of social media companies to remove certain accounts or venues where those ideas incubate.

AK: I think it goes back to the financial issue, in that we look for individual platforms, rather than necessarily looking at the dynamics across them. There’s been nice investigations, for example, tracing political slogans and their origins across various social media platforms. Often what happens is something will perhaps start on Twitter. It might get onto Reddit and then mutate and be amplified, go on to Facebook, back onto Twitter, into the media, and then a series of steps across platforms and amplifications. And of course, if you can’t see any one of those steps, then it will look like the outbreaks just come from nowhere.

So I think we need to understand more about this ecosystem of things going across platforms, being amplified in different directions. Because a lot of groups who do try and get these things seeded—and there are a lot of subcultures online that try and get these messages to rise to prominence—can complete that system quite effectively and can actually get messages into the view of prominent people on social media, or get them into the view of journalists and media outlets. And then a member of the public will see that message from a source they recognize and won’t necessarily realize the origins of that.

I think that reflects how we’d go about targeting a contagion, whether it’s a biological one, whether it’s malware, whether it’s financial. A really coordinated effort to make sure you don’t end up with unseen routes of transmission can make a big difference. There have been studies looking at some of these different approaches to controlling malicious information online and suggesting that actually single-platform approaches may have little effect, or even a negative effect, unless you have coordination across platforms, because of how information can spread.

WIRED: How is that unseen transmission factoring into Covid-19?

AK: The really notable feature with this outbreak is the delays before you see the severe impact. So often in the early stages—you saw in Europe, for example, a lot of skiers getting ill, mild infections, younger groups not seeing the severe impact. And then it spreads and spreads, and it starts getting to the worst group, you start to see the hospitalizations. And by the time you get to the point where your hospitalizations and deaths are rising, you’ve already got that month of infections that have occurred. And those are still going to appear as a future impact. So it really suggests that if you look at the situation and it seems like a few mild infections in younger groups, that may well be the point to to think about what’s going to come further down the line, rather than waiting for a situation where you’re almost overwhelmed.

WIRED: That’s a rather grim picture, especially looking at a lot of states here in the US right now.

AK: Unfortunately, I think it’s a similar situation in many areas of Europe and other countries. We’re going to have to find a way to keep this burden of infection down. We’ve seen what can happen. It can overwhelm health systems. But unfortunately, there’s no way of having very light control measures and everything back to normal without very quickly getting these flare-ups and new outbreaks.

WIRED: What else are you thinking about as countries respond to the pandemic?

AK: One thing that has been noticeable in terms of different countries’ responses is their use of data and their balance of privacy. I mean, certainly a lot of countries in Asia, because of SARS and other outbreaks, have the ability to trace contacts in more detail and access location data and use GPS to enforce quarantine. And it has been noticeable that countries in Europe and the US haven’t been using data in that kind of way, even though a lot of those places have tech companies collecting arguably similar levels of intensive personal data. But they’re using it for other purposes.

That’s something we should be focusing on more. There are data sets out there that could be very valuable in helping us control this. I think, as a society, we’re going to have to navigate the privacy trade-offs and maybe accept that, if we want to bring this under control, we need certain sorts of data. This is a question that’s been debated a lot in our field: How do you get that data in an ethical way, in a way that balances privacy and disease control?

WIRED: What’s an example of a data set that those companies might either have access to now or could easily collect and would be particularly useful for combating the pandemic?

AK: I’m not at this stage suggesting that companies should be handing over data—but more that in our lives we have the infrastructure and we have that data bouncing around. One feature of a lot of these outbreaks is how important location can be. And a lot of tracing efforts in places like Japan and Korea focus very much at the cluster level. So it’s: What was the nightclub? What was the workplace? Obviously, having information on location and where people have been and where people visited is very useful in that respect and can speed up tracing and an ability to get people to go and get tested. Certainly also countries like Thailand and New Zealand have floated getting people to check into venues with QR codes and that sort of thing.

WIRED: Do you feel like the countries that have been more successful in using data feel like they’ve been able to crack that privacy balance?

AK: I guess there’s different levels of cultural acceptance and—just from a government point of view—what can and has been implemented. There have been situations where there’s been a bit of tension, for example, in Korea. One of the flare-ups was in an area with LGBT bars, and that made some people reluctant to come forward if testing wasn’t anonymized, because they didn’t want to be linked, because of stigma, to those kinds of places. And so they responded to that by making testing anonymous, so it could encourage people to come forward. So there is that trade-off, that if you’re giving up location information, that’s actually quite personal data about where you’ve been. But I think it is a balance that we’re going to have to deal with, because if the alternative is we have widespread lockdowns, then that’s very disruptive to a large city in a different way.


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