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Published onApr 09, 2020

It starts with a grain of rice on a chessboard. This is Grain Zero. The craftsperson makes an offer to the monarch. “I have made this beautiful chessboard and I will give it to you for some more rice. I have placed a grain on the first square. I want you to add grains to each of the remaining squares in turn. Two on the next one. Four on the one after that and so on until all 64 squares have been covered.” The monarch feels they can spot a good deal and so accepts the offer.1

Suffice it to say, it was not a good deal and accepting it would surely bankrupt the monarch’s land. The reason why it is bad is that it is very clear what is going on and only a lack of willingness to do the math would allow you to think otherwise. Put simply, the total amount of rice being asked for was not some mystery. It was the solution to this equation:

1 + 2 + 4 + … + 9,233,372,036,854,775,808 = 18,446,744,073,709,551,615

That, it turns out, is a lot of rice. If you laid the grains end to end you would go from the Earth to Alpha Centauri and back twice.2 Ultimately, there isn’t enough rice in the world, let alone the land, to pay out the contract. I’m no lawyer so I have no idea what the outcome of this all would really have been had it ended up in the courts.

Obviously, this fable isn’t about contract law, it is about our ability to use mathematics to understand the world around us. If you base your decisions on what you can see with little effort, then you might miss the underlying processes at work. Alternatively, if you understand the underlying processes and see them to their ultimate conclusion, you will make a better decision. Those conclusions may be surprising but, paradoxically, they are predictable.

The COVID-19 pandemic came as a predictable surprise to most people. While the mathematics are not as clear as the rice and the chessboard, they were present and the same disconnect between what you could see immediately and what the math told you about where this was heading was there. The tough challenge was how to make some very costly decisions based on the mathematics alone.

The Inevitability of Pandemics

Pandemics have some of the mathematical properties of the rice on a chessboard but also some important differences. The main similarity is that it has to start somewhere. The SAR-CoV-2 or novel coronavirus that causes the disease COVID-19 infected one person initially. That person then housed the virus as it spread throughout their body and then, most probably by leaving it on surfaces, transmitted the virus to others. This can seem like a fluke. However, when you realize there are millions or billions of viruses out there, it was just a matter of time. One of them is going to stick and spread.

The mathematical key here is to realize that we care about how likely one or more of those viruses will become a problem. For any given virus that might be out there, there is a low – perhaps one in a million chance – of it becoming a problem. That sounds comforting until you realize there are a billion such viruses. So, yes, you are rolling a million-sided dice but you are rolling it a billion times and you are hoping never to “win.” The probability that one of those rolls will come up the wrong way is so hard to calculate that it is easier to calculate the probability that there won’t be a problem (i.e., you’ll lose a billion times) and subtracting it from 1:

1 - (1 - 1/1,000,000)1,000,000,000 = 0.9999999999999999999999999 ….

This is the probability a virus will become a problem. It isn’t 1 but it is very close to 1. If there are only a million viruses we still get a 63 percent chance one of them is going to be a problem. The point is that it is inevitable; so inevitable that one would be forgiven if you never wanted to go near another person again.

But we do. And if we do nothing, then, at some point, a pandemic catastrophe will happen. Now it has and, to anticipate a later chapter, the probability that another problematic virus emerges in the future remains close to inevitable.

Our approach to viruses continues to be to accept the inevitable and hope to mitigate and adapt when the time comes. But that strategy relies critically on our ability to accept the mathematics and act quickly. That means we need to know what is going on as early as possible.

In this, I am reminded of this scene from the Cold War movie class, Doctor Strangelove. In it, the Soviets have described a Doomsday Machine that will be triggered should they be subject to a nuclear attack by the US. Strangelove, modelled loosely on the game theorist and mathematician Jon von Neumann, remarks on how “essential” it is to deterrence as no one would attack the Soviet Union if they knew it would end up destroying the world and them with it. However, he then exclaims: “but the … whole point of the doomsday machine … is lost … if you keep it a secret! Why didn’t you tell the world, eh?” Suffice it to say, that lack of common knowledge ended up [SPOILER ALERT] destroying the world.

If we are going to act as if viruses are not a concern most of the time, we have to be able to recognize when they become a concern. Secrets or a lack of knowledge can push us away from sensible behavior. In other words, we need to know and then realize the implications when Grain Zero is placed on the chessboard.

The Degree of the Problem

Pandemics are better than a rice/chessboard process in a very important way: once the first grain of rice is placed, there are ways to stop the process before Square 64 is reached. The key, therefore, to any mitigation strategy that modifies the mathematics of that process is a willingness to do it.

Before getting to that, it is worthwhile to review that mathematics. When a person contracts an infectious virus, they can pass it to others by contact. This isn’t true of all viruses nor of all infectious diseases but, at the time of writing, this is the most plausible infection path for the novel coronavirus. Sometime in December 2019, someone, most likely in Wuhan, China, contracted the virus and began passing it on to others. The question was: how many others? Not specifically for that person but for any random person who might carry the virus.

In epidemiology this has a number, R0 (the basic reproduction number). R0 is the expected number of people on infectious person is likely to infect with a particular virus at the outset.3 In the past, with enough knowledge, R0 for other viruses or infectious diseases could be measured. Absent any interventions, the critical threshold number is 1. If each infected person infects at most one other person, then the total number of infections might rise initially but it will progress very slowly and, because eventually you are meeting more and more people who have had the virus and are, hopefully, immune, the infection rate will die off fairly quickly. For an R0 > 1, an epidemic is possible with a much higher share of the population likely to become infected. This is why the number one goal in pandemic management is to create conditions so that the basic reproduction number is moved to less than 1.

The most infectious disease in modern times was measles with an R0 between 12 to 18.4 This is because it could spread in the air. The usual influenza we experience each year is between 0.9 and 2.1. Some years are good while others are bad. The SARS outbreak was between 2 and 5 while Ebola which came from direct bodily fluids was between 1.5 and 2.5. You can see both significant variation but also significant ranges of uncertainty. For Ebola, this was likely related to population density. At the time of writing, COVID-19 has an estimated R0 between 1.4 and 3.9. It is for this reason that many predicted that, left unchecked, 70 percent of all people would eventually contract the virus.

The Human Equation

The interesting thing about R0 is that it is not just a biological number — that is, related to how a virus can move and bind itself to others — but also a social number.5 If a hermit gets the measles, then R0 is zero. If a party goer gets it, R0 is much higher. The estimates of R0 are averages which is a guide to decision-making but not what you want to know. In principle, you want to know everyone’s specific R0 and you likely want to draw your attention to reducing the R0s of those who are at the top of this list.

Rather than individual-R0s, the best we can hope for are group-R0s. For instance, children move about, keep personal hygiene, etc., in a very different way than other beings. As any parent with young kids knows, there are years in which your house turns into the town from Albert Camus, The Plague, sans any widespread epidemic. This is why, in many countries, the first step in social distancing was to shut down schools. This wasn’t because children were especially at risk — they weren’t, thank goodness — but because they were ‘vectors’ — an identifiable group known to have potentially high R0s. The same was true of college students. If most students stayed in dorms then they were more isolated from the outside community than those who commuted from home. The latter were likely to be strong vector for infection because they spent their days going from numerous gatherings of one hundred people or more before bringing it all back to others. By contrast, office workplaces are potentially lower risks although once people attend conferences that becomes another matter.

The epidemiological models consider who might interact with whom when they try and predict the spread of an infection, but those assumptions are ‘hard-wired’ into their models. Economists (and other social scientists) typically shy away from predictions based on such hard-wired behavior. Instead, when considering how people might interact with one another, they look to their choices. People do not blindly react to pandemics and continue to go about their daily business. Nor do they hide out for the duration. What they do is balance the risk of interactions as the pandemic progresses, based on information they have at hand. In other words, what epidemiological models can miss is that humans change their behavior over time, and this can impact on the mathematics of the infection.

The research that integrates economics into epidemiology is very much nascent. However, from the work that has been done to date, some important insights can be drawn. First of all, we can expect that when people are concerned about the costs of being infected, they won’t necessarily need to be told to socially distance themselves from others.6 In particular, as the infection rate starts to climb, more people will reduce their economic activity which has the effect of moderating the spread of any virus. During the 2009 H1N1 epidemic, people in the US reduced their time spent amongst others7 and similarly in Mexico although there the behavior differed amongst different socio-economic groups with poorer groups adjusted less.8

Second, it is possible that the behavioral response to a pandemic can cause the peak infection level to be lower than what might otherwise emerge from a standard epidemiological model.9 This is because, as the infection rate increases, people will perceive greater risk from interacting with others. While that reduces the infection rate, this back and forth will slice the top off the peak but spread the length of the pandemic further; that is, it will ‘flatten the curve’ (something that will be discussed in more detail in Chapter 3).

This has another important implication that can test our usual epidemic intuition. If a virus is more virulent (that is, can be more easily passed between people), the usual prediction is that there a larger share of the population will become infected (as R0 is relatively high). However, once the human element is taken into account, this could go the other way. If it was known that a virus was particularly virulent, people would fear going out and would socially distance. The more virulent it is, the more people will self-isolate to avoid others. This could well mean that virulent outbreaks have a lower total number infected than less virulent ones. This is, of course, just a theoretical possibility at this stage but there was anecdotal evidence in the COVID-19 outbreak that certain groups – particularly, younger people who had less to fear from the consequences of being infected – did not practice social distancing as much as others.10

While people might reduce their social interactions out of fear, it is important to emphasize that this may still be too little relative to what we might all agree would be in the collective interest. That is because people take into account their own fear in refraining from social interactions but not the impact those actions might have on others. In other words, fear is not necessarily enough and governments may have to take heavy-handed actions to influence R0.

The good news is that policy actions designed to change the behavior of many can have an impact. This was starkly demonstrated in a comparative study of the Philadelphia and St Louis responses to the Flu Pandemic of 1918.11 As Figure 2-1 (drawn from that paper) demonstrates, St Louis had a milder and prolonged epidemic compared with Philadelphia which had the majority of cases in just one month. The difference between the two was that Philadelphia held a parade of returning soldiers from World War I while St Louis, armed with the same health warnings, closed schools and even churches and banned gatherings of more than 20 people. As network economist, Matthew Jackson notes, being able to reduce the number of highly connected clusters within a network of social relationships can dramatically reduce R0.12

Figure 2-1: Pandemic of 1918. Source:

While we understand the general science behind disease transmission, the mix of biological and social factors for each new disease means that we have broad ranges for R0 and scant details about what any particular measure might do to the spread of the virus. That said, we know that if we shut everything down, then we can minimize any given R0. To be sure, in doing so, we maximize the R0 within a given household, but the idea is to keep the spread between households at a minimum. How much we want to do this depends both on the degree of the problem — how high R0 would otherwise be — and also on the costs of becoming infected versus the costs associated with trying to reduce R0.

Willingness to Act

This leads us to the costs. The potential health costs of COVID-19 are of primary interest. As I apply my economist filter to what I understand of the biomedical properties here, I see those health costs in four buckets. The first are the people who contract the virus but have no important symptoms. They create no health costs at all. The second are people who contract the virus and have symptoms akin to a severe flu. The health costs here are primarily in terms of lost ability to work and function. The third are those who have severe enough symptoms to require hospitalization with the obvious associated costs. The final category is those for whom COVID-19 proves to be fatal. Early estimates from China suggested that 81 percent of those who tested positive for COVID-19 were in the first two categories. Of the remainder, 14 percent were severe, and 5 percent were critical. The remaining 2.3 percent had died.13

The problem we face is that the mix of people in the third and fourth category potentially depends critically on the ability of the health care system to manage their infections and resulting consequences. It is economizing on this dimension that is the focus of policymakers in minimizing the health costs associated with COVID-19.

There are two ways to achieve this. The first would be to ensure there was sufficient capacity in the health care system to handle cases when they are at their most intense. That will be the subject of Chapter 3. The second is to reduce the intensity of critical COVID-19 cases at any point in time. In other words, that means taking actions to reduce R0.

Let’s consider ways of reducing R0 in terms of their costs. The least costly ways are good health practices. This includes vigorous hand washing and regular cleaning of surfaces. These are the types of things that occur within hospitals that become of high value during a pandemic. There are also a related set of protocols for the operation of health care facilities themselves so as to protect health care workers. Not surprisingly, these were the first set of things that were enacted in most countries.

The second set of actions was to limit the spread of the virus across national boundaries. The logic here is that, if the virus has not infected significant numbers of a country’s population (and in the case of COVID-19 that would have to be a very small number), then by limiting travel between countries, the virus might be kept out. Some countries, notably Taiwan did this very quickly while most others did it in a somewhat ad hoc way. For instance, the US closed travel to any foreign nationals coming from China but not their own citizens. In March 2020, Israel took the, at the time, unusual step of requiring any person coming in to self-quarantine for two weeks using cell phones to track infractions. At the time of writing, it is safe to say that the ability to contain the spread across national boundaries was limited. Obviously, restricting travel would start to impact negatively on certain industries, especially tourism, hotels and airlines.

The third set of actions came under a catch-all term of ‘social distancing.’ Initially, this involved cancelling large gatherings. In Australia this was 500 people initially while at a similar time Austria banned gatherings of more than five people which might have given pause to households with four or more children. This, however, led to more extreme actions such as canceling school, college classes, instituting work from home practices and eventually closing restaurants and bars. Finally, in some jurisdictions there were orders to ‘shelter in place’ (including China, Italy, and parts of the US).

The first two measures — hygiene and travel restrictions — are disruptive, potentially, very disruptive. However, they pale in comparison to the costs associated with social distancing. To achieve social distancing in a manner that would prevent the health care system from exceeding capacity requires a reduction in economic activity that would plunge any economy into an immediate recession. This is why there is a reduction in economic activity if you choose to hold the line on health. How to handle that is the subject of Chapter 4. Suffice it to say, however you cut it, the costs are significant, perhaps of the order of 10 to 20 percent of GDP of any country. And this is just the economic cost. You are also asking much of the population to remain at home. To be sure, thanks to the Internet, in many places it has never been more comfortable to do this. Nonetheless, it is unknown just how long such social distancing could last.

So herein lies the basic trade-off. We want to reduce R0 as it is very costly to have a high number of sick people at one time. The reason we have to do this is because of the limited health care system capacity. If R0 is too high, health care capacity becomes quickly overwhelmed and doctors have to engage in triage, which in the context of COVID-19, often means choosing who will live and who will die. This outcome has to be balanced against the significant economic cost associated with spreading infections over time. To be effective, social distancing has to go the distance. But with every week or month of low economic activity the costs rise.

If that weren’t a tough enough trade-off, it is actually worse than that. Whether or not people can develop immunity from COVID-19 is still an open scientific question but, let’s assume that is more likely to be true than not. If you reduce R0 too far, initially, then most of the population do not become infected and that means that once you stop policies such as social distancing the virus can emerge once more, and we all have to do this again. It is a reasonable assumption that we only want to intervene once.

The reason there is a cost to this is that you are actually more socially useful if you get the virus and are no longer a carrier for the purposes of infection. That means that other people and society do not have to fear interactions with you. In other words, achieving ‘herd immunity’ is an investment in the future. It is like a vaccine but alas you have to actually get the virus rather than an injection. For an understandably short time, the British response to the pandemic, reflecting this idea, was to embrace the idea of ‘taking one for the team.’ That said, a week in bed is one thing. Dying is another. How you conduct this policy without getting significant people in the latter category is hard to see.

How Will I Know?

Thus, governments face a real quandary: when to institute social distancing and how intensive it should be? The problem is that there is uncertainty. When a virus first appears, we know soon after what its R0 is likely to be. But we don’t know immediately. In a situation like COVID-19 where many infected people are asymptomatic, that information can be even harder to get.

We also know that time can be of the essence. For COVID-19, wait a day to act and you might have 40 percent more cases 21 days later than if you acted immediately.14 As time goes on, that 40 percent becomes a very large number. The more limited your information, the harder it is to act and achieve results. So, for a country where the outbreak commences, this is a very difficult choice. Moreover, given that today’s travel possibilities can lead to transmission out of a country very quickly, placing the onus of that decision on the country of origin may not be enough. In any case, it was over a month before China started to impose travel restrictions in this case.15 In retrospect, the price to be paid by the world was very high. However, what we were asking China to do was pay a price themselves. These types of decisions are rarely pursued optimally. Moreover, for countries who could observe outbreaks elsewhere and failed to act quickly, even in terms of their own self-interest, excuses could run out.

The point of this is to demonstrate just how hard it is to pull the trigger on measures to reduce R0 when an outbreak has just begun. There is uncertainty and, moreover, the costs of actions are felt disproportionately. However, the notion that delaying a day or two will have much in the way of real benefits is a false comfort. If you choose to shut down your economy on Wednesday rather than Tuesday, a day’s work and economic activity is lost. But that is peanuts relative to the costs associated with a shutdown at all. The take-a-way, therefore, is that if you know you are going to shut down the country eventually, there are huge returns to doing it quickly.

One reason to delay is to gather more information. If you will learn by Wednesday that you could safely keep schools open, you might do well to continue to keep them open on Tuesday. What is more, it may be that shutting down early causes you to miss that critical information altogether. Thus, while ‘the drift’ told us that if we do not hold the line on health, we may take away options we could use, it is nonetheless true that we can learn about different ways of containing the virus based on the actions we take.16

Delaying a decision in order to gather more information has a value in economics called the real option value. Suppose you need to consider when to shut down the economy for a month or more. You know that will have potential costs, but those costs are like an investment in terms of the benefits associated with reducing R0. The decision to shut down will be the same on Tuesday versus Wednesday unless you learn something in the interim. Suppose you are predisposed to shut down on Tuesday but there is more information to be accumulated.17 Should you wait?

It turns out the answer depends on the type of information you are expecting to receive. As Ben Bernanke, chair of the Federal Reserve during the 2008 financial crisis, found if you are expecting news that will justify and reinforce the decision you were already predisposed to make on Tuesday, there is no reason to wait.18 That information will not change your mind. Instead, the reason to wait is if you receive news that will convince you not to shut down. The only information that gives you an option value of waiting to pull the trigger is news that would cause you to remove your finger from the trigger entirely.

As I write this, it is hard to imagine the information governments were expecting to receive that would have caused them not to act on some type of social distancing. If there was hope, it was not articulated or in the data. Thus, we are left to speculate. My speculation is that waiting was driven by receiving political news rather than scientific or economic news. Governments may have decided not to shut down if they found that a large proportion of the population would resist those efforts. In many cases, this is why governments implored citizens to engage in social distancing in the hope that they could achieve a reduction in R0 without stronger measures. Those stronger measures included legal requirements to stay at home which could potentially then be enforced with penalties associated with violations.

In summary, it is important to realize that acting decisively is very challenging. It is more challenging depending upon the style of government, the transparency of information and the competence of the decision-makers. In the end, most governments eventually made strong moves to reduce R0 and they did so that, upon reflection, was relatively fast compared to decisions of far less consequence. In retrospect, with situations like this, we may always conclude that governments should act earlier. The question we need to address is what changes we need to make so that it is possible for decisive action to be taken when it needs to be.

Steve Petrov:

It’s interesting to note that decision making was delayed in both China and in the USA for virtually opposite reasons! In China due to repression of information and local communist bureaucrats fearful of being blamed messengers and in the US with its free flow of information and debate over the value of various sources of info etc.

Steve Petrov:

Your speculation is actually backed up by a debate within the political science community. They talk about utility versus prospect theory when politicians deal with risk when making decisions. See for example

Indeed, one can discern from as far back as the Federalist Papers that politicians have a distinctly different type of rationality than economists.

Steve Petrov:

Interestingly, the Korean Office case study suggests that offices are especially high risk but also that duration of close proximity is more important than proximity itself.

“Despite considerable interaction between workers on different floors of building X in the elevators and lobby, spread of COVID-19 was limited almost exclusively to the 11th floor, which indicates that the duration of interaction (or contact) was likely the main facilitator for further spreading,” the Korean CDC wrote in its report.”

Steve Petrov:

The efficient market theory says that we should learn quickly. But China has spent vast resources to stop information from moving freely. Academics will hate this but The US President does have a point re China.

Steve Petrov:

Probably a dismissal of the complaint given Sovereign or Crown immunity. And, given that this is a fable set in olde fashioned times, a beheading for being a smart-ass by exposing the King’s Innumeracy.

Benson Ng:

This is a very good point indeed! Like an overprotective parent mindset, we cannot ensure their safety forever. Eventually, the citizens will have to resume normal lifestyle and we can’t prevent them from infection of the world. For a country like Singapore which have no natural resources and GDP depends greatly on our geographical advantage and services, it is a hard decision for the government to make. While some are fast to judge that the Singapore government are slow in their ”lockdown”. I think they are actually doing to good job of controlling the spread while maintaining our healthcare standard. After all, a “controlled level of immunity” for our citizens will make more economical sense and advantage than to become overprotective.

Tom Davidson:

Why do you say “probably” via surface contamination rather than via respiratory droplets?

WHO says: “According to current evidence, COVID-19 virus is primarily transmitted between people through respiratory droplets and contact routes.”

Michael Jones:

I think more discussion would be helpful on the properties of R0. It seems that R0 is not just a function of the virus itself, but of the social nature of the people infected as well.

In a country with significant levels of physical touching like countries in Europe, the R0 would be higher than say the US. But it’s the same virus.

Michael Jones:

I think you meant to write “and this can have an impact…”

Eric Rasmusen:

You dismiss that idea too quickly. The way to conduct the policy would be to spend 10% of GDP you’d lose by lockdown for a long time on protecting the most vulnerable people for a short time till the epidemic burns out— test all the nursing home employees, deliver food to retirement communities via special sterilized means, etc.

Steve Petrov:

Excellent point Eric. I added several comments in Chapter one along these lines but I talked about economic surpluses and subsidies and their comparative efficiencies. Your metric of using a % of GDP is simple and elegant.