“Social distancing” – reducing the number of daily close contacts individuals have – is being encouraged by policymakers and epidemiologists. Why it works, and why now rather than for other diseases, is often left unstated. Economists have two important contributions here. First, game theoretic models of behavior are great for thinking through where government mandates are needed and where they aren’t. Second, economists are used to thinking through tradeoffs, such as the relative cost and benefit of shutting down schools versus the economic consequences of doing so. The most straightforward epidemiological model of infection – the SIR model dating back to the 1920s – is actually quite commonly used in economic models of innovation or information diffusion, so it is one we are often quite familiar with. Let’s walk through the simple economics of epidemic policy.
We’ll start with three assumptions. First, an infected person will infect B other people before recovering if we make no social changes and no one is immune. Second, people who have recovered from coronavirus do not get sick again (which appears to be roughly true). Third, coronavirus patients tend to be infectious before they show up in a hospital. We will relax these assumptions shortly. Finally, we will let d represent the amount of social distancing. If d=1, we are all just living our normal lives. If d=0, we are completely isolated in bubbles and no infections transmit. The cost of distancing level d is c(d), where distancing grows ever more costly the more you do it – for the mathematically inclined, c(1-d)>0, c'(1-d)>0, c(1)=0.
In the classic SIR model, people are either susceptible, infected, or recovered (or “removed” depending on the author). Let S, I, and R be the fraction of the population in each bubble at any given time t. Only group S can be infected by someone new. When an infected person encounters someone, they pass the disease along with probability dB, where d is distancing level and B is the infection rate. If an infected person interacts with another infected or recovered person, they do not get sick, or at least not more sick.
Define one unit of time as the period someone is infectious. We then can define how the proportion of people in each group change over time as
dS/dt=-dBSI
dI/dt=dBSI-I
dR/dt=I
The second equation, for instance, says the proportion of population that is infected is the infection rate given distancing dB times the number of possible interactions between infected and susceptible people SI, minus the number of people infected in the current period I (remember, we define one unit of time as the period in which you are sick after being infected, so today’s sick are tomorrow’s recovered). If dI/dt>0, then the number of infected people is growing. When the infection is very young, almost everyone is in group S (hence S equals something close to 1) and no distancing is happening (so d=1), so the epidemic spreads if (dBS-1)I>0, and therefore when B>1. Intuitively, if 1 sick person infects on average more 1 person, the epidemic grows. To stop the epidemic, we need to slow that transmission so that dI/dt<0. The B in this model is the "R0" you may see in press, incidentally. With coronavirus, B is something like 2.
How can we end the epidemic, then? Two ways. First, the epidemic dies out because of “herd immunity”. This means that the number of people in bin R, those already infected, gets high enough that infected people interact with too few susceptible people to keep the disease going. With B=2 and no distancing (d=1), we would need dI/dt=(dBS-1)I=2S-1<0, or S<.5. In that case, the number of people eventually infected is half of society before the epidemic stops growing, then smaller numbers continue to get infected until the disease peters out. Claims that coronavirus will infect “70%” of society are based on this math – it will not happen because people would pursue serious distancing policies well before we reached anywhere near this point.
The alternative is distancing. I use distancing to mean any policy that reduces infectiousness – quarantine, avoiding large groups, washing your hands, etc. The math is simple. Again, let B=2. To stop coronavirus before large numbers (say, before 10% of society) are infected requires (dBS-1)I=(2dS-1)I<0. For S roughly equal to 1, we therefore need d<1/2. That is, we need to reduce the average number of infections a sick person gives to others at least in half. Frequent handwashing might reduce infection by 20% or so, though with huge error bars. That is, to stop the epidemic, we need fairly costly interventions like cancelling large events and work-from-home policies. Note that traditional influenza maybe has B=1.2, so small behavior changes like staying home when sick and less indoor interaction in the summer may be enough to stop epidemic spread. This is not true of coronavirus, as far as we know.
Ok, let’s turn to the economics. We have two questions here: what will individuals do in an epidemic, and what should society compel them to do? The first question involves looking for the d(s,t,z) chosen in a symmetric subgame perfect equilibrium, where s is your state (sick or not), z is society’s state (what fraction of people are sick), and t is time. The externality here is clear: Individuals care about not being sick themselves, but less about how their behavior affects the spread of disease to others. That is, epidemic prevention is a classic negative externality problem! There are two ways to solve these problems, as Weitzman taught us: Prices or Quantities. Prices means taxing behavior that spreads disease. In the coronavirus context, it might be something like “you can go to the ballgame, conditional on paying a tax of $x which is enough to limit attendance”. Quantities means limiting that behavior directly. As you might imagine, taxation is quite difficult to implement in this context, hence quantity limits (you can’t have events with more than N people) are more common.
It turns out that solving for the equilibrium in the SIR epidemic game is not easy and generally has to be done numerically. But we can say some things. Let m be the marginal cost of distancing in the limit, normalized to the cost of being infected (m=c'(d)/C for d=0, where C is the cost of being infected). If distancing is not very efficient (m is low) or if transmission is hard (B is not much more than 1), then in equilibrium no one will distance themselves, and the epidemic will spread. People will also not distance themselves until the epidemic has already spread quite a bit – the cost of social distancing needs to be paid even though the benefit in terms of not being infected is quite low, and you can always “hide from other people” later on.
Where, then, is mandated social distancing useful? If individuals do not account for the externality of their social distancing enough, they will avoid some contact to prevent getting sick right away, but not enough to prevent the epidemic from continuing to spread. If the epidemic is super dangerous (very high costs of sitting sick a la ebola or B very high), in equilibrium individuals will distance without being forced to. If the cost of being sick is high relative to the economic and social disruption of distancing, it is better even from a social planner’s point to view to just to risk getting sick. We don’t attempt to prevent the common cold with anything more extreme than covering our mouths.
However, if B is not too high, and the cost of being sick in high relative to the cost of distancing but not too high, it can be optimal for the government to impose social distancing. In the case of coronavirus, we need d<1/2. Since some people cannot reduce contact like doctors, the rest of us need to reduce the number of close contacts we interact with every day by even more than half. That is a bigger reduction than the workday versus Sunday number of interactions for the average person!
This model can be extended in a few useful ways. Three are particularly relevant: 1) what if we know who is sick before they are contagious, 2) what if people have different costs of being sick, and 3) what if the network of contacts is more complex than “any given person is likely to run into any other”.
If we observe people before they are sick, then quarantine can reduce d. Imagine a disease where everyone who gets sick turns blue, then they can only infect you two days later. Surely we can all see the very low cost method of preventing an epidemic – lock up the blue people! Ebola and leprosy are not far off from this case, and SARS also had the nice property that people are sick well before they are infectious. It seems coronavirus is quite infectious even when you only feel mildly ill, so pure testing plus quarantine is unlikely to move d – the distancing parameter – enough to reduce the number of infections caused by each sick person enough. This is especially true once the number of infected to too large to trace all of their contacts and test before they become infectious themselves.
If people have different costs of being sick, the case for government mandates is stronger. Let young people be only mildly sick, and old people much more so, even as each group is equally contagious. In equilibrium, the young will take only minor distancing precautions, and old major ones. Since the cost of distancing is convex, it is not efficient nor an equilibrium for the old to pursue extreme distancing while the young do relatively little. This convexity should increase the set of parameters where government mandated distancing is needed. As far as I am aware, there is not a good published model explicitly showing this in an SIR differential game, however (economists trapped home this weekend – let’s work this out and get it on ArXiv!).
Finally, the case of more “realistic” networks is interesting. In the real world, social contacts tend to have a “small world” property – we are tightly connected with a group of people who all know each other, not randomly connected to strangers. High clustering reduces the rate of early diffusion (see, e.g., this review) and makes quarantine more effective. For instance, if a wife is infected, the husband can be quarantined, as he is much more likely to be infected than some random person in society. “Brokerage” type contacts which connect two highly clustered groups are also important to separate, since they are the only way that disease spreads from one group to another. This is the justification for travel restrictions during epidemics – however, once most clusters have infected people, the travel restrictions no longer are important.
A Fine Theorem 