Ah, shun the horrid gulf! by Scylla fly. ’Tis better six to lose, than all to die.’
Homer, The Odyssey (trans.: Alexander Pope)
Thus Circe warns Odysseus that it is better to lose only a few of his sailors to Scylla's six horrid necks and six terrific heads than to risk losing his entire ship in the dire thunders and boiling seas of Charybdis' roaring whirlpools.
This article will use a simple mathematical model to argue that in navigating between the economic Scylla of a strict extended lockdown and the calamitous Charybdis of a health service collapse, governments will be hard pressed to avoid both.
As I have argued elsewhere, these models are too simple to use for policy decisions. In particular, tracking and contact tracing may give us a way to navigate these direst of straits. Nonetheless, the simple model provides a useful frame for the discussion and helps to explain and illuminate much of the policy argumentation we read in the press.
What are you saying, SIR?
The Susceptible - Infected - Recovered (SIR) model is the simplest model we have that captures the basic dynamics of the spread of contagion. I have described it in detail in two previous blogs, one completely without equations, and another with just a couple. This blog is also without equations, but I will provide an appendix a derivation of key numbers presented here.
In the SIR model, the rate at which the number of people infected with COVID-19 grows is proportional to the number of people already infected. The more there are infected, the more there are to infect. The "constant" of proportionality is the rate at which the infection spreads multiplied by the proportion of the population not yet infected less the rate at which people recover.
To start with, the proportion of the population not yet infected is most of it, so the growth rate is just the difference between the rate of infection and the rate of recovery and death. As more and more people become infected, the proportion of the population not yet infected falls and the "constant" of proportionality (which isn't constant at all) also falls until it eventually gets to zero. At this point people are recovering (or dying) faster than they're getting sick and the number of infected people falls.
Peak infections and final immunity
There are two critical numbers we would like to estimate from our model. The number of infected at the peak and the number of people yet to be infected when it is all over.
The number of infected at the peak determines the maximum burden on the health service. A given proportion of infected people will be symptomatic, of them some will be severely ill and require hospital treatment and of those, some will need intensive care. We can work back through those ratios from the number of places in intensive care to work out how large the peak can be before the health service is overwhelmed.
In Denmark, for example, we have around 1200 intensive care places. If 5% of symptomatic cases require intensive care (as was the case in Wuhan) then we can cope with 24,000 symptomatic cases. If 50% of people infected develop symptoms then that's roughly 50,000 we can manage at the peak, around 1% of the population.
When this first wave is done, we would like to start alleviating measures in place to reduce the rate of infection. The more people that have been removed from the pool of susceptible people yet to be infected, the smaller the proportion of those to the total population and, critically, the greater the scope we have for scaling back measures to reduce the rate of infection.
Depending what the uncontrolled rate is - and it seems to vary somewhat from country to country depending on population density and community contact behaviours - if the immunity (a euphemism for those who have been through anything from an asymptomatic free ride to gruesome illness to death) is substantially greater than 50%, then we may be able to manage without measures altogether (See section below on R0 and the appendix).
The total population exposed to the virus through the entirety of the contagion is much higher than the proportion at the peak. There are all the people who have already recovered when we get to the peak as well as all those yet to be infected. It turns out that the SIR model allows you to calculate this proportion. If the number of infected people peaks at 1% then around 25% of the population will have been exposed when the contagion has finally passed. See the graph below
Tweaking the tiller of R0
The SIR model is seductively simple. It turns out that all of this: the height of the peak, the extent of the infection in the population is controlled by a single parameter called the Reproduction number, which sometimes written R0. It's essentially the number of people infected people will, on average, go on in turn to infect. (Note it doesn't say anything about how long it takes for them to do it - that's encoded in the other parameter in the two-parameter SIR model). R0 in the model is calculated by dividing the average rate at which the infection spreads by the average rate at which people recover (or die) or, equivalently, multiplying it by the average length of time they're infectious.
All the measures we have seen around the world - lockdowns, institutional closures, outlawing of large gatherings and so on - these are all trying to reduce R0. If you can get R0 down under 1 then the contagion collapses: the rate at which people are infected is less than the rate at which they recover (or die) and stop infecting.
The height of the peak (red curve above) is determined by R0, but so is the proportion of the population removed (immune - grey curve above) at the end of the epidemic. And here's the rub. Small peak, low immunity; large peak, high immunity. Governments must steer between an R0 that is too high, leading to overburdened health systems and too low, leading to low immunity and long lockdowns.
They must do these through measures whose efficacy - to put it mildly - abound with uncertainty. The figure above is from the Imperial College COVID response teams article on the efficacy of measures. The bars show a confidence interval. It's not an exact science.