How do you control an epidemic? In that context, let me discuss the SIR model of disease and demonstrate it in action. It’s the one that gives rise to those curves we’ve seen so many of, the ones that we’re trying to flatten.
We classify people into three categories: 1) susceptible (S) haven’t got the disease yet, so they’re vulnerable to it; 2) infected (I) have got it, so they might even give it to others; 3) recovered (R) either got over it or died, and presumably won’t get it again. The mathematical variables are S = fraction of the population which is susceptible, I = fraction infected, and R = fraction recovered, hence the name “SIR model.”
They change in two ways. First, new people get infected. This depends on S, how many potential victims are available to infect, on I, how many are available to infect them, and a constant factor that I’ll denote by the letter a. If you’re infected, it’s how many people you infect per day (on average) until you recover, if everyone is available to be infected. Second, infected people either recover or die, which depends on I, how many are infected, and a constant factor I’ll denote by the letter b.
Those constants, a and b, are the control knobs on the spread of a disease. Sometimes folks use other (but equivalent) variables, but the model is the same and its results unchanged. Particularly interesting is their ratio . When is less than 1, b is bigger than a so the infected recover faster than new cases arise, the disease can’t spread fast enough to sustain itself, and the whole thing fizzles out.
If I set the constants to values a=0.5, b=0.1, begin with only 1 infected person per million population, and let the model run for a 120 days, then I get the graph (of infected fraction I) shown above.
At first, progress of the desease is exponential, dominated by new cases, with the difference between the two control constants determining the speed of its exponential growth. But the available pool of victims S shrinks, and eventually there are so few victims to be found that recovery dominates and it turns to exponential decline, with the constant b determining its exponential rate.
We can make this more plain by graphing the same thing, but on a logarithmic plot:
The first phase, exponential growth, shows as straight-line growth in the logarithmic plot, while phase two, exponential decline, shows as straight-line decline.
In the first graph we see that at one time, about 50% of the population is infected. Eventually nearly everyone (over 99.5%) gets the disease, so if 1% of infected die, that’s about 3.4 million dead Americans. The whole thing is over in a few months, but with half the population sick at one time, it will overwhelm the health care system.
In order to change course, we must change the growth constant a or the recovery constant b or both. There’s not much we can do about the recovery constant, but we can start in on the growth constant by noticing that it has (at least) two important factors. We can say that , or to put it into words, the growth constant is the probability p that you infect someone you come into contact with, times the number c of contacts per day. And that tells us how to reduce the spread rate:
1: Reduce the probability of infection for each contact. Wear a mask, keep your distance (2m or 6ft).
2: Reduce the number of contacts. Close non-essential businesses, prohibit large gatherings.
When we do that, we decrease the spread rate a. It’s a fact. If we stop doing it, the spread rate a will increase. That too is a fact.
Let’s run the model again, but this time we’ll change the control settings in mid-stream. After 25 days, we’ll lower the spread rate a, dramatically in fact. Then we let things continue. What’s notable is that the new value of a is less than b, so the disease can’t grow, it starts to die out.
We won’t see that right away. That’s because we don’t really know the number I of infected. We often see graphs of “total cases,” but that’s the infected plus the recovered. What we see more often is new cases per day.
And we don’t even see that; we see new detected cases per day. Cases don’t get detected right away; for most people they are never detected at all, and those we do find are often those very sick, long after they contract the disease. I computed the number of new cases per day, then applied a Gaussian smooth to simulate spreading the detections out over a considerable time span. Here’s the result:
This is what it looks like if you get hit hard by the disease at the outset, then put strong measures in place to reduce the epidemic. They work. Like they did in New York state:
That’s how you control an epidemic.
Let’s do another simulation. At the start, we’re hit hard but not as hard. Then we institute control procedures, but not as tight as they did in New York. As a result, we keep the new-case level under control, but we barely get it to decline.
After a couple of months of that, we loosen controls. We have to save the economy! Let’s go to the beach! Disney World!! I need a haircut!!! It’s not as loose as it was when we did nothing — but it’s no longer tight enough to keep the lid on. Here’s the result, compared to what happened in the state of Florida:
That’s how you lose control of an epidemic.
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