Next phase, next stage, next craze, next wave

Let me tell you 'bout a thing, gotta put it to the test

This is one of those “someone is wrong on the Internet” things, but it probably has implications for COVID numbers in the UK over the next month or so. It’s ever so slightly more technical than recent posts, but I promise not to go totally maths crazy.

It’s all prompted by a Twitter post from user JPWeiland who wrote

It'd be wise to consult a modeler first. Every modeler know that on a declining background, a fast growing variant needs to break through 50-60% before it generates a wave.

Now maybe I’m not a modeller, but this is really a question about mathematical functions, so it’s fairly squarely in my wheelhouse.

As usual, it all comes back to questions of exponential growth. But in fact, we don’t really need any fancy mathematical tricks, no logarithms or anything like that. Imagine the following setting: we have a virus which is growing by a constant factor every day¹ . If on Day 1 there were n people infected, then if the daily growth rate was r then on Day 2 there would be n(1+r) of them.

The nice thing about exponential growth (when it happens) is that it’s predictable. If we know r, then we can run this equation out into the future, and see what the effect of a certain period of growth would be, by calculating the right power of (1+r). So knowing r is really useful².

Luckily, it’s clear that, using the data we have, we can directly find r. If we just calculate (Day 2 number)/(Day number) - 1 , then we have n(1+r)/n - 1, which simplifies to give us just r itself, so we can predict the short-term future. (Told you the maths wouldn’t be too bad).

Of course, in practice, things are a bit messier than that. We never know the exact numbers of people infected, because we only ever infer them from reported data like positive tests, hospital admissions and deaths. Since (as I describe in Numbercrunch) all of these numbers are subject to all kinds of sampling effects, including day-of-the-week fluctuations, it’s better not to look simply at ratios of successive days’ figures to estimate r. People have developed a bunch of ad hoc tricks involving things like weekly “case ratios” to get around this, but as far as we’re concerned we can just pretend that we know the daily infection numbers exactly.

However, what I’ve described so far is very much the “spring 2020” version of the calculation. That is, we’ve imagined that we only have a single variant with a single exponential growth rate. Of course, this is very far indeed from the “winter 2023” position, with its bewildering cast of different strains of omicron, each with their own set of mutations and hence their own growth rate.

But in fact things are doable even in that scenario, from a mathematical point of view. If we imagine the situation with two strains, then you’ll see how the calculation will work in general. Suppose that we have a second variant, with a growth rate of s. If on Day 1 there are m people infected with this second strain, then on Day 2 there will be m(1+s), in the same kind of way as before.

So, now we can see what happens overall. On Day 1, there’ll be n+m infections of all types. By Day 2 this has grown to n(1+r) + m(1+s). So if we do our (Day 2 number)/(Day number) - 1 calculation again, we’ll get something interesting. What we’ll see is nothing but n/(n+m) r + m/(n+m) s.

That is, the following Fundamental Principle holds: the overall daily growth rate is the weighted average of the individual growth rates, with the weights being the population proportions of the respective strains.

You can do the same sort of calculation for a sum made up of lots of strains, but exactly the same thing will happen, and the Fundamental Principle will hold. This explains the phenomenon that Thomas House and I wrote about in Plus Magazine, that as a strain becomes more and more prominent, its weighting becomes bigger and so the overall growth rate grows. It’s why we see overall growth rates accelerating at the beginning of a variant-driven wave (as we saw in December 2021 or June 2022 for example), at least until it starts to run out of people to infect and the growth rate drops.

Something interesting is that, while I’ve been acting like r and s are positive numbers, there’s absolutely no reason that needs to be true. In fact, everything I’ve said would go through when r and s are both negative. Our (1+r) factor would be less than 1, so numbers would fall by a constant proportion each day, and we’d observe an exponential decay phase. But more interestingly, it’s also true when (for example) r is negative and s is positive, which is the kind of scenario we are interested in now.

What you can see from our Fundamental Principle is that we would be in overall case growth exactly when the weighted average of the individual growth rates is positive. So, in theory, by tracking the percentages of each strain and estimating their individual growth rate we can tell when we are about to move into overall growth. (Of course, this is tricky because there are lots of strains to consider, but in practice the average tends to be dominated by a few big hitters, so we can focus on those).

But what you’ll notice is that this doesn’t match up with JPWeiland’s statement at the top. There’s nothing about 50-60% here. It’s entirely possible that with say 75% of cases made up of roughly constant strains and 25% of cases made up by a fast enough moving variant, then you’ll move into overall growth. And while the “50-60%” thing may be reasonable as a rule of thumb, it’s worth noticing that (data from the dashboard) the UK spent a lot of the late summer and early autumn in a growth phase despite no individual variant going above a 40% market share.

The relevance of all of this is that, while the UK has been seeing very welcome falls in hospital admissions since the start of October (helped I am sure by the recent booster campaign, targetted to the elderly and vulnerable), that may shortly come to an end.

There is a new variant called JN.1, an offshoot of the BA.2.86 strain that people got prematurely excited about in late August. While I was urging against panic then, it seems likely that JN.1 has a big enough growth rate and has now reached a sufficient market share that the kind of calculations I’ve described suggest we might shortly return to overall growth.

This is not to say that disaster, lockdown, human sacrifice, dogs and cats living together, mass hysteria is in the offing. While the estimated growth rate is probably higher than anything we’ve seen in the UK for a while, it’s not as high as the BA.5 strain which caused the last really big wave of admissions (in summer 2022). And while there’s likely to be a fair amount of mixing in the lead up to Christmas, as we saw a year ago that tends to drop off in the New Year, so JN.1 might well find it harder to spread then. Since admissions are currently at a pretty low level by December standards, any wave has a fair amount of work to do to even reach the admissions levels we saw last winter.

So, as usual I’d say that if you have the offer of a booster then now is a good time to take it. And, as much as anything, this post is just to explain the general principles of growth rather than to sound an enormous alarm bell. But at the very least if we do return to growth and you see the usual suspects harvesting retweets with lines about “nobody could have predicted this”, then perhaps you can remember that these phenomena are indeed somewhat predictable, at least with sufficient data and maths.

1. Yes, I know this doesn’t happen indefinitely. You don’t need to tell me that. But it (or something very close to it) can happen for a few weeks at a time, which is all we care about.

2. This isn’t r in the sense that you might have heard it, of capital R (the number of people infected by each person). You can get from one to the other if you are prepared to make some assumptions about how long infection takes, as described here, but actually we don’t need to do that.