**COMMENT**

As we see a resurgence of the Covid-19 pandemic around the world, particularly in Europe (which still includes the United Kingdom) and the United States, people are talking about a “second wave”. I am not sure that this is an accurate way to characterise what is happening.

New Zealand suppressed community transmission and then had a small resurgence from new imported cases: that is fair to call a “second wave”. What we are seeing in the other places is suppressed community transmission that is resurging because measures were relaxed too soon or too much.

The danger of thinking of this as a second wave is it obscures the fact that suppression measures have slowed the spread and relaxing them prematurely can make numbers take off again.

A number of things complicate the picture.

In the initial surge, with the exception of a few places like Taiwan and South Korea, there was inadequate test capacity so the initial surge missed many of the milder cases. In many countries, this situation has improved. A way to check for this is “test positivity” — the fraction of tests that come back positive. If you have high test coverage, test positivity tends to drop. If there is sufficient capacity, it will generally be below 10%. In the US, for example, test positivity nationwide (Johns Hopkins data) peaked at more than 20% in April and has since dropped to well below 10%. So though the current surges look bad compared with the initial numbers in places like New York, the true number of infections is not being underestimated to the same extent. Even so, Johns Hopkins shows some states with positivity rates that are above 20% — though these are mostly rural, less populous states like Iowa and South Dakota.

Another factor differentiating the new surge from the initial one is improved understanding of treatment — fatality rates are generally a lot lower than they were at the start.

Even so, I argue that it is wrong to see this resurgence as another wave. To understand this, I ran some highly simplified models at https://ncase.me/covid-19, authored by Marcel Salathé and Nicky Case. (Their site is a good resource for understanding the basics and goes into more detail than I have space for here.)

A commonly-used epidemiology model divides the population into susceptible (S: not yet infected by the virus and with no immunity), exposed (E: infected but not yet contagious), infective (I: past the exposed state) and recovered (R: at this point the assumption is that you are no longer susceptible). In the simplified model I use, the population is treated as uniform and no account is taken of deaths, nor of variation in severity or duration of the disease.

A model like this is called SEIR. The terminology used is: *R*_{0}, the basic reproduction rate — the average number of people an infected person infects when everyone they meet is susceptible. As the disease progresses, the reproduction rate drops because the odds of encountering people who are no longer susceptible increases. The current value of *R* is often referred to as *R** _{t}*, reflecting that it is time-dependent. In the online model I run,

*R*

_{0}is set by varying three features of the disease: how many days on average pass between an infected person passing it on to someone else, how long it takes post-infection to become infectious and how long someone stays infectious. If you set these numbers to infecting a new person every four days, three days to become infectious and 10 days to cease being infectious, you get

*R*

_{0}= 2.5, which is a fair approximation. Since the degree of social mixing varies between societies,

*R*

_{0}can vary a lot but, if we keep it simple, it’s easier to understand how interventions change the situation.

First, I run the model with no interventions. If you do this yourself, watch how *R*_{0} and *R* (that I prefer to call *R** _{t}*) vary.

*R*

_{0}stays the same because the properties of the disease and social mixing stays the same, but

*R*decreases steadily and at the point where

*R*=1, the number of new cases drops off quickly. That happens when the total who have ever been infected is 60% of the population; that’s where herd immunity starts. Watch it run to completion. The grey curve reflecting those who recovered (and in the real world, this includes those who die) ends up at almost 90% of the population. The fraction who are infected above the herd immunity level is called overshoot — here, nearly 30% of the population.

Now, I run a simplified model of interventions. When about 20% is infected, I stop it and adjust the days between new infections from four to 10. This fakes the effect of a disease where *R*_{0} = 1. The value of *R* at this point is well below one because a significant fraction of the population is no longer susceptible. I resume the run from that point and numbers start to drop until new infections appear to be well under control.

I stop the simulation again when the total fraction who’ve been infected is a little above 40% and put the days to new infection back to four. This puts *R*_{0} back to 2.5, though the current value of *R* is only 1.43 because of the fraction who are already infected. With *R* above one, numbers take off again though not as fast as before.

If I run the simulation to completion (it stops at the end of a year), I end up with almost 75% of the population who have had the disease — better than 90% but still a lot considering that the fatality rate is higher than that of flu and that long-term effects remain a big unknown.

Because relaxing interventions completely put *R*_{0} back to 2.5, the herd immunity level was also put back to 60% of the population. But overshoot was smaller because arriving at the peak (where *R*=1) was slower.

These adjustments to the model approximate the effect of an instantaneous introduction of strong containment measures and an equally instantaneous total relaxation of those measures. This is not very realistic but is easier to understand than more gradual or less society-wide variation.

Aside from the omissions noted earlier, this model assumes that everyone who recovers remains immune. There is increasing evidence that this is not always true. Though most of the small number of cases reported so far have it mildly the second time, there has been at least one report of a second infection more severe than the first.

The level of antibodies in someone who has recovered starts to fall off two to three months post-infection. That is soon enough that those infected at the start could be reinfected before the disease runs its course, even when there are no interventions that slow it down.

That brings me to the value of a vaccine. Even if a vaccine only confers temporary immunity, if enough of the population is immune simultaneously, a resurgence is much less likely.

What about South Africa? We retain some measures to contain the spread including universal masking — imperfectly though that is implemented. As the weather warms, people are less likely to congregate in unventilated spaces. But there is a real danger of complacency leading not to a second wave but to a resurgence. We should not forget that community spread is still with us and until we have evidence that infections have reached the herd immunity level or higher, we can still have a resurgence.

A YouTube summary of my argument is available here.