Recently I’ve seen multiple variations of the idea that since Covid is now widespread it doesn’t help anybody else for you to mask or stay home when you’re contagious. The claim is that anybody you would have infected will be infected very soon anyway. Here’s how one person expressed the idea on Nate Silver’s Substack.
If you take the daily chance of transmission from 5% to 1% what exactly have you accomplished? If you're only planning on living for a month or so I suppose this could make a difference. But for everyone else catching the virus eventually is a near certainty.
It’s really hard to evaluate whether a claim like that is true or almost true by just playing with words. In contrast, even the crudest mathematical modeling can clarify under what conditions if any claims like that make sense, and even let us make a crude estimate of how much difference your actions make under the current endemic conditions. That’s what I’ll do here.
The basic assumption behind the makes-no-difference claim is that since Covid is endemic, non-pharmaceutical interventions (NPI) such as staying home or wearing a good mask when sick don’t help because everyone will get sick anyway. Even if that were true rather than just roughly true, it ignores how often people get sick. Every time someone gets sick is bad for them and for the people who depend on them at home and on the job. Each time raises the risk of various long-Covid sequelae. Each time can increase the load on a strained healthcare system. So let’s try to understand how often people get sick in terms of basic input factors, including NPIs.
In the simplest endemic situation, the average number of sick people doesn’t change with time. (I warned that we were going to use simplified models to get only a qualitative and semi-quantitative result.) If the number falls below that steady-state level, the number of people with illness-acquired immunity drops, and that raises the rate of getting new infections. Likewise a transient increase in the number who are sick leads to an increase in the average level of immunity, causing the illness rate to fall. Either way it ends up settling back to the steady-state level. The basic conclusion of the simple math I’ll do is that the steady-state illness rate itself depends on NPIs, and that under realistic conditions people staying home or masking well when they’re contagious will significantly reduce the number of illnesses.
Here's the notation.
N: the population size
NC: the number of contagious people
f: the fraction of contagious people using NPIs to avoid spreading the disease
S: the fraction of the population that are approximately immune due to fairly recent illness
T2: the typical duration of that enhanced immunity
T1: the typical duration of a contagious disease
R1: the reproductive number taking into account some vaccine immunity and long-term illness-induced immunity but not recent illness-induced immunity
Re: the effective reproductive number, taking everything into account
Obvious complications, e.g. that immunity is never absolute and that it wanes gradually, will not be relevant to the rough calculation to follow. The point is not that there’s a clear line between immune and susceptible but that there is important illness-induced immunity that is proportional to the number of illnesses within about the last T2 period.
First let’s look at the steady state, in which without the NPIs on average the number with illness-induced immunity doesn’t change, i.e. Re=1.
1= Re = R1*(1-S) so S=1-1/R1 , a familiar result from the most simplified models.
What happens when some people use NPIs to protect others? We reduce the original R1 by a factor of (1-f).
Then S=1-1/(R1(1-f)).
If enough people use NPIs, the steady-state S drops to zero, meaning the disease stops. For some diseases, such as the original SARS, that has been attainable. Realistically, for Covid it isn’t. So let’s look at the opposite limit. What is the average effect of just one person staying home or using a good mask even if nobody else does so? If that’s a one-time event, it’s true that there’s no effect on the net number of infections. A few might get slightly delayed. Nevertheless, even a small fraction of people consistently using NPIs has an effect.
In the limit where there’s just one person at any time practicing NPIs, the new steady-state S is reduced by (1/NC )*dS/df since one contagious person can change f by 1/NC. How many people are contagious? At the level of approximation needed for our results here, we have NC= S*(T1/T2). Differentiating S with respect to f gives
dS/df = -1/(R1(1-f)2). The key point of this argument is that this derivative does not go to zero for f near zero but rather to -1/R1.
So if one contagious person is practicing good NPIs at any time that reduces S by
T2/(NST1R1). Over the whole population of size N, that reduces the number of recently sick people by T2/(ST1R1) = (T2/T1)/(R1-1).
What would that mean with realistic numbers? Infectiousness lasts a week or so. The short-term immunity boost lasts 20 weeks or so. We can make a crude estimate of R1 under current conditions by looking at how rapidly new variants grow. It seems that a typical R1 is less than 3. So if in a population of arbitrary size there’s always one contagious person avoiding spreading the disease, roughly T2/T1 people will be saved from having gotten sick in each T2 period. That’s about the same as the total number of NPI users in that period. In this parameter range the effect of contagion from non-NPI-users does reduce the effect of NPIs, but not to a negligible level. The number of subsequent illnesses is reduced by something like one for each contagious person using NPIs.
Clearly the benefit from using NPIs will vary a lot between people. It’s more important for people in crowded situations or dealing with the medically vulnerable. The cost of any particular NPI, such as staying home vs. wearing a well-fitted N-95, will also vary. My point is just that the argument that it doesn’t matter is wrong. It does matter even for a highly infectious disease like Covid. It matters even more for infections like the flu that have lower R1. That’s why widespread NPI use nearly eliminated flu in 2020-2021.
In practice, that means that anyone with flu-like symptoms should stay home regardless of initial rapid antigen test results, which are typically false negatives at the start of symptoms. RATs are much more reliable later for determining when viral loads have gone down sufficiently to minimize contagion.
So far I’ve focused on the limit in which few people are using NPIs to avoid infecting others. Since dS/df = -1/(R1(1-f)2) each individual’s NPIs tend to matter more when others are also doing them. Institutional actions (e.g. encouraging employees to stay home when sick, encouraging masking instead of forbidding it,…) should work better than scattered individual choices.
The most obvious neglected institutional action (other than allowing sick days!) is to improve air quality, especially by use of easily available commercial HEPA filters. While these never give an effective f of 1, even cheap homemade MERV-13 box fan units can do better than f=0.5. In practice air filters may offer the best way of reducing disease frequency at low cost without needing to rely on individual choices, although they can be used with the other NPIs we’ve discussed. They do require actions by administrators of schools, libraries, offices, etc. Such actions have been painfully rare. I suspect that the “everybody’s going to get it anyway” rationale is part of the explanation. The important question that ignores is: “How often are they going to get it?”
We’ve seen that the net number of illnesses that occur can be reduced regardless of whether in the long run everybody ends up getting sick. That’s a real benefit.