Understanding the Relationship between R, Herd Immunity, and Attack Rate

The end goal for dealing with Covid-19 is either complete elimination (unlikely at this point) or "herd immunity", which is the state wherein enough people are immune to the disease (either through already having had it or through a vaccine) so that the disease can no longer easily spread through the population.

How many people need to be immune in order for us to achieve herd immunity?  This is an important question for policy decisions because--given that some relatively fixed percentage of people who get the disease will die*--the answer to this question will also answer the question of how many people will need to die if you want to achieve herd immunity to a disease without having a vaccine.

It turns out that the number of people you need to achieve herd immunity is directly related to the speed at which the virus naturally propagates through your population: the basic reproduction number, or the now-infamous "R".  This can actually be shown to be the case with a little bit of thought and not much math.

Setup for the Thought Experiment

R is simply defined as the number of people that one infected person, on average, will pass the disease to for the complete time during which he is infected. Now this average number will obviously depend on two things: how naturally infectious the disease is, and how many occasions an average person in a society will have the opportunity to infect some other person.

In the middle of a pandemic, which many societal measures in place to prevent the spread of the disease and with many people voluntarily altering their behaviors so that they don't spread the disease, R is going to depend on a lot of situationally specific factors: what are the local societal restrictions, how compliant the public is with them, how have people changed their behavior due to the disease, etc.  

However, if we're talking about herd immunity, we can kind of ignore those factors, because what we are trying to get at is the percentage of the population who need to be immune in order for the disease to die away without artificially imposed behavioral changes.  In other words, what we want to know is, when can life go back to normal without fear of infection?

Infection Spread in Normal Societal Conditions

So for our purposes now, we can assume normal societal mixing in our thought experiment as a constant.  The number of people who will be infected by a single infected case, or R, is then going to depend only on how infectious the disease is, and not on abnormal societal factors.

Now, as we said, R is a value that combines the factors of how naturally infectious the disease is and how much social mixing an infected person will do during the course of the disease.  You can think of this as multiplying two numbers:  an infected person will come into close enough contact with a certain number of people during the course of the infection, and each time will have a certain percent chance to infect that person.  Multiply those two numbers and that's the total number of people whom that person will infect.  

So if an average person in a society comes into close contact with 100 people during the course of his illness and has a 2% chance of infecting someone each time, he will on average infect 2 people in total and the R of the disease will therefore be 2.

How Immunity Changes the Picture

What if, in the same scenario above, an infected person comes into contact with 100 people, but 10% of those people have already had the illness and are now immune?  Well, in that case that same 2% chance of each person getting infected will only apply to 90 out of the 100 people the infected person meets.  So that means that the R is going to drop down to 1.8.  Another way of saying this is that the baseline R of the disease (the R(0) or "R naught") needs to be multiplied by the fraction of the population that is susceptible to the disease (0.9 in our example) in order to get the current effective R value.

Now it's obvious that in order for a disease to grow in a population, R must be higher than 1.  If it's less than 1, it will begin decreasing in the population.  It can still spread if R is less than 1, but it cannot grow.  That is, if there are 1000 infected people in your population and the R is 0.7, 700 additional people will be infected in the next disease "generation", and then 490, and so on and so forth.  But as long as the R is less than 1, the disease will be progressively dying and not growing. And this is what you want from herd immunity.

**Therefore, the point at which you can claim herd immunity is exactly the point at which the fraction of people who are still susceptible to the disease would multiply with the R(0) of the disease to equal 1.**

So for measles, which has a ferocious R(0) of about 10, you would need a very low 10% of your population to be susceptible in order to reduce the R to 1, which means that you don't get herd immunity until 90% of the population is immune.  Seasonal flu, on the other hand has an R(0) of something like 1.3.  1/1.3 is around 0.77, so you can get herd immunity against the current seasonal flu with just 23% of the population already having been infected (or otherwise immunized).

Covid-19 has a very high R relative to the flu.  Estimates started at around 2.4, but have gone up with better analysis and is probably closer to 3.8.  Supposing it is 3.8, what percentage of population would need to be infected in order to confer herd immunity? 1/3.8 is about 0.26, so that means the answer is about 74% of the population.

Relationship to Attack Rate

The "attack rate" of a disease is the total percent of the population that will get the disease before it runs its course.  From a policy standpoint, it would be good to know what this number will be if we pursue achieving herd immunity via letting the infection run its course.

This total number of infected is not equivalent to the percentage of the population required to be immune in order to achieve herd immunity.  This is because even when you achieve this number--if you achieve it by letting people get infected--at that time you still have a certain percentage of people infected with the disease, and (as we said) the disease will still spread from those infected people at ever decreasing rates as it is dying out.

You can actually calculate the likely number of people who are still going to get the disease in this situation, but this is not so simple as what we have done so far: it requires differential calculus, and I said there wasn't going to be much math.  So I'll skip that part and just say: there will be some further infections after herd immunity is reached, if it is reached by lots of people getting infected.

Conclusion

Reaching herd immunity is not the same for every disease.  The number of people who need to be immune in order for herd immunity to work depends on the native infectiousness of the disease.  Because Covid-19 is quite a bit more infectious than the seasonal flu, we need to understand that getting to herd immunity with Covid-19 involves a far larger percentage of the population getting sick than is typical for a bad flu season, around 75% at a bare minimum.

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* Note: I said above that the number of people who will die from the disease is a relatively fixed proportion of the total attack rate.  I should acknowledge that this is not a completely safe assumption, given that various treatments could be developed or discovered that would make Covid-19 less fatal.