Armchair Ecology

Herd immunity: how vaccines work for all of us

‘tis the season to think about getting your flu shot, so let’s talk a little bit about herd immunity. Vaccines are amazing because they work at two levels. They protect the individual – by stimulating your immune system, they make you more likely to resist the virus when or if you encounter it. But vaccines also protect the entire community, through the “herd immunity” effect. To give you the short version, the most vaccinated people in a group, the less likely we are to see an epidemic happen.

Want the long version? Buckle up, we’re about to write equations!

Pathogens are only really dangerous at a large scale when they can spread through a population, which happens when an infected individual can create new infection cases faster than they are “removed” from the pool of contagious people. Removal from the pool can happen through a variety of mechanisms, including the natural end of the contagious period, development of immunity against the pathogen, and mortality.

A pathogen that would kill its host instantly, for example, would not be very good at starting an outbreak, because infected individuals would be removed before they can get the chance to create new cases. On the other hand, a pathogen that is very good at invading the immune system would possibly stay active within its host for a long time, creating more cases and triggering an epidemic.

Newly infected cases are created from a pool of susceptible individuals, i.e. those that because they have never encountered the pathogen, or because they have encountered it a long time ago, have no immune response to it. For common infections like the common cold (which is a family of about 200 strains of pathogens with similar symptoms), the overwhelming majority of the population is susceptible at all time. We mostly recover within a week, during which we create new cases, and this is the story of why we get on average 2 to 6 common colds (from different strains) every year.

So let’s write this up, and we will make the important simplifying assumption that there is no natality, and no mortality. The population of infectious individuals is $I$, and it can vary over time because of two mechanisms: the infection (which creates new infected individuals), and the recovery (which removes them). So we can write this as:

$$\dot I = I\left(a\times S - b\right)$$

$S$ is the number of susceptible individuals. The rates $a$ and $b$ represent characteristics of the pathogen, namely the rate at which new infections happen (for every contact between and individual of $I$ and of $S$), and the rate at which individuals recover. The $\dot I$ quantity is the absolute change in number of infectious individuals over a given time period. If we assume that our time period is a day, then we can guesstimate the value of the parameters. For example, if the infectious period lasts for four days, we can write this as a chance of not being infectious anymore of $b \approx 0.25$, every day.

Reasoning about the infectious population this way makes it somewhat easier to think about what happens in the other. The susceptible individuals can lose part of their population to the infectious group, at rate $a$ (on every contact):

$$\dot S = - a\times S \times I$$

This is an interesting formulation, because we can verify two things. First, if we have no infectious individuals ($I = 0$), this whole term collapses to 0, and the susceptible population remains constant – outbreaks do not happen ex nihilo, and they require some number of infectious individuals to start (keep this in mind, this is a crucial piece of information). The second thing we can see from this expression is that $\dot S$ is always go to be negative, or strictly equal to 0. Either the population of susceptible individuals remains stable ($S = 0$, there are no susceptible individuals left; $I = 0$, there is no infectious individual), or it is decreasing. Susceptible individuals will end up somewhere else.

This somewhere else is the pool of recovered individuals (after going through the infectious step). The other term we have not yet used from $\dot I$ is $-b\times I$, which the flux of individuals that are removed from the infectious pool, into the recovered pool:

$$\dot R = b\times I$$

Why all of this mathematical nonsense? In short, because it is one way to determine whether an outbreak will persist. The way we represent the change in the number susceptible individuals ($\dot S = -a\times S\times I$), it is clear that any number of infectious larger than 0 will result, at first, in a loss of susceptible individuals. Yet this does not guarantee that the outbreak will occur, because the newly infectious individuals must remain infectious long enough to keep this cycle going. In short, the infection will spread if infectious cases are produced faster than they are recovered.

This question can be very slightly reframed as, how many new infectious cases can a single infectious individual create? If the answer to this question is “some quantity larger than one”, the disease will spread. If not, it won’t. The answer to this question is an important quantity in epidemiology, called $\mathcal{R}_0$, the basic reproduction number, pronounced “R nought”.

So of course, no one agrees on how to measure it. There are several methods, that do not always agree on the result, and require different mathematical tools or assumptions to apply. There is one way that works remarkably well for our $SIR$ model, and it goes as follows:

$$\mathcal{R}_0 = \text{transmissibility}\times\text{rate of contact}\times\text{duration of infectiousness}$$

Luckily, we already know the duration of infectiousness, which is $b^{-1}$. Transmissibility is the chance to see a successful infection of a susceptible individual per contact (which is $a$), and the rate of contact is the number of encounters between individuals from $I$ and $S$, i.e. $I\times S$. In practice, this is done on a model describing the proportion of individuals ($i=I/N$, $s=S/N$, $r=R/N$, $N=S+I+R$), with the parameters scaled appropriately. Taking this together, we are interested in the sign of $i\times\left(a\times s - b\right)$, which is positive when

$$\frac{a}{b}\times s\times i > i$$

And of course, we should rewrite this as

$$\frac{a}{b}\times s> 1$$

But because we are really only concerned about the very beginning of the outbreak, we can apply a neat little trick, and say that $N$ is somewhat large, and $R = 0$, and $I$ is small enough that $S\approx N$ is a good approximation. This matters immensely because it removes the population size from the equation – our pathogen can spread if, and only if,

$$\frac{a}{b} > 1$$

Thanks to our model, we can have a criteria that will help us predict when the pathogen will spread, and this is $\mathcal{R}_0 = a/b$. If this quantity is greater than one (or “greater than unity”, to be all fancy about it), we have a problem.

But, you may ask, what about vaccination?

Good question. Vaccination is a shortcut, to bring us from the susceptible to the recovered step, without having to go through the infectious one. We can tweak two of our equations (remove individuals from $S$, put them in $R$). But we don’t need to!

Remember, that we used a neat little trick to simplify the expression of $\mathcal{R}_0$? This is because in practice, what we derived was the expression $\mathcal{R}_0\times s$, which is to say the proportion of individuals that are susceptible. But what if vaccinated a proportion, say $p$, of this population? Well, because we can still use the $s \approx 1$ simplification, we can say that our epidemic will happen when

$$\mathcal{R}_0 \times (1 - p) > 1$$

In short, there is a value of $p$ (larger than 0, smaller than 1) for which we can get this entire value to drop below 1, and the epidemic to stop spreading. To find this value (the herd epidemic threshold), we re-order things a little bit, and we end up with

$$p^\star = 1 - \frac{1}{\mathcal{R}_0}$$

And the beauty of it, is that this works regardless of the expression of $\mathcal{R}_0$. Influenza viruses have an estimated $\mathcal{R}_0$ value between 2 and 3. To control an epidemic, we would need to vaccinate between one half and two thirds of the population.

One of the greatest benefit of vaccination is not that one, as an individual decreases the chance of becoming sick (though that is, admittedly, pretty cool), but rather that individual actions protect the entire group, or the entire herd. In fact, willingly unvaccinated people (as opposed to those who cannot be vaccinated) benefit from the actions of others (while still representing a risk for public health, since they can contaminate people around them). It’s a really cool situation to see a solution that works, effectively, at two different scales.

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