Over at my friend Pal's blog, in a discussion about vaccination, a commenter came up with the following in an argument against the value of vaccination:
100% – % of population who are not/cannot be vaccinated – % of population who have been vaccinated but are not immune (1-effective rate)-% of population who have been vaccinated but immunity has waned – % of population who have become immune compromised-(any other variables an immunologist would know that I may not)
What vaccine preventable illnesses have the result of that formula above the necessary threshold to maintain herd immunity?
I don’t know if the population is still immune to Smallpox, but I would hope that that is just a science fiction question. Smallpox was eradicated, but that vaccine did have the highest number of adverse reaction (I’m sure PAL will correct me if that statement is wrong)
It's a classic example of what I call obfuscatory mathematics: that is, it's an attempt to use fake math in an attempt to intimidate people into believing that there's a real argument, when in fact they're just hiding behind the appearance of mathematics in order to avoid having to really make their argument. It's a classic technique, frequently used by crackpots of all stripes.
It's largely illegible, due to notation, punctuation, and general babble. That's typical of obfuscatory math: the point isn't to use math to be comprehensible, or to use formal reasoning; it's to create an appearance of credibility. So let's take that, and try to make it sort of readable.
What he wants to do is to take each group of people who, supposedly, aren't protected by vaccines, and try to put together an argument about how it's unlikely that vaccines can possibly create a large enough group of protected people to really provide herd immunity.
So, let's consider the population of people. Per Chuck's argument, we can consider the following subgroups:
- \(u\) is the percentage of the population that does not get vaccinated, for whatever reason.
- \(v\) is the percentage of people who got vaccinated; obviously equal to \(1 - u\).
- \(n\) is the percentage of people who were vaccinated, but who didn't gain any immunity from their vaccination.
- \(w\) is the percentage of people who were vaccinated, but whose immunity from the vaccine has worn off.
- \(i\) is the percentage of people who were vaccinated, but who have for some reason become immune-compromised, and thus gain no immunity from the vaccine.
He's arguing then, that the percentage of effectively vaccinated people is \(1.0 - u - nv - wv - iv\). And he implies that there are other groups. Since herd immunity requires a very large part of the population to be immune to a disease, and there are so many groups of people who can't be part of the immune population, then with so many people excluded, what's the chance that we really have effective herd immunity to any disease?
There's a whole lot wrong with this, ranging from the trivial to the moderately interesting. We'll start with the trivial, and move on to the more interesting.
In real modeling, we usually describe two groups of people: the portion of people who are immune to the disease (\(q\)), and the portion who are susceptible (\(S\)). Obviously, \(q + S = 1\), because everyone is either susceptible or immune. What he's done is play games with this.
He wants to make \(q\) look as small as possible, so he basically wants to turn \(S\) into a complex list of things. There's really no good reason to break this up: for most diseases, we have a pretty good idea of what the actual immunity provided by the vaccine is; the effectiveness rate of the vaccine incorporates his \(n\), \(w\), and \(i\) factors. But it makes herd immunity look a lot more unattainable when you see the percentage of vaccinated individuals minus this, minus that, minus the other thing. The entire division into sub-categories is solely for the purposes of deception.
Even if you did insist on doing that... the groups in the list aren't mutually exclusive. The set of people who are immune compromised and the set of people who were vaccinated but didn't get immunity will definitely overlap; so will the set of people who are immune compromised, and who's immunity has waned. He doesn't consider that - quite deliberately - because it reduces the appearance that there are all of these people out there who are unprotected by vaccines.
Getting to the more interesting stuff: In terms of modeling, the real model of a disease is actually pretty simple. There's a factor called \(R_0\), which is the infection rate of the disease. \(R_0\) is the probability of any individual suffering from thedisease exposing someone else to the disease in a sufficient quantity toinfect them if they're succeptible. The actual infection rate of a disease is called \(R\); and \(R = R_0 times S\). A disease can survive in a population when \(R ge 1\) - that is, if the probability of exposure times the percentage of the population that's susceptible to the disease is greater than one. If it's less than one, then the number of infected individualswill steadily decrease until there's no-one infected; if it's greater than one, then you've got an epidemic where (at least for some period of time), you've got exponential growth in the number of people infected; and if it's exactly one, then you've got a steady state, with a constant number of people infected.
So, for the disease to survive in a population, \(R_0 times S\) must be greater than 1. We can rephrase that in terms of the immune population: for a disease to be sustained in a population, \(R_0 times (1 - q) geq 1\). If \(q\) gets large enough that \(R_0 times (1 - q) < 1\), then the disease will die out. So that's the threshold for herd immunity.
For a disease like the flu, \(R_0\) appears to be highly variable. Different studies produce numbers ranging from 2 to 5. Suppose it's at the high end: that is, \(R_0 = 5\). Then herd immunity will occur when \(5times (1 - q) 0.8\). If 80% of the population is immune, then in normal circumstances, flu is non-sustainable.
Now, who is that group, \(q\)? They're people who've either had the flu, or people who've been successfully immunized against it. According to the CDC, the highest documented effectiveness of the flu vaccine is around 85% effectiveness - meaning that 85% of the people who get it will have effective immunity from the flu. In practice, it's generally a lot lower; the actual rate is very hard to determine, because there are so many influenza like illnesses, and so few people actually get tested to determine whether they actually have the flu versus something flulike. But still - even with the highest potential effectiveness, we would need to immunize 94% of the population to get herd immunity. Real vaccination rates for flu are nowhere close to that. (Of course, this is all rather fudgey; we're using a very high rate for \(R_0\), and a very high rate for the effectiveness of the vaccine. The real rate needed for herd immunity could be as low as around 80%, or it could be above 100%, meaning that it's impossible to produce herd immunity by vaccination.)
But this whole discussion so far is, in some sense, rather deceptive. It's vastly oversimplifying reality. It's looking at herd immunity as a simple yes/no: is the disease sustainable in the population according to a simple model?
When we look at a disease like whooping cough, or polio, or meningitis, we can, absolutely, get a vaccination rate which exceeds the critical threshold. When's the last time you saw someone with polio? How many of us have ever seen a child with whooping cough? Those are incredibly rare precisely because we've got a high enough rate of immunity within the population (thanks to vaccines!) that when an exposure to one of those occurs in the population, it burns itself out: it's \(R_0\) is simply too low for it to be sustained in an immunized population.
So to answer the original question: what vaccine preventable diseases are there where effective herd immunity is really possible? Lots. For many diseases, it's not particularly difficult to attain herd immunity: mumps, measles, chicken-pox, smallpox, polio, whooping cough, meningitis - for all of these, we can (and in most places, do) have really solid herd immunity. But while it's not difficult to have herd immunity against these diseases, it's also not too hard to break it. There've been recent outbreaks in many places where the vaccination rate has fallen.
When it comes to things like the flu, we're not getting complete herd immunity from vaccines. In fact, it's quite probable that we can't, because the \(R_0\) for seasonal flu is high enough, and the population-wide effectiveness of the flu vaccine is low enough that even with 100% vaccination, \(R_0 times S > 1\).
But even in cases like flu, where we can't produce perfect herd immunity, there's still a herd immunity benefit from vaccination. Remember that the rate of infection is determined by both \(R_0\) and \(S\). If you cut \(S\) in half, that you cut \(R\) rate in half as well! If we can get the immune population up to 75% - which is quite doable, if healthy children and adults all get immunized, we can reduce the effective rate of spread of the flu by a factor of 4! And that's nothing to sneeze at. The number of lives that could be saved by cutting the number of cases of flu by 3/4 is huge!
And in some sense, this is all a distraction. "Perfect" herd immunity doesn't mean that no one will ever catch the disease. There will be exposures; people will sometimes catch the disease when there's a new exposure. Herd immunity means that when that happens, the disease won't be sustainable - it will burn out.
The real question that we care about when it comes to vaccination isn't "is it possible for people to get infected?", because the answer to that will always be yes, as long as there's any possible source of exposure.The real question is: "What is the risk of a susceptible individual being exposed and as a result coming to harm?" And that's a much fuzzier question, but it's very clear that as we reduce the population of people who can carry it - as we reduce the pool of people in \(S\) who can be infected enough to expose others - we reduce that risk. When we get a vaccine, we dramatically reduce our own chances of getting sick; and when enough of us get vaccinated, we also dramatically reduce the chances of other people getting sick.