Update: tweaked the descriptions of nuclear physics to be a little more specific.

I’m not sure that anything fills me with despair more than the trend of parents refusing to vaccinate their children. A couple of weeks ago, an article in The Hollywood Reporter described how affluent Hollywood schools are experiencing outbreaks of whooping cough and measles that haven’t been seen since, well, before vaccination.  These are nasty diseases, debilitating and potentially fatal.  It is shameful and not a bit terrifying that people are more or less deliberately bringing back illness that predominantly targets the very young.

But why do they do it?  From the article, we have this depressing tidbit:

According to more than a dozen area pediatricians and infectious disease specialists THR spoke to, most vaccine-wary parents have abandoned autism concerns for a diffuse constellation of unproven anxieties, from allergies and asthma to eczema and seizures.

In other words: once the link between vaccines and autism was shown not only to be mistaken but in fact fraudulent, people found other reasons to rationalize their actions.

Another statement from the same article left me utterly flabbergasted:

Experts on both sides of the issue say these families seem unconcerned about herd immunity — often questioning the legitimacy of the very concept…

Reading such things is genuinely painful to me.  For those unfamiliar with the term, “herd immunity” is the — uncontroversial to science and medicine — idea that a properly vaccinated population provides additional protection to everyone in the community, vaccinated and unvaccinated alike.  This is extremely relevant to the question of “anti-vaccination,” because it suggests that the group benefits pretty much disappear when enough of the population stops vaccination.

I find it pretty much unthinkable that people wouldn’t believe in herd immunity; I can only hope that they don’t completely understand how it works.  With this on my mind, it occurred to me on the drive home the other day that herd immunity can be readily explained by analogy with a phenomenon in physics — nuclear chain reactions and critical mass.  In short, we can argue that group vaccination is akin to keeping a nuclear substance below its critical mass — and failing to vaccinate is mathematically akin to setting off a nuclear bomb.

Let’s start with a simplified discussions of radioactivity, nuclear reactions, and nuclear chain reactions.  A mass of radioactive material, of course, is simply a big group of radioactive atoms stuck together.   The atoms are “radioactive” because their nuclei are unstable; occasionally, one of them will break apart, resulting in a slightly less massive atomic nuclei (which may still be radioactive, or not) and a decay product.  We illustrate this below, imagining that the decaying atom lies directly in the center of the spherical chunk of material.

The decay product has a lot of energy — typically measured in millions of electron-volts, where the amount of energy required to ionize an atom is typically measured in at most tens of electron-volts.  The decay particle has so much energy, in fact, that it can collide with the nucleus of another atom and knock another high energy particle from it, in the process known as nuclear fission.

However, the secondary product — typically a neutron — also has a lot of energy, and for some radioactive materials (fissile materials such as plutonium) this secondary particle can also knock a neutron free from another atom.  And so on, and so on.  This is what is meant by a nuclear chain reaction — a single (or small number) of decays lead to a rapidly increasing number.

But under what circumstances will a chain reaction form?  The condition is surprisingly simple to state.  If, on the way out of the material, any energetic particle creates less than a single secondary fission, on average, then nothing special will happen.  This is the subcritical condition.  In the very special case where an energetic particle creates exactly a single secondary fission on average, then each radioactive decay will produce a persistent sequence of secondary decays, but not a growing effect — this is the critical condition.

If, on average, more than one secondary decay is sparked by an energetic particle, then the sequence of fissions rapidly, in fact exponentially, increases.  The nuclear reactions become a runaway, creating a massive burst of radiation, heat, and — if the conditions are right — a nuclear detonation.  This is the supercritical condition.

So what determines whether a given chunk of fissile nuclear material is subcritical, critical, or supercritical?  This is determined by the total mass of the chunk, the density of the material and, to a lesser extent, even the shape of the chunk.  An easy way to understand this is to imagine that you are dropped into the middle of a forest blindfolded, and try and find your way out without being able to see.  How many trees will you bump into along the way?  Clearly, you will run into more trees in a bigger forest, as you have to avoid more trees as you escape.  When the size of the forest is just large enough that, on average, a person will collide with one tree on the way out, the forest is of “critical size.”  This is analogous to the critical mass of nuclear material.

Also, given two forests of the same size, a forest with more dense foliage will result in more collisions.  Similarly, a nuclear material that is compressed will have its critical mass effectively reduced.

Most nuclear reactors, in my understanding, exploit the idea of critical mass, manipulating the positions and shielding of nuclear material in order to continuously oscillate from slightly subcritical to slightly supercritical.  In this way, they are able to generate a lot of heat while avoiding the potential destruction of a fixed supercritical situation.  This can be risky, as physicist Louis Slotin found in 1946 while working at Los Alamos.  Slotin was working with a sample of plutonium surrounded by a pair of beryllium hemispheres, which reflect neutrons back into the radioactive sample.  By manipulating the separation of the hemispheres, Slotin was able to adjust the criticality of the sample.  During an experiment, however, the screwdriver he was using to keep the hemispheres apart slipped, and they came together, initiating a critical reaction. Though he pulled them apart immediately, the reaction gave him a fatal dose of radiation; he died nine days later.

Slotin was working on the first atomic bomb.  Fission atomic bombs often achieve supercriticality by a change of density: a conventional explosive detonated around the core compresses it, pushing it to the point of a chain reaction.   And then, the runaway reaction releases a devastating amount of radiation and heat.  Within an atmosphere, this heat results in a massive explosion.

Boom.

The takeaway lesson from this?  When the conditions are right, it takes very little change to take a benign situation and make it malignant.

This brings us back to vaccines and herd immunity.  To describe infectious disease, we can draw a picture strikingly similar to that of radioactivity.  However, the exterior circle no longer represents the spatial size of a material, but the amount of time a person infected with a disease is contagious.

When a person become contagious, often not even showing symptoms of the disease*, they encounter a number of other people along the way.  Some of those people are vaccinated, and are effectively “ghosts,” in that there is no passing of the disease to them.  If the contagious person interacts with a vulnerable person, however, they can become infected as well, and then can pass along the disease to others themselves.

How quickly the disease spreads, if it spreads at all, depends on the number of people vaccinated.  Again, we find very simple math: if, on average, an infected person encounters less than one unvaccinated person while he/she is contagious, the disease will die out.  If, however, an infected person encounters more than one unvaccinated person while he/she is contagious, the disease will multiply: each new infected person infects new ones.  The disease spreads rapidly, and we end up with an epidemic, if not a pandemic.

The number of people vaccinated is a crucial parameter: the density of unvaccinated people is analogous to the density of atoms in a radioactive material.  When people are vaccinated, there is a low density of vulnerable people.  If enough people refuse to vaccinate their children, however, the population becomes supercritical: any small outbreak of disease will spread until it has touched almost everyone.

Herd immunity, then, in my model, is equivalent to keeping the density of unvaccinated people at a subcritical level, so that any incidental infections quickly die out.  The “herd” of vaccinated people does not provide a path for the disease to spread before it loses its contagiousness.

What is striking to me about such a model is what I might call the brutal statistics associated with it.  There are tweaks I can make to the model to improve it, but in the end it really comes down to a numbers game: how many vulnerable people does an infected person encounter? The critical threshold is the difference between minor illness here and there and large amounts of disease, suffering, and death.

So what is the critical threshold?  Estimates for most diseases require somewhere around 80% of the population to be immunized.  The fact that we are flirting with these levels for many preventable diseases is utterly shameful.  As a society, we have the power to keep this bomb from going off.

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* Evolution at work: if a disease is only contagious during the period of the worst symptoms, people can avoid the infected.