The Number Field

Elena has been sick for more than a week now. She had simple cold symptoms, but last weekend she started having periodic coughing fits that went on for what seemed like a long time and were almost uncontrollable. I have a hard time describing the cough--sort of short and full of mucous, and while she's in a coughing fit she hardly has time to breathe between the coughs. We have two effective ways to stop the coughing: give her some juice, or steam up the bathroom with hot water and take her in there until it's over. The coughing fits only happen a few times a day, often after she's been doing something more strenuous while playing, but they have woken her up in the middle of the night.

On Tuesday we decided to get some medical help, so Jenny called the nurse line and they had her make a doctor's appointment. The doctor decided to treat her for whooping cough based on her symptoms and the fact that there has been a recent outbreak of whooping cough around here. The tests for the whooping cough take several days to run and are not entirely reliable, and in this case it's best to start a course of antibiotics without waiting for results. Another name for whooping cough is pertussis, which is the P in the DTaP vaccination. Since Elena has followed the recommended immunization schedule, she's already had three of these shots, so you might wonder why she still got sick. We discussed this briefly in my mathematical biology class so I can give an overview of the dynamics. I'll try not to get too mathematical or launch into too much of a rant, but be forewarned, because I can't make any guarantees.

Like almost all immunizations, the DTaP shot is only partially effective. Studies of effectiveness are always done at the population level, not on individuals, so it's unclear what "partially effective" means. It could mean that most people are completely immunized while a few are completely unprotected, or it could mean that everyone immunized has a smaller chance of falling ill, but is still slightly susceptible. Of course, it's probably some combination of the two: inoculated individuals are protected to varying degrees, depending on a wide set of complicated factors. This matters when combined with the idea of how contagious a disease is. Pertussis has a contagion rate of 70-100%, so if an non-immunized person is exposed, it's almost a certain thing that they will fall ill.

The most basic model used by epidemiologists to study the spread of disease is called SIR, an acronym for "Susceptible, Infected, Recovered," which is the progression that individuals go through. In the model, you look at the percentage of the population in each of those categories over the time of a disease outbreak. In order to do so, information about contagiousness, population susceptibility, and recovery rates is plugged into a matrix, which then can be multiplied against the population percentages. The resulting system can be analyzed to see whether the disease is likely to die out, maintain a steady level, or explode into an epidemic. All of that depends on some things called the eigenvalues of the matrix. You don't need to know exactly what that means, but you do need to know that matrix multiplication doesn't work like regular multiplication. Two matrices that look very similar could bring about very different behaviors, because they have qualitatively different eigenvalues. Specifically, any eigenvalues bigger than \( 1 \) mean the outbreak will grow, while eigenvalues smaller than \( 1 \) mean that it will die out. So even though there's a much bigger difference between \( 1.05 \) and \( 3 \) than there is between \( .95 \) and \( 1.05 \), the first pair would imply similar similar behavior and the second pair gives very different behavior.

If we want to reduce the spread infectious disease (at the demographic level of policy and public health), there are only a few things that we can try to do, such as improving the effectiveness of our vaccines, but that's likely to take a long time. There's really just one thing that can be done on a large scale in a short time: increase the level of vaccination in the population. When you model these kinds of changes, the non-linearity of the system becomes clear. If you guess that 50% of the population is vaccinated, and see what happens when you move that up to 65%, it might be that there are no visible changes--the disease goes on as before. But then if you move it up just a little bit more, say to 70%, then you cross over the threshold and change the eigenvalues, which makes the outbreak less likely to persist. Sometimes this is called "herd immunity" because it works only on a large scale, and means that it's actually okay for a certain number of people to remain unvaccinated, because they're effectively protected from exposure by the immunity of the rest of the group.

So what does this all mean? First, it's okay for vaccines to be only partially effective, because a second layer of protection kicks in when enough people get vaccinated. Second, it's unrealistic to vaccinate every person in a population. Some people just don't go to the doctor, some are still too young for the vaccine, some have legitimate allergies (eggs are the foundation of many vaccines, and some people are allergic). But that's okay, because those people get the protection of the entire group. It's even okay for a few crazy people to not want immunization for irrational reasons. What's not okay is when those people write about it on blogs or appear on Oprah --then this influences others and it becomes a movement, which changes the dynamics and threatens everyone's immunity.

Vaccine schedules are created by smart people for a reason, which is to optimize the health and safety for each individual. But they optimize in two dimensions--the miniscule risks presented by the vaccine itself, and the much greater risk of catching the disease if not enough people are immunized. The second dimension is the important one, but it's the one that people ignore, because they don't understand how it works. They don't understand that the risks are nonlinear. By not vaccinating, they take advantage of the group, but this only works up to a point, after which outbreaks and sustained epidemics become likely.

When a non-vaccinator makes their choice, they probably don't accurately consider the risk that their child will get sick, because serious, dangerous diseases are extremely rare in our modern world. We've eradicated many of the terrible diseases of yesteryear. The way we did it was--no surprises here--through vaccinations! If the vaccination campaigns hadn't worked so well, and terrible diseases like polio and smallpox were still prevalent, I can guarantee that nobody would worry about nonexistent links between autism and vaccination. But these things can come back when large numbers of people opt out of vaccination. There was a tragic story about measles on This American Life some time ago.

I'm not worried about Elena. She's happy and strong, the cough seems not to bother her too much and the coughing spells have been far shorter in the past few days. She may not even have whooping cough; we still haven't received the results. But if she does, it's a little bit frustrating to know that she might have a cough that lingers for months, when we did everything we could to avoid it. So my next step is to try to use this tiny platform of mine to sway the public discussion in the right direction, and help people understand why non-vaccinators are misguided. I hope that all of you vaccinate your children on the recommended schedule, and can encourage people to get the real facts when they are trying to understand how it works.

Updated: I changed a few small things at the beginning for clarity and accuracy, based on Jenny's first-hand account of Elena's doctor's visit. She also summarized things succinctly for me:

When a person decides to not vaccinate their child, or even to follow a delayed scheduled, they are not just deciding to take on the risks for their own child of not vaccinating. They are putting every person their child comes into contact with at increased risk. This could be compared to the unfairness of people affected by second hand smoke - but at least in those cases you can see and smell the smoke.