How are vaccination rates like water freezing?

A quick primer on the non-linear nature of the effect of vaccination rates

Troy Tassier is a professor of economics at Fordham University and the author of The Rich Flee and the Poor Take the Bus: How Our Unequal Society Fails Us during Outbreaks.

There’s a famous scene in the movie Jurassic Park where a scientist, played by Jeff Goldblum, explains chaos theory to another scientist, played by Lauren Dern. He begins by stating that “a butterfly flapping its wings in Peking can lead to rain in Central Park instead of sunshine.” Goldblum’s character is a smug ass. The scene that contains this line begins with another character referencing Goldblum and stating, “I hate that man.” Goldblum does better with his second try to explain the concept. He places a drop of water on Dern’s hand and the drop rolls off her hand to the left. He then repeats the experiment, dropping the water in almost exactly the same location, but this time the water rolls another direction. Goldblum is making the point that sometimes small changes can produce unexpected results.

Consider another example. If you start with water at a cold temperature of 40 degrees and drop the temperature by a degree, the water feels a bit colder but it doesn’t essentially change. However, we all know that if we continue to drop the temperature a degree at a time, the water continues to get colder but once it hits 32 degrees something extreme happens. The water turns to ice. Physicists call this a phase transition. While this is different than what Goldblum is describing with chaos theory it is another example of how small changes (like a butterfly flapping its wings) can lead to large effects - and sometimes these effects are unexpected.

Often we tend to think of the world in a linear fashion. If we push a toy car twice as hard, we expect it to roll twice as far across the floor. However, there are many examples in the world of science and technology where this linear thinking fails us. One of them concerns how vaccines keep us safe.

Vaccination rates for viruses like measles don’t behave in a linear manner. If we can set a vaccination rate high enough we can fully eliminate the spread of a disease like measles in an area. Imagine you live in a city with a million people and 99% of them are effectively vaccinated. If you happen to catch measles there are only 10,000 people who are not immune to infection. The chance that you happen to interact with one of them and pass on your infection is pretty small, almost zero. Imagine that we drop the vaccination rate by 1% to 98% of the population. Now 20,000 people are susceptible to you infecting them. We are still in a situation where it is unlikely that you will pass your infection to anyone else in the city. As long as we keep the vaccination rate in this city pretty high, our linear thinking matches reality. Small decreases in vaccination rates only lead to a small increase in the risk of infections spreading.

However, like dropping the temperature of water, vaccination rates can experience a phase transition too. There’s a magical level where vaccination rates drop low enough that infections begin to spread much more easily than a linear view of the world would suggest. It happens when an infected person is almost certain to contact a person to whom they can pass on their infection. When this happens a single case can turn into an outbreak and potentially seed a large scale epidemic.* This phase transition isn’t as stark as water turning to ice. We don’t go from no additional cases to a million cases by dropping the vaccination rate by a hundredth of a percent. However, the ascent of cases is far from linear and more steep than most expect.

We are starting to see some of this behavior in the West Texas outbreak. If enough people in the local area where the outbreak started were vaccinated, the outbreak wouldn’t have cascaded to other towns in Texas or other states like New Mexico and Oklahoma. We wouldn’t have hundreds of cases developing from one initial case.

Even more concerning, yesterday, a new research study was published in the Journal of the American Medical Association. The authors estimate the expected number of measles cases over the next 25 years as a function of the US vaccination rate. Measles vaccination rates have dropped slightly in recent years. If rates stay at the current level of today, the authors estimate that the US will suffer about 850,000 cases of measles in the next 25 years - about 34,000 cases per year. This is a dramatic increase from what we have experienced recently. Since 2000 there has only been one year with over 1,000 measles cases. If the vaccination rate were to fall an additional 10 percent, the authors estimate that the total would increase to 11.1 million cases over 25 years. This would be an average of about 444,000 cases per year. On the other hand if we were to increase our measles vaccination rate by just 5%, they estimate that only 5,800 cases of measles would occur over the next 25 years. This is an average of 232 cases per year, about what we have experienced over the past quarter century in the US.

Incidence of infectious diseases like measles don’t follow a simple linear pattern in relation to vaccinations. We are sitting on the precipice of a phase transition and the world we live in with respect to measles could look very different with even a slightly further decrease in vaccination rates.

*A more technical description would state that this happens when the reproduction number rises above one, meaning that each infected person passes their infection to one additional person on average. I have a more thorough non-technical description of this concept in the first chapter of The Rich Flee and the Poor Take the Bus.