Health and Science News for Parents

Guest Post: A math lesson in vaccines and infectious disease

written by Tara Haelle

Today’s post is a guest post by Craig Egan, fearless fighter of anti-vaccine myths and misinformation. Craig tends to have a bit more bite and snark than I do in these posts, but he does an excellent job here of using math to break down exactly why it’s not ethical to choose not to vaccinate your children while claiming that you’re not putting others at risk.

Anti-vaccine advocates are always asking, “If vaccines work so well, why are you concerned about my unvaccinated kid? Aren’t yours protected?”

Original image by Lusi (Sanja Gjenero).

Original image by Lusi (Sanja Gjenero).

For this discussion, let’s leave out the usual list of people who cannot be vaccinated (such as those who are immuno-compromised or have a medical condition that contraindicates vaccination), the list of people the anti-vaccine advocates will callously consider not their problem. Let’s not get into the idea that maybe it’s okay for me to worry about their kids too, kids who, through no fault of their own, are the poker chips these parents are wagering while they confidently try to call a bluff on virtually every expert on the planet. Let me, for a moment, be as thoughtless and selfish as they are, and answer the question being asked: Is their unvaccinated child putting my child at risk, if the vaccines I believe in work so damn well?

No. Probably not. But they are putting somebody’s vaccinated child at risk.

Let’s do some real simple math here. The variables for infection rates and vaccine effectiveness vary by disease, and the equations get pretty complicated as they interact with each other, so I’m going to round off and keep it as simple as possible. Statistically, if you expose 100 unvaccinated people to measles, 90 of them will catch it. If you expose 100 people who have had one dose of the vaccine, 5 of those partially vaccinated people will still catch measles. If you expose 100 people who have had two doses of the vaccine, 1 of those fully vaccinated people will still catch measles. Those are the numbers, roughly. Different diseases are different. Measles is on the extreme end of both factors: It is extremely contagious, but the vaccine also works extremely well.

Let’s say a child goes on a trip to some exotic place where measles is common due to low vaccination, like India or Southern California. If that child comes in contact with measles, an unvaccinated child will almost certainly (90% chance) catch the disease, where a vaccinated child will almost certainly not (1% chance) catch the disease. The vaccinated child comes back from his vacation with a statue of Vishnu or some Mickey Mouse ears, while the unvaccinated child comes back as Patient Zero.

Let’s assume the kid has become contagious before returning from his trip. He gets on a Boeing 737 with 200 other people. Even if every single person on that flight is vaccinated, 2 of them will contract measles. Anyone who is unvaccinated will almost certainly (90% chance) catch it. You go through the airport, get your luggage, and possibly infect a few more on the way.

Monday comes around, and it’s time to go back to school. Let’s go with the national average, and say 90% of the students are vaccinated, and that child comes in contact with 500 students. Of the 50 students that are unvaccinated, 45 of them will catch measles. Out of the 450 students that are fully vaccinated, 4 of them will still catch measles.

You can adjust these numbers how you see fit: for how infectious the disease is, how effective the vaccine is, or how many people the child will come in contact with, but you will never come to an equation that justifies this kind of reckless endangerment.

I think we can stop there, and not get into shopping malls, taking a subway, visiting our newborn cousin in the hospital, etc. And we still have that list of people who cannot be vaccinated that I generously ignored for this exercise, but sure as anything would not be ignored when one of them died in the real world.

Statistically speaking, my kids are a hundred times more likely to be one of the 446 vaccinated children at that school who were protected than the unlucky 4 the vaccine did not work for. So yeah, I guess you got me there. I’m not worried your unvaccinated child is going to infect mine. But that doesn’t mean I can’t be pissed on behalf of the 49 students your bluff call did infect. I can be concerned about the people on the plane you infected who went off and started their own school outbreaks. And I can do everything possible to hold you accountable for the damage you’ve caused for absolutely no good reason. When you try to call a bluff like that and are wrong, you are expected to pay up.


11 Responses to “Guest Post: A math lesson in vaccines and infectious disease”

  1. Andrew Lazarus

    The highly effective measles vaccine is NOT the best example. Anti-vaxers thing there is something significant when a majority of the sick cohort was vaccinated, ignoring the important measure of relative risk. And for that, a better example is something like the pertussis vaccine that’s only 80% effective (probably less), and has 95% coverage. Now when you do the numbers, a majority of the sick kids are vaccinated, even though the relative risk is 4x as high. (In the latest California outbreak, the observed risk was about 2x—and anti-vaxers are so innumerate, they feel validated by that!)

    • Andrew, you make a valid point. I used measles as an example to show that even when a vaccine is highly effective, you are still putting people at risk. The situation you are describing is one that I also find myself addressing as such:

      To best explain the numbers, take the unvaccinated out of the equation for a minute. Let’s say you have a disease that infects 10,000
      people a year. Then you create a vaccine that is 90% effective, and everyone takes it. The next year you only have 1,000 cases.

      People who understand statistics will see a 90% drop in cases and call the vaccine a success.

      People who do not understand statistics will see that 100% of the cases were in people who got the vaccine, and call it a failure.

      Before the vaccine for pertussis, for example, there were between 4000-9000 deaths from the disease every year in the US alone.
      Now that we are vaccinated, there are maybe 10 deaths in the worst years. How many of those were vaccinated is irrelevant. What
      matters is how drastically the number of cases is reduced.

      • Andrew Lazarus

        I do the exact same computation arguing with innumerate antivaxers, except 1000 kids, 80% effective vaccine. It doesn’t change any minds, but at least it gets the statistical idiocy off the table.

        • Yes, directly changing the mind of an anti-vaxxer is a Sisyphean task to be sure, but hopefully it minimizes any onlookers thinking they might have a point. Keep up the fine work.

        • Chris

          That is a good article, just as is the one on this page.

          A few years ago I made up a similar explanation on why it seemed that more who were vaccinated get pertussis. I tried very hard to make it as simple as possible. The anti-vaxers have tried to mock it, but without any substantial reason why it should be mocked:

          Some community immunity arithmetic:

          Take 1000 people (ignoring the infants under 2 months who cannot be vaccinated, or babies under a year who can only be partially vaccinated), if 5% refuse vaccines then the numbers are:

          950 vaccinated persons (assuming full schedule)
          50 unvaccinated persons

          The pertussis vaccine is actually only 80% effective at worse, so the numbers are:

          760 protected persons
          190 vaccinated but vulnerable persons
          50 unvaccinated persons

          There is an outbreak and it gets spread to 20% of the population, then:

          760 protected persons without pertussis

          38 vaccinated persons get pertussis
          152 vaccinated person who may still get pertussis

          10 unvaccinated persons get pertussis
          40 unvaccinated persons who may still get pertussis.

          This is how more vaccinated persons get the disease than unvaccinated. Even if the infection rate was at 100%, there would still be more of the vaccinated getting the diseases because there are more of them!

          (and yes, I have a text file that I keep references and tables I have made up over the years, including the US Census data on measles incidence in the 20th century… which I made up when someone on ScienceBasedMedicine claimed the pro-vaxers only used data after 1950)

          • Tara Haelle

            I created a similar one with the simplest numbers I could use that’s generic (not tied to any particular illness) and doesn’t necessarily assume a constant for the total population infected applying to all sub populations equally. It lets me show how relative risk looks among unvaxed vs. vaxed:

            “Regarding the numbers who get infected, you cannot compared the number of vaccinated people who get ill to the number of unvaccinated people who get ill. You have to compare the percentages because in any outbreak, FAR FAR FAR more of the individuals involved will have been vaccinated, so of course more of them in *number* will get ill.

            Here’s an example: You have a community of 10,000 people. Let’s say 9,600 of them are vaccinated. An outbreak occurs. Let’s say that 200 of the people who are unvaccinated get sick and 1,000 of the people who were vaccinated get sick. WOW – that means FIVE TIMES as many people who were vaccinated got the illness than weren’t vaccinated, right?

            But wait. 1,000 out of 9,600 vaccinated people is 10%. Meanwhile, 200 out of 400 unvaccinated people is 50%.

            So, just one out of every ten vaccinated people got sick, but HALF of the unvaccinated people got sick. So in reality, being unvaccinated put you at five times greater risk for the illness.”

  2. Teddifish

    Just one thing to add: Maybe some of us are also worried about *your* kid catching diseases. Maybe we don’t think that having idiot parents makes it ok to suffer from a preventable disease.

  3. Chris

    I have also noticed that many who will not vaccinate claim that the possibility of the getting diseases very low in the USA. They seem to neglect that is because they live in a area where the vast majority do vaccinated.

    They don’t like it when I ask them if they have thanked their responsible neighbors for protecting their own children by vaccinating.

  4. […] the case that for a given individual they're less likely to get sick if vaccinated than not.…tious-disease/ prosciencemum is online now   Quote Quick […]

  5. If a parent had a vial of a deadly disease and they opened it and intentionally exposed their children or someone else’s children to this disease on purpose, I don’t think there would be much debate as to whether this was child abuse. Yet antivaxxers essentially do this all the time to both their own and other people’s kids and “everything is fine.” Refusal to vaccinate your children IS child abuse.

  6. mike stevens

    A good analogy I make is to say this:
    “You know more people are killed while driving sober than there are while drink-driving? So is that a rational argument for telling people they are safer while driving pissed out of their heads?”
    Using antivax logic, that would seem to be the case.


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