Quick Vaccine Math

A friend of mine asked me to write about the math of vaccines. A lot of people have been talking about it lately, so I’m not really sure if I’ve got anything new to add, but I can at least give my usual mathy spin to it.

Vaccines have been getting a lot of attention lately, for good reason. There are a lot of people in America who’ve bought in to a bogus line about the supposed danger of vaccines, and the supposed benign-ness of the diseases that they can prevent. That’s led to many children not getting vaccinated as they should, which has culminated in a recent outbreak of measles caused by a contagious but not yet symptomatic child at DisneyLand.

When it comes to vaccines, there’s two things that a lot of people don’t understand. One is herd immunity; the other is probability of infection.

Herd immunity is a really important concept. In an ideal world, if you got vaccinated against measles, there’d be no chance that you’d ever catch it. But it doesn’t work that way. What a vaccine does is reduce the probability that you’ll catch the disease. Due to lots of random factors involving the way that a given individual’s immune system works, the vaccine can’t be perfect. Beyond that, there are also many people who either can’t be immunized, or whose immune system is not functioning correctly. For example, people who are getting chemotherapy for cancer have severely depressed immune systems, and even if they’ve been immunized, their immune systems aren’t capable of preventing the disease.

So just relying on the fact that you’ve been immunized isn’t really enough. To prevent outbreaks of the disease, we rely on an emergent property of a vaccinated population. If enough people are immune to the disease, then even if one person somehow gets infected with it, they won’t be able to cause it to spread.

Let’s walk through a simple example. Suppose we’ve got a disease where the vaccine is 95% effective – that is, 95 out of every hundred people who received it are completely immune to infection by it. Let’s also suppose that this is a highly infectious disease: out of every 100 non-immune people who are exposed to it, 95 will become ill. If everyone is immunized, how many people need to be exposed to a sick person in order for the disease to spread?

Infections turn into outbreaks when the number of infected people grows – if each sick person infects more than one other person, then the infection will start to grow exponentially. The severity of the outbreak will depend on how many people get infected by each sick person.

Suppose that the first sick person has contact with 20 people while they’re contagious. 95% of them are immune – which means that only one out of that twenty is succeptible. There’s a 95% chance that that person will get infected. This isn’t good, but if it’s kept to that rate, we won’t have an outbreak: each sick person will probably infect one other person on average – and not always even that. So the infection will die out without exploding into a significant outbreak.

What happens in 5% of the population doesn’t get vaccinated? Then the pool of infected people grows to 10%. And in our contrived example, we now have a 90% probability of the sick person making two other people ill. That’s more than enough to cause a major outbreak! On average, each sick person will cause 1.8 other people to become sick!

When the population’s immunity rate (either through vaccine, or through prior infection) gets to be high enough that an infection can no longer spread, the population is said to have herd immunity: even individuals who can’t be immunized no longer need to worry about catching it, because the population doesn’t have the capacity to spread it around in a major outbreak.

(In reality, the effectiveness of the measless vaccine really is in the 95 percent range – actually slightly higher than that; various sources estimate it somewhere between 95 and 97 percent effective! And the infectivity of most diseases is lower than the example above. Measles (which is a highly, highly contagious disease, far more contagious than most!) is estimated to infect between 80 and 90 percent of exposed non-immune people. So if enough people are immunized, herd immunity will take hold even with more that 20 people be exposed by every sick person.)

Moving on: there’s a paradox that some antivaccine people use in their arguments. If you look at an outbreak of an illness that we vaccinate for, you’ll frequently find that more vaccinated people become ill than unvaccinated. Therefore, they say, it’s not the fault of the unvaccinated. We’ll look at the math to see the problem with that.

Let’s use the same numbers as above: 95% vaccine effectiveness, 95% contagion. In addition, let’s say that 2% of people choose to go unvaccinated.

That means thats that 98% of the population has been immunized, and 95% of them are immune. So now 92% of the population has immunity.

If each infected person has contact with 20 other people, then we can expect expect 8% of those 20 to be infectable – or 1.6; and of those, 95% will become ill – or 1.52. So on average, each sick person will infect 1 1/2 other people. That’s enough to cause a significant outbreak. Without the non-immunized people, the infection rate is less than 1 – not enough to cause an outbreak. The non-immunized population reduced the herd immunity enough to cause an outbreak.

Now, within the population, how many immunized versus non-immunized people will get sick?

Out of every 100 people, there are 5 who got vaccinated, but aren’t immune. Out of that same 100 people, there are 2% that didn’t get vaccinated, or 2. So we’d expect that in 100 cases of the disease, about 70 of them to be vaccinated, and 30 unvaccinated.

The vaccinated population is much, much larger – 50 times larger! – than the unvaccinated, so we’d expect more vaccinated people to become ill, even though it’s the smaller unvaccinated group that broke the herd immunity! The easiest way to see that is to take those numbers, and normalize them.

In the vaccinated community, those 70 sick people are – in the worst possible case, where every single non-immune vaccinated person became ill! – the 5% non-immune from a population of 1400 people. So the worst possible infection rate in the vaccinated population is just 5% – and in reality, it’s more like 4.75%. But those 30 sick people from the unvaccinated pool are 30 out of about 32 non-immunized people who were exposed. The unvaccinated people were more than 20 times more likely to be infected.

The reality of vaccines is pretty simple.

  1. Vaccines are highly effective.
  2. The diseases that vaccines prevent are not benign.
  3. Vaccines are really, really safe. None of the horror stories told by anti-vaccine people have any basis in fact. Vaccines don’t damage your immune system, they don’t cause autism, and they don’t cause cancer.
  4. Not vaccinating your children (or yourself!) doesn’t just put you at risk for illness; it dramatically increases the chances of other people becoming ill. Even when more vaccinated people than unvaccinated become ill, that’s largely caused by the unvaccinated population.

In short: everyone who is healthy enough to be vaccinated should get vaccinated. If you don’t, you’re a despicable free-riding asshole who’s deliberately choosing to put not just yourself but other people at risk.

14 thoughts on “Quick Vaccine Math

  1. davidjnichollsDave Nicholls

    Hi,

    I’m not sure if I’m missing something here, but the following section does’t make sense to me:

    “Let’s use the same numbers as above: 95% vaccine effectiveness, 95% contagion. In addition, let’s say that 2% of people choose to go unvaccinated.

    That means thats that 98% of the population has been immunized, and 95% of them are immune. So now 92% of the population has immunity.

    If each infected person has contact with 20 other people, then we can expect expect 8% of those 20 to be infectable – or 1.6; and of those, 95% will become ill – or 1.52. So on average, each sick person will infect 1 1/2 other people. That’s enough to cause a significant outbreak. Without the non-immunized people, the infection rate is less than 1 – not enough to cause an outbreak. The non-immunized population reduced the herd immunity enough to cause an outbreak.”

    If 98% of the population are immunised with a 95% effective vaccine, doesn’t this mean that:

    98%*95% = 93.1% are immune
    98%*5% = 4.9% vaccinated but not immune
    2% unvaccinated

    Therefore the percentage of people at risk is 6.9%, rather than the 8% you give in the final paragraph. I think this makes the average number o people infected 1.311 rather then 1.52

    Dave

    Reply
  2. Sean Roark

    I know quite a few of the vaccine conspiracy theory crowd and am very happy to have this math based retort!

    However, I can tell you this part is going to be contentious:

    “Vaccines are really, really safe. None of the horror stories told by anti-vaccine people have any basis in fact. Vaccines don’t damage your immune system, they don’t cause autism, and they don’t cause cancer.”

    Because lots of people do not trust the messenger when it comes to vaccine safety. You can point to all the studies you want, they will find where either the government or a big corporation has its fingers in there. Then they assume the data is manipulated for some devious purpose. Some people think vaccines are a massive profit center so big companies are manipulated data to cover up issues with them. Others fear the government wants to use vaccines as a way to give people mind control drugs (or other such things).

    Others are just driven by fear and a ‘where there is smoke there is fire’ mentality. And its not hard to find smoke about vaccines as plenty of influential voices are spreading lies.

    This paranoia is hard to overcome.

    Reply
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  8. douglasjebarnes

    “Therefore, they say, it’s not the fault of the unvaccinated.”

    Unfortunately, explaining the math has little effect. These people sit at the intersection of Innumeracy and Dunning-Kruger.

    Reply
  9. Anonymous coward

    “What happens in 5% of the population doesn’t get vaccinated? Then the pool of infected people grows to 10%.”

    That’s not how percentages work.

    Reply
    1. markcc Post author

      Actually, it is how percentages work. If you think I made a math error, feel free to point it out. I’ve always been willing to admit when I make a mistake on this blog.

      As far as I can see, the only error is the use of the word “infected” instead of “infectable”, which I admit is a silly mistake. But in the ongoing reasoning from there, I am reasoning from them being infectable.

      If you have a pool with 5% vaccinated but non-immune, and 5% unvaccinated, then you’ve got a total pool of 10% infectable. That’s exactly how percentages work.

      Reply
      1. StephanF

        Some of the 5% unvaccinated people may have been from the pool of the 5% of all people who would not be immune even if vaccinated.

        Reply
  10. EMF

    (A) If 5% are unvaccinated, and 5% of the vaccinated aren’t immune, then 9.75% total aren’t immune; percentages often multiply (95%^2, not 100%-(100%-95%)*2).
    (B) If it’s the flu that you don’t get vaccinated for, and you’re not in health-care, you’re not really free-riding; even with 100% vaccination for the flu, we’d still have an epidemic every winter.

    Reply
  11. PavelS

    Statistics is weird beast. It usually highlights majority and hides minority.
    People often don’t care much about it until you start to play this statistics game with their children.
    And mass vaccination is just this sort of game.
    Mass vaccination is a thing of type “risk few to save many”.
    Everybody is nodding and agree with this principle until one realize that the game is played also with his children.
    At that very time, herd is forgotten, because for parents, herd is nothing. It isn’t real, it doesn’t exist. Only their children do.
    Then parents tends to ask questions like: “Is it safe to vaccinate MY one week old daughter?”
    or “Hey doctor, MY first daughter had 40’C fever for two days after vaccination, is it really good idea to vaccinate my new born son in the same way like her?” and alikes.
    Notice that there is no statistics in these questions. No herd involved. Just real people.
    I just want to say that I really understand the parents’ fear.

    Reply

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