Wednesday, October 28, 2009

How getting vaccinated protects others

In my last post I explored a possible model for the flu and how different levels of immunity had social impact. Here I want to drill down more on how getting vaccinated not only protects yourself, but even those who aren't immune.

Here I have a graph of the expected number of people to be infected for various immunization levels.

Here we see if no-one is immunized we have an expected infection level of 61 but if 30 people are immunized that number drops to 17.

Perhaps this isn't very illuminating. I mean of course the more people who are immunized the smaller an outbreak will be, since those who are immune won't get infected. But even more than that is happening. Let's look at the chances of a non-immune person getting infected given various immunization levels.

So here we see that an individual has a 61% chance of infection given no immunity in society. But as her neighbours get immunized her chances of getting infected drop, down to 9% when 50 other people are immunized. This is because each person who is immune (a) reduces the chances of an outbreak and (b) increases the chances that the outbreak will burn out sooner.

There's an even more powerful way to see this by looking at the multiplier effect of getting vaccinated. The multiplier is how much the expected number of people infected drops divided by the amount that the personal risk drops. For example wearing a seat belt has a constant multiplier effect of 1. If wearing a seat belt reduces my chance of harm from 1% to 0.5% then we expect the expected amount of people harmed to drop by 0.005. You wearing a seat belt protects no-one but yourself.

However, not drinking and driving has quite a high multiplier since the accident caused by a drunk driver is quite likely to harm more than just the drinker.

Here's the graph for the multiplier effect of getting vaccinated in the model.

For most of the graph the multiplier effect is somewhere between 2 and 4. So by this model, you getting vaccinated passes on the protection in at least equal levels. By making yourself safe you are protecting others in the process.

Lowering the transmission level also has a similar effect to raising immunization. Stuff like washing your hands helps in this regard (see the post from the spring).


Ben Piper said...

What's really interesting is how this model can be used not only to encourage vaccination, but also give people an excuse not to get vaccinated. I actually saw a tweet from someone who said they wanted everyone else to get vaccinated, but not themselves. People are very divided on the vaccination issue, with some saying they absolutely will not get vaccinated, while others are standing out in freezing rain for hours to receive the vaccine. Assuming vaccination is 100% effective, your model gives comfort to those who don't want to get vaccinated because they can rely on reduced prevalence of the virus. It looks like the infection risk drops to almost 0% once around 74 people are vaccinated. Can this be extrapolated out to say that if 74% of the population gets vaccinated, the risk of an unvaccinated individual getting infection is also near 0%?

Steven H. Noble said...

alas it is in the numbers where the model shouldn't be trusted. I have an inkling that you may be right that 75% is big enough to say that the other 25% don't have to worry that much. But I really don't have a good reason to believe that.

That said, the vaccine is certainly not 100% effective (another flaw in my model). I've heard numbers ranging from 90% to 35%. Vaccine efficacy is not something that I know anything about.

Christian Nally said...

Another factor not included in this model that would weigh on the discussion is any negative effect of the vaccine.

Great comment on the best option being that everyone else gets the vaccine but you. Ha ha!