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).