In my last post, I wrote about the 60% effectiveness rate for flu shots that news media commonly report. The effectiveness is actually a relative measure of the reduction in your flu risk if you’re vaccinated. Relative measures are hard to interpret without additional information. With that in mind, I reanalyzed the data to put it in absolute terms. I found that if you get a flu shot, your average annual risk of getting the flu drops from 7.0% to 1.9%, which is a 5.1% reduction.

I’ve received several requests to look at this over a longer timeframe. After all, flu shots aren’t a one-time thing. Last time, I concluded that if you regularly get the flu shot, you’ll probably spare yourself a week of misery at some point. Let’s use Minitab statistical software to quantify the long-term probabilities!

## How We Will Model the Flu Outcomes Over 20 Years

Flu shots are like a retirement investment where you keep investing year after year, and it’s the cumulative effect that magnifies the differences in the annual percentages.

Financial planners often show you two scenarios based on two different courses of action. Return rates, like influenza rates, fluctuate over the years. Consequently, financial planners use a reasonable long-term average that gives you a ballpark idea of the different outcomes.

That’s what I’ll do in the probability distribution plots below. The graphs show different outcomes for influenza infections over the course of decades. The underlying assumptions are that the average infection rate is 7.0% for the unvaccinated and 1.9% for the vaccinated. We’ll see what that 5.1% difference translates into over decades.

I hope regular readers of my blog realize that these data are binary (you’re either infected with the flu or not) and that we can use the binomial and geometric distributions to model the outcomes.

## Long-Term Comparisons: No Flu Shot versus Regular Flu Shots

The two scenarios that I will compare are never getting the flu shot and always getting the flu shot. We’ll answer two questions to illustrate the different outcomes:

- How long until I can expect my first case of the flu?
- How many times can I expect to get the flu?

For both graphs, the left panel is for the no flu shot scneario and the right panel is for the regular flu shot scenario.

## How long until I first get the flu?

The first graph uses the geometric distribution to model the number of years until you first get the flu. Each bar represents the probability of getting the flu for the first time in that specific year. The probability for any given year is small, but I’ve shaded the graphs to indicate the number of years until your cumulative probability of getting the flu for the first time is 50/50. I’ve cut the graphs off at 65 years because studies show that flu shots become less effective above that age.

If you never get flu shots, you have a 50/50 chance of getting your first case of the flu in 11 years. However, if you always get the flu shot, you have a 50/50 chance after 38 years. By way of comparison, if you don’t get vaccinated, you only have a 6.8% chance of going 38 years or longer without getting the flu.

You can also see how much more your probabilities are front-loaded with the vaccinations, while the right tail is much thicker in the vaccinated panel. You want higher probabilities in the later years, not the earlier years!

## How many times will I get the flu?

The second graph uses the binomial distribution to model the number of times you can expect to get the flu over 20 years. Each bar represents the probability of getting the flu a specific number of times over 20 years. I’ve shaded the graphs to indicate the cumulative probability of getting the flu two or more times over 20 years.

The big difference that jumps out is the large bar that represents zero cases of the flu if you’re vaccinated. If you’re vaccinated regularly, you have a 68% chance of not getting the flu within 20 years. However, if you’re never vaccinated, you only have a 23% chance of not getting the flu in the same time frame.

You can also see how much the distribution spreads out if you’re not vaccinated. If you’re vaccinated, you only have a 5.4 percent chance that you’ll get the flu two or more times over 20 years. On the other hand, if you never get vaccinated, you’ll have a 41.3% chance of getting the flu two or more times.

## Closing Thoughts on the Long-Term Effectiveness of Flu Shots

When you flip a coin 20 times, you’ll never know for sure how many times you’ll get the heads side up. However, using the binomial distribution, you can model the most likely outcomes. The same is true with the flu.

With the flu, there are some additional complications. There are plenty of other viruses circulating that can make you sick. The probabilities in this post are strictly related to you getting the specific flu viruses that the flu shots are designed to protect against—three per shot. Let’s assume that you got the flu shot every year for 20 years, and that you never got a flu that you were vaccinated against. Unfortunately, you may well get another strain of the flu that was not included in a shot, or a different type of virus altogether that causes “flu-like” symptoms.

Also, the graphs show that even if you never get a flu shot, it’s not too unlikely (50/50) that after 11 years you will not yet get the flu. However, don’t take this to mean that there are no benefits to vaccinating yourself. If you are regularly vaccinated, you can expect your first case to occur significantly later and you can expect to have the flu a fewer number of times.

As I said in my last post, I’ll continue to get the flu shot because it’s a simple way to prevent some misery!