Vaccines and herd immunity

Many of us have heard of “herd immunity,” but few of us understand it. I didn’t either until I got curious, did a little reading, and figured some stuff out. The basics aren’t hard to understand if you understand exponential functions. With vaccination rates going down and disease rates going up, it’s important that we understand this stuff so we can make rational choices. Remember asking, “When are we going to use this stuff?” Now’s the time. Let’s go.

For our example, let’s look at measles.

1. A typical person with measles will infect about 15 other people while infected, assuming none of those people are immune. That number, about 15 for measles, is called the basic reproductive rate, or R₀, of the disease. (R₀ depends on how easily the disease is transmitted and how many people a typical infected person comes into contact with.) Most of those infected will be infected within about two weeks. To keep things simple, let’s assume that each infected person infects exactly 15 people in exactly two weeks. Let’s also assume that after infecting 15 new people, an infected person ceases to be infected. Of course, each of those 15 newly infected people will infect 15 other people in the next two weeks, and so on. Assuming nobody is immune, after one person is infected, how many people will be infected after 4 weeks (2 two-week periods)? 8 weeks (four two-week periods)?
2. Define a function f(t) which gives the number of people infected after t weeks, assuming nobody is immune.
3. Still assuming nobody is immune, how many weeks would it take to infect the entire population of the United States, around 350,000,000? (Hint: if you know about logarithms, use ’em! Otherwise, try graphing software or a graphing calculator. Or use the brute force approach: keep multiplying by 15, representing two more weeks, until you top 350,000,000.) (Of course, the more people who are infected, the more unrealistic our assumption that nobody is immune. Still, those are sobering numbers!)
4. The average number of people actually infected by each person with the disease, called the effective reproductive rate, or R, is less than R₀ if a fraction of the population is immune (either vaccinated or already had the disease). Let H be the fraction of the population that is immune (so if, say, 2/3 of the population were immune, H would be 2/3). Based on R₀ and H, what is a formula for R? (Hint: what fraction is not immune?)
5. No longer assuming everybody is immune, now based on R, define a function g(t) which gives the number of people infected after t weeks.
6. What values of R make g(t) an increasing function? A decreasing function? What does that question have to do with the spread of the disease?
7. At least what fraction, which we’ll call H, of the population must be immune for g(t) to be a decreasing function?
8. That fraction is called the herd immunity threshold for the disease, or HIT. Do you see how the herd immunity threshold is important for public health?

Answers below. Please try the problems on your own first!

1. Four weeks: R₀^4/2 = 15² = 225. Eight weeks: R₀^8/2 = 15⁴ = 50625.
2. f(t) = R₀^t/2 = 15^t/2
3. Solve for \(t\): \[\begin{eqnarray*} f (t) & = & 350, 000, 000\\ R_0^{\frac{t}{2}} & = & 350, 000, 000\\ \frac{t}{2} & = & \log_{R_0} (350, 000, 000)\\ t & = & 2 \log_{R_0} (350, 000, 000)\\ & \approx & 14.5 \end{eqnarray*}\]
   That’s 14.5 weeks to infect all of the US! Sobering.
4. R = R₀(1-H) (Note that 1-H is the fraction of the population that is not immune.)
5. g(t) = R^t/2 = (R₀(1-H))^t/2. Notice that this is the same function as f, but using the effective reproductive rate R instead of the basic reproductive rate R₀.
6. Increasing: R > 1. Decreasing: R < 1.
7. We need to solve R₀(1-H) < 1. So
   \[\begin{eqnarray*} R_0 (1 – H) & < & 1\\ 1 - H & < & \frac{1}{R_0}\\ - H & < & \frac{1}{R_0} - 1\\ H & > & 1 – \frac{1}{R_0}\\ H & > & \frac{14}{15} \\ H & > & 93 \% \end{eqnarray*}\]
   The lower bound for the value H, in this case 14/15 or about 93%, is the herd immunity threshold (HIT) for measles.
8. When more than the HIT, about 93%, are immune, R < 1 so each infected person infects, on average, fewer than one person, so disease outbreaks are unlikely. When fewer than 93% are immune, R > 1 so each infected person infects, on average, more than one person, so the number of infected people grows over time. And increasing exponential functions are relentless.