When an epidemic rages through a population, at first it faces no immunity at all. The disease constantly encounters fresh meat. There is nothing that can stop it.
Eventually, the population gains some immunity, yada yada yada, the disease doesn’t spread as easily and the epidemic slows down. There is a point when the disease, on average, infects just one more person per infected person. You have heard this before. It is the “herd immunity threshold.”
What is more surprising is that even once the population has hit this immunity threshold, the epidemic continues to grow — for a time. Epidemics have a kind of momentum that pushes infections even past this threshold. If you know about this already, you probably learned about it the way I did: from reading experts discuss the COVID-19 pandemic.
I immediately liked the “momentum” but found myself having a difficult time thinking precisely about it. Then, while reading about mathematical ecology the other day, I learned something that helped it all snap into place for me: this happens with people too.
The notion of population momentum makes a lot more sense to me in a human case. Probably if I was a virus the epidemic case would be easier, but I am what I am. Wikipedia has a great exposition of it, including this handy chart:
In the first generation, the fertility rate is 4 and the 200 fertile people give birth to 400 children — some pretty robust population growth, given the age distribution of the population. Then, at time = 1, the fertility rate drops and parents have only two kids each, merely replacing the fertile population as the old population ages out (dies). Even though the fertility rate has dropped, there are still the result of the previous fertility boom at t = 0. Those 400 children are going to have two children each, and that’s going to help the population grow for a bit longer. Soon enough, though, the fertile population will just be replacing itself.
This phenomenon was first described by Nathan Keyfitz in 1971. He directs the idea to policymakers who are reluctant to offer contraception for fear that their countries will stop increasing in population. “In some countries hesitation in making contraception available is rationalized by the view that the country is not yet “full,” he writes. “Concern that total numbers will taper off prematurely is misplaced.” He goes on to explain how to calculate the total “ultimate” population once fertility reaches replacement levels.
It’s this exact same phenomenon that governs the growth of a virus, even after (say) a vaccine is introduced that brings the rate of infection down to 1. I find it interesting that some of the same population dynamics govern both humans and viruses. It suggests to me that a path towards better educating others about epidemic dynamics would be to start with human stuff.