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Herd immunity depends on how the herd behaves

September 2, 2020

As most now have read, a contagious disease continues to spread within a population so long as each individual infected on average passes the disease to at least one other. That measure, the “r-nought,” determines whether the actively infected population declines, stays constant, or increases, as it is below 1, equal to 1, or higher. Everything else remaining the same, the r-nought decreases as the immune fraction of population increases. Which is why contagious diseases with short periods of infection come and go. “Herd immunity” is the notion that a disease cannot propagate in a population once there is a large enough fraction immune to it.

Recently, I have seen quite a few posts and articles that take as an assumption that the fraction of population where that obtains for a particular disease is a constant. It isn’t. That fraction depends on the behavior of the population. A disease that is under control in a herd of cows that has been grazing on open land may run rampant when they are collected into a feed lot. A nation that seems to have achieved herd immunity regarding Covid-19 through rigorous shutdown may see that reverse when those controls are lifted. A state where millions have shifted to working from home, emptying skyscrapers, cannot assume herd immunity remains if that is reversed.

Herd immunity depends on social behavior. At what fraction it occurs naturally will vary between different cultures. And can be lost or gained with changes in social practice.

I am not going to link to any authority stating the above, because it simply is inherent in the math. My explanation above likely isn’t needed by those of my friends who studied computer science or physics, or who otherwise are mathematically adept, because they are accustomed to thinking about problems involving graphs and probability. Be leery if you are wont to write on this topic, and you are not so inclined.

Update: Julia Kriz Dzierwa makes a terminology correction: “R-nought, or R0, is the initial reproduction rate. Rt is the reproduction rate at a given time. R is the reproduction rate in general. It would be accurate to say R decreases as the immune fraction of population increases.”

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