## “Herd Immunity” Varies With The Herd

In today’s New York Times, N. Popovitch and M. Sanger-Katz wrote an article about how The World Is Still Far From Herd Immunity for Coronavirus, in which they treat herd immunity as if it were a characteristic of the disease only, achieved when 60% of the population has antibodies.

## The CDC On Herd Immunity

The US CDC website defines herd immunity it as “a situation in which a sufficient proportion of a population is immune to an infectious disease (through vaccination and/or prior illness) to make its spread from person to person unlikely.”

The CDC  makes it clear is that the “sufficient proportion” depends on both the disease and the population in which it spreads. In other words, for a given disease herd immunity varies with the herd. The same proportion of immune individuals will not achieve it in populations with different lifestyles. It is higher if they commute in crowded buses to work shoulder-to-shoulder on assembly lines; lower, if they move in individual cars and work in private offices.

The CDC’s definition fails to say what they mean by unlikely. To reopen factories without making them COVID-19 hot spots, we need the workforce to have herd immunity. It means that its members must be unlikely to infect each other, not that 60% of them must have immunity.

## Herd Immunity In The SIR Model

Two of the charts from my previous post on this subject can clarify the issues. The first one shows the generic pattern of an epidemic over time in a population, in the classic SIR model.

The key parameter often mentioned today by people like Angela Merkel is R_{0} , pronounced “are-nought,” which can be interpreted as the expected number of people an infected person would transmit the disease to while infectious in a population where no one else is infected. In a population of size N The number of infected people peaks when the number S of susceptible individuals drops enough to have R_{0} = N/S . Some authors call the ratio r = 1- S/N of recovered people to the entire population at the peak of the epidemic the “herd immunity threshold.” Past this point, the epidemic ebbs, but infection can still be likely.

At the right side of the curve, where the number of infected people drops to 0 , the limit r_{\infty} of r varies between 0% and 100% depending on R_{0} . r_{\infty} describes the proportion of the population with acquired immunity that is necessary to confer herd immunity on the entire population. You don’t have to say how unlikely transmission is.

r_{\infty} is a final score directly observable only when the epidemic is over. R_{0} , on the other hand, can be estimated early on, albeit with wide margins of error. With a model of the epidemic,  r_{\infty} can then be inferred from R_{0} .  The second chart plots s_{\infty}  = 1- r_{\infty} as a function of R_{0} in the basic SIR model, with the ranges of R_{0} estimates published for the seasonal flu and COVID-1.

## When The Population Is The Workforce Of A Factory

All the practices introduced into a factory to prevent contagion at work lower the R_{0} of the disease within the workforce while working, which lowers both the herd immunity threshold and the level of actual immunity required to achieve herd immunity in the long run, and this is quantifiable.

Of course, outside of work, the employees of the factory are within society at large. They are subject to its contagion dynamics. The main problem of today, however, is factories turning into epidemic hot spots.