This page is a companion to my The Conversation article on “herd immunity”, providing more mathematical details and comments which could not fit into the publisher format.

The term “herd immunity” is used in three different ways. Firstly, in a more generic sense, it simply means the proportion of immune individuals in the population.

Secondly, for an ongoing epidemic, it means a threshold proportion of immune individuals at the point when the incidence starts declining because of the diminishing supply of susceptibles (as opposed to the reduction in the transmission due to lockdown).

Thirdly, it is the level of immunity after the outbreak that prevents the emergence or re-emergence of the disease.

Interestingly, the formula that determines the level of immunity in both the second and third sense is the same. For a simple SIR model with the reproductive rate , it is given by

For more complicated models, including spatial and social heterogeneities and other factors, the formula is the same, except that we need to be careful at how is defined and what it actually means.

The final epidemic size, that is the number of individuals who have been infected at some point (including those who recover or die), is given implicitly as a solution to the equation

with representing the population size. This equation has not got an analytical solution, but can be solved numerically to obtain the graph:

It should be stressed that this result is only valid for diseases that result in long-lasting immunity, at least at time scales comparable with the outbreak duration. If the immunity is not permanent, individuals will be able to reinfect and no level of herd immunity prevents the recurrence of the disease.

Unless, we manage to completely eradicate the virus world-wide. How difficult this is, is shown by only two diseases that we managed to successfully eradicate, smallpox and rinderpest, with polio perhaps close.