The state of equilibrium (where R=1) can help us explore the immunisation level required for the eradication of an infectious agent.

From the definition of R (see previous slide), it is:
R = R₀ (S/N) [S: number of susceptibles; I: number of immunes; N: total population].

In the state of equilibrium, it is:
R = R₀ (S/N)eq = 1 [(S/N)eq: the fraction of susceptibles in the state of equilibrium] or
R = R₀ [1 - (I/N)eq] = 1 [(I/N)eq: the fraction of immunes in the state of equilibrium; we assume that, at equilibrium, the number of infected cases is negligible, therefore S+I=N or S/N+I/N=1] or
R₀ = 1 / [1 - (I/N)eq].

If we solve for (I/N)eq, we get:
(I/N)eq = 1 - (1/R₀).

On reflection, we realise that (I/N)eq is the necessary proportion of immunes in the population in order to have equilibrium (R=1). In other words, if the proportion of immunes were higher than this, R would be smaller than 1, and the microorganism would be eventually eradicated.

If this level of population immunity is achieved by mass vaccination, we can consider (I/N)eq as the critical proportion of vaccination level above which we would eventually accomplish eradication of the microorganism (P_C). Therefore, P_C = 1 - (1/R₀).