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|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),
In the state of equilibrium, it is:
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 (PC). Therefore, PC = 1 - (1/R0).