Figure 1.

Hypothetical, visual model for evolution of an attenuated virus to exceed the epidemic threshold. (A) For each individual vaccinated, there is a distribution of average transmission chain lengths that depends on the number of secondary contacts infected and how many contacts they infect. Thus, the vaccinated individual may transmit to (a) no one (chain length 1), (b) to one secondary contact but they fail to transmit (length 2), (c) to two secondary contacts, one of which does not transmit further (length 2) and one of which transmits to a single contact who does not transmit (length 3), and so on for all possibilities. The average chain length increases with the average R₀ of the virus. (B and C) For the viruses that have attained a particular chain length, there will have been some evolution occurring during that chain, resulting in a distribution of R₀ values increasing with chain length. If the chain is long enough, there may have been enough evolution that some viruses now have $R_0>1$, and if they get into the next patient, an outbreak may occur (depending on other factors). The net probability of evolving a virus that exceeds the epidemic threshold combines the probability of each chain length with the probability that viruses in that patient have $R_0>1$ and escape to the next patient. The net probability of escape will be vanishingly small if chain lengths are short and the virus either never evolves $R_0>1$ or is slow to do so.