  Analytical expressions for the basic reproduction number, R0, have been
obtained in the past for a wide variety of mathematical models for infectious
disease spread, along with expressions for the expected final size of an
outbreak. However, what has so far not been studied is the average number of
infections that descend down the chains of infection begun by each of the
individuals infected in an outbreak (we refer to this quantity as the "average
number of descendant infections" per infectious individual, or ANDI). ANDI
includes not only the number of people that an individual directly contacts and
infects, but also the number of people that those go on to infect, and so on
until that particular chain of infection dies out. Quantification of ANDI has
relevance to the vaccination debate, since with ANDI one can calculate the
probability that one or more people are hospitalised (or die) from a disease
down an average chain of infection descending from an infected un-vaccinated
individual. Here we obtain estimates of ANDI using both deterministic and
stochastic modelling formalisms. With both formalisms we find that even for
relatively small community sizes and under most scenarios for R0 and initial
fraction vaccinated, ANDI can be surprisingly large when the effective
reproduction number is >1, leading to high probabilities of adverse outcomes
for one or more people down an average chain of infection in outbreaks of
diseases like measles.
