The Outcome of a Stochastic Epidemic—a Note on Bailey's Paper

In a recent paper (1953) N. Bailey has considered a stochastic epidemic model of the type set up by Bartlett (1949), and has shown that the probability distribution (Pw) of the ultimate number of infected individuals (w) may be calculated by solving a certain set of doubly recurrent relations. I propose to show that for quite a general case these same probabilities may be obtained by the solution of a set of singly recurrent relations (eqs. (17) and (24)). Furthermore, an expression may be derived (eqs. (38) and (40)) for the probability that an infection introduced into a large population will ‘take’—this provides a stochastic equivalent to Kermack & McKendrick's threshold theorem (1927 and later).