The simple stochastic epidemic curve for large populations of susceptibles

Abstract

one assumes that each of the x susceptibles at a given time can receive the infection from any of the y infectives then in circulation; thus, the rate of accrual of new cases is proportional to the product xy-we shall take the factor of proportionality (the 'infection-rate') to be unity, as may clearly always be done through a proper choice of the time scale. One feels that the accrual rate should be construed as a probabilistic transition rate: under this construction, the notifications of new cases, which one would read in the newspapers from day to day during the course of an epidemic, constitute a histogram which is the statistical image of the underlying 'epidemic curve'; we may therefore regard the epidemic curve as the frequency function of a certain random variable, viz. the time of occurrence of a new case. Owing to the presence of quadratic factors in the state equations, the mathematics involved in determining the epidemic curve turns out to be surprisingly difficult, even in the 'simple' case which we shall consider here, i.e. where there is no removal of discovered infectives from circulation. Haskey (1954) has indeed given an ingenious expression for the simple stochastic epidemic curve in the case of one initial infective and n susceptibles; and Bailey (1963) has since obtained a rather easier derivation of Haskey's result. The formula is, however, sufficiently complicated to obscure the properties of the solution; and the arithmetic entailed in evaluating it becomes extremely heavy when n is large (which is, after all, the connotation of the word 'epidemic'). One might attempt to side-step the issue by considering instead the corresponding deterministic problem, where the rate of accrual of new cases is interpreted as a growth rate and is identified with the epidemic curve itself: the solution is now quite straightforward, but unfortunately also quite different from the stochastic one, as we shall see before long. In this note we shall solve the stochastic problem for an arbitrary initial number, a, of infectives, on the assumption that the initial number, n, of susceptibles is large; we shall find that the asymptotic form of the epidemic curve in this (very natural) limit is extremely simple and elegant, and that only when a is also large does it go over into the deterministic case. We begin by considering the latter, since it provides a convenient frame of reference. Calling r the number of susceptibles still uninfected at time t, we have