The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.