The Stochastic Reproduction Rate of a Virus

We consider an SIR model where the probability of infections between infected and susceptible individuals are viewed as Poisson trials. The probabilities of infection between pairwise susceptible-infected matches are thus order statistics. This implies that the reproduction rate is a random variable. We derive the first two moments of the distribution of Rt conditional on the information available at time t-1 for Poisson trials drawn from an arbitrary parent distribution with finite mean. We show that the variance of Rt is increasing in the proportion of susceptible individuals in the population, and that ex ante identical populations can exhibit large differences in the path of the virus. This has a number of implications for policy during pandemics.