Extinction thresholds in deterministic and stochastic epidemic models

The basic reproduction number, R-0, one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if R-0 > 1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if R-0 > 1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1 - (1/R-0)(i). With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i(j), j = 1,...,n, and on the probability of disease extinction for each group, q(j). It follows from multitype branching processes that the probability of a major outbreak is approximately 1 - q(1)(i1) ... q(n)(in). In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.