Extinction thresholds in deterministic and stochastic epidemic models

The basic reproduction number, R₀, one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if R₀>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if R₀>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1-(1/R₀)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. 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.