Extinction thresholds in deterministic and stochastic epidemic models

Abstract

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.