# Threshold phenomena in epidemic theory

Barbour, A D (1994). Threshold phenomena in epidemic theory. In: Kelly, F P. Probability, statistics and optimisation. Chichester: Wiley, 101-116.

## Abstract

The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.

## Abstract

The threshold theorem for deterministic epidemics in mixing populations can usually be rewritten in such a form that a large epidemic results from trace infection if and only if $R_0>1$, where $R_0$ can be interpreted as a basic reproduction ratio for an associated population model. The Whittle stochastic threshold theorem replaces certainty with probability: if $R_0\leq 1$, a large epidemic is highly unlikely to result from the introduction of one or two infectives, whereas, if $R_0>1$, the probability of having a significant epidemic is no longer trivial. In this paper, the Whittle approximation to a model for parasitic infection in a mixing population is analysed. A feature of the model is that $R_0$ is well defined, but for certain parameter values the threshold is not at $R_0=1$. Thus to have $R_0=1$ as threshold for epidemics in mixing populations is by no means a universal rule. A related birth and death process with drift is also investigated.