Extinction thresholds in deterministic and stochastic epidemic models

Abstract

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 [Formula: see text]. 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.

Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease

“…In these studies, the basic reproduction number was used to predict disease persistence or extinction. However, R 0 is an expected value and the predictions based on R 0 may not be valid for populations with a small initial number of infectious hosts or a low initial level of environmental contamination [2,3,33,48]. For instance, if one infectious host is introduced into a completely susceptible population, that host may die or recover before transmitting the infection to a susceptible host.…”

“…In stochastic epidemic theory, there are also thresholds for a major outbreak or disease extinction [2,3]. These stochastic thresholds are similar to the basic reproduction number, but depend on the initial number of infectious hosts and the initial level of environmental contamination.…”

“…For instance, if one infectious host is introduced into a completely susceptible population, that host may die or recover before transmitting the infection to a susceptible host. Similarly, if a small number of free-living pathogens are introduced into the environment with susceptible hosts present, those pathogens may die or be removed from the environment through decontamination prior to an indirect transmission [2,3,33,48]. Therefore, in order to determine the probability of an outbreak or disease extinction for these infections, it is necessary to consider a stochastic model.…”

“…These stochastic thresholds are similar to the basic reproduction number, but depend on the initial number of infectious hosts and the initial level of environmental contamination. The theory of multitype branching processes is used to determine the stochastic threshold [2,3]. Whittle [67] showed that for the Susceptible-Infectious-Recovered model with an exponentially distributed infectious period, the probability of disease extinction can be determined by the basic reproduction number.…”

“…Therefore, in order to determine the probability of an outbreak or disease extinction for these infections, it is necessary to consider a stochastic model. For more information related to stochastic epidemic modelling and stochastic fade-out of epidemics, see [1][2][3]7,27,33,41,48].…”