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.

Environmental heterogeneity, spatial connectivity, and movement of individuals play important roles in the spread of infectious diseases. To account for environmental differences that impact disease transmission, the spatial region is divided into patches according to risk of infection. A system of ordinary differential equations modeling spatial spread of disease among multiple patches is used to formulate two new stochastic models, a continuous-time Markov chain, and a system of stochastic differential equations. An estimate for the probability of disease extinction is computed by approximating the Markov chain model with a multitype branching process. Numerical examples illustrate some differences between the stochastic models and the deterministic model, important for prevention of disease outbreaks that depend on the location of infectious individuals, the risk of infection, and the movement of individuals.

Salmonella Typhimurium (STM) infection in pigs represents a considerable food safety concern. This study used mathematical modelling to evaluate the effectiveness of cleaning (faeces removal) as a measure to control STM spread among grower-finisher pigs. A modified Susceptible-Infected-Recovered-Susceptible (SIRS) model of STM transmission through a contaminated environment was developed. Infected pigs were divided into three states according to the pathogen level being shed in their faeces. Infection transmission was evaluated using the basic reproduction number (R 0) and the prevalence of infectious pigs at slaughter age. Although increased frequency and efficiency of cleaning did reduce the prevalence of STM shedding at the time of slaughter, these efforts alone were not capable of eliminating the infection from the population. The level of STM faecal shedding by infectious pigs strongly influenced the infection spread and prevalence at slaughter. To control STM in pigs, cleaning should be combined with vaccination and/or isolation of high-level shedders.

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Indirect transmission through the environment, pathogen shedding by infectious hosts, replication of free-living pathogens within the environment, and environmental decontamination are suspected to play important roles in the spread and control of environmentally transmitted infectious diseases. To account for these factors, the classic Susceptible-Infectious-Recovered-Susceptible epidemic model is modified to include a compartment representing the amount of free-living pathogen within the environment. The model accounts for host demography, direct and indirect transmission, replication of free-living pathogens in the environment, and removal of free-living pathogens by natural death or environmental decontamination. Based on the assumptions of the deterministic model, a continuous-time Markov chain model is developed. An estimate for the probability of disease extinction or a major outbreak is obtained by approximating the Markov chain with a multitype branching process. Numerical simulations illustrate important differences between the deterministic and stochastic counterparts, relevant for outbreak prevention, that depend on indirect transmission, pathogen shedding by infectious hosts, replication of free-living pathogens, and environmental decontamination. The probability of a major outbreak is computed for salmonellosis in a herd of dairy cattle as well as cholera in a human population. An explicit expression for the probability of disease extinction or a major outbreak in terms of the model parameters is obtained for systems with no direct transmission or replication of free-living pathogens.