This paper is concerned with a stochastic model for the spread of an epidemic with a contact tracing scheme, in which diagnosed individuals may name some of their infectious contacts, who are then removed if they have not been already. Traced individuals may or may not also be asked to name their own contacts. The epidemic is studied by considering an approximating, modified birth-death process with intersibling dependencies, for which a threshold parameter and expressions from which extinction probabilities may be calculated are derived. When all individuals can name their contacts, it is shown that this threshold parameter depends on the infectious period distribution only through its mean. Numerical studies show that the infectious period distribution choice can have a material effect on the threshold behaviour of an epidemic, while the dependencies help reduce spread.

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

The general stochastic SIR epidemic in a closed population under the influence of a term-time forced environment is considered. An ‘environment’ in this context is any external factor that influences the contact rate between individuals in the population, but is itself unaffected by the population. Here ‘term-time forcing’ refers to discontinuous but cyclic changes in the contact rate. The inclusion of such an environment into the model is done by replacing a single contact rate λ with a cyclically alternating renewal process with k different states denoted {Λ(t)}t≥0. Threshold conditions in terms of R⋆ are obtained, such that R⋆>1 implies that π, the probability of a large outbreak, is strictly positive. Examples are given where π is evaluated numerically from which the impact of the distribution of the time periods that Λ(t) spends in its different states is clearly seen.

A stochastic epidemic model is defined in which infected individuals have different severities of disease (e.g. mildly and severely infected) and the severity of an infected individual depends on the severity of the individual he or she was infected by; typically, severe or mild infectives have an increased tendency to infect others severely or, respectively, mildly. Large-population properties of the model are derived, using branching process approximations for the initial stages of an outbreak and density-dependent population processes when a major outbreak occurs. The effects of vaccination are considered, using two distinct models for vaccine action. The consequences of launching a vaccination program are studied in terms of the effect it has on reducing the final size in the event of a major outbreak as a function of the vaccination coverage, and also by determining the critical vaccination coverage above which only small outbreaks can occur.

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.

We consider a stochastic model for the spread of a susceptible–infective–removed (SIR) epidemic among a closed, finite population, in which there are two types of severity of infectious individuals, namely mild and severe. The type of severity depends on the amount of infectious exposure an individual receives, in that infectives are always initially mild but may become severe if additionally exposed. Large-population properties of the model are derived. In particular, a coupling argument is used to provide a rigorous branching process approximation to the early stages of an epidemic, and an embedding argument is used to derive a strong law and an associated central limit theorem for the final outcome of an epidemic in the event of a major outbreak. The basic reproduction number, which determines whether or not a major outbreak can occur given few initial infectives, depends only on parameters of the mild infectious state, whereas the final outcome in the event of a major outbreak depends also on parameters of the severe state. Moreover, the limiting final size proportions need not even be continuous in the model parameters.

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.