A random graph model for the final-size distribution of household infections

Predecessors and Successors in Random Mappings with Exchangeable In-Degrees

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

Application of an Individual-Based Transmission Hazard Model for Estimation of Influenza Vaccine Effectiveness in a Household Cohort

Household cohort studies are an important design for the study of respiratory virus transmission. Inferences from these studies can be improved through the use of mechanistic models to account for household structure and risk as an alternative to traditional regression models. We adapted a previously described individual-based transmission hazard (TH) model and assessed its utility for analyzing data from a household cohort maintained in part for study of influenza vaccine effectiveness (VE). Households with ≥4 individuals, including ≥2 children <18 years of age, were enrolled and followed during the 2010-2011 influenza season. VE was estimated in both TH and Cox proportional hazards (PH) models. For each individual, TH models estimated hazards of infection from the community and each infected household contact. Influenza A(H3N2) infection was laboratory-confirmed in 58 (4%) subjects. VE estimates from both models were similarly low overall (Cox PH: 20%, 95% confidence interval: -57, 59; TH: 27%, 95% credible interval: -23, 58) and highest for children <9 years of age (Cox PH: 40%, 95% confidence interval: -49, 76; TH: 52%, 95% credible interval: 7, 75). VE estimates were robust to model choice, although the ability of the TH model to accurately describe transmission of influenza presents continued opportunity for analyses.

Nonstationarity and randomization in the Reed-Frost epidemic model

The purpose of this paper is to determine the exact distribution of the final size of an epidemic for a wide class of models of susceptible–infective–removed type. First, a nonstationary version of the classical Reed–Frost model is constructed that allows us to incorporate, in particular, random levels of resistance to infection in the susceptibles. Then, a randomized version of this nonstationary model is considered in order to take into account random levels of infectiousness in the infectives. It is shown that, in both cases, the distribution of the final number of infected individuals can be obtained in terms of Abel–Gontcharoff polynomials. The new methodology followed also provides a unified approach to a number of recent works in the literature.

The effects of local spatial structure on epidemiological invasions

Predicting the likely success of invasions is vitally important in ecology and especially epidemiology. Whether an organism can successfully invade and persist in the short-term is highly dependent on the spatial correlations that develop in the early stages of invasion. By modelling the correlations between individuals, we are able to understand the role of spatial heterogeneity in invasion dynamics without the need for large-scale computer simulations. Here, a natural methodology is developed for modelling the behaviour of individuals in a fixed network. This formulation is applied to the spread of a disease through a structured network to determine invasion thresholds and some statistical properties of a single epidemic.