Moment closure approximations are used to provide analytic approximations to nonlinear stochastic models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and thirdorder moment closure approximations which we apply to the stochastic SI and SIS models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial. In addition, a novel closure approximation is developed in order to capture the behaviour of the stochastic SIS model at the critical point of persistence or extinction of the process. This mixture approximation, is a third-order approximation and comprises a probability distribution designed to capture the behaviour of the system conditioned on non-extinction (quasi-equilibrium) and a probability mass at 0 which represents the probability of extinction. Two versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results show: 1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and 2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model in marked contrast with existing approaches.

We aimed to determine (1) the prevalence of premenstrual syndrome in a sample of twins and (2) the relative contribution of genes and environment in premenstrual syndrome. A group of 193 subjects inclusive of same gender twins (n = 176) and females from opposite sex twin sets (n = 17) entered the study. Heritability analysis used same gender twin data only. The probandwise concordance rate for the presence or absence of premenstrual syndrome was calculated and the heritability of premenstrual syndrome was assessed by a quantitative genetic model fitting approach using MX software. The prevalence of premenstrual syndrome was 43.0% and 46.8% in monozygotic and dizygotic twins, respectively. The probandwise concordance for premenstrual syndrome was higher in monozygotic (0.81) than in dizygotic twins (0.67), indicating a strong genetic effect. Quantitative genetic modeling found that a model comprising of additive genetic (A) and unique environment (E) factors provided the best fit (A: 95%, E: 5%). No association was found between premenstrual symptom and the following variables: belonging to the opposite gender twin set, birth weight, being breast fed and vaccination. These results established a clear genetic influence in premenstrual syndrome.

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.

Abstract. Hand, foot and mouth disease (HFMD) is endemic in Sarawak, Malaysia. In this study, a geographical information system (GIS) was used to investigate the relationship between the reported HFMD cases and the spatial patterns in 11 districts of Sarawak from 2006 to 2012. Within this 7-years period, the highest number of reported HFMD cases occurred in