Computation of final outcome probabilities for the generalised stochastic epidemic

This paper is concerned with methods for the numerical calculation of the final outcome distribution for a well-known stochastic epidemic model in a closed population. The model is of the SIR (Susceptible→Infected→ Removed) type, and the infectious period can have any specified distribution. The final outcome distribution is specified by the solution of a triangular system of linear equations, but the form of the distribution leads to inherent numerical problems in the solution. Here we employ multiple precision arithmetic to surmount these problems. As applications of our methodology, we assess the accuracy of two approximations that are frequently used in practice, namely an approximation for the probability of an epidemic occurring, and a Gaussian approximation to the final number infected in the event of an outbreak. We also present an example of Bayesian inference for the epidemic threshold parameter.     

Bayesian Inference for Stochastic Epidemics in Populations with Random Social Structure

A single-population Markovian stochastic epidemic model is defined so that the underlying social structure of the population is described by a Bernoulli random graph. The parameters of the model govern the rate of infection, the length of the infectious period, and the probability of social contact with another individual in the population. Markov chain Monte Carlo methods are developed to facilitate Bayesian inference for the parameters of both the epidemic model and underlying unknown social structure. The methods are applied in various examples of both illustrative and real-life data, with two different kinds of data structure considered.     

A class of stochastic epidemic models and its deterministic counterpart

A general stochastic model for the spread of an epidemic developing in a closed population is introduced. Each model consisting of a discrete-time Markov chain involves a deterministic counterpart represented by an ordinary differential equation. Our framework involves various epidemic models such as a stochastic version of the Kermack and McKendrick model and the SIS epidemic model. We prove the asymptotic consistency of the stochastic model regarding a deterministic model; this means that for a large population both modelings are similar. Moreover, a Central Limit Theorem for the fluctuations of the stochastic modeling regarding the deterministic model is also proved.     

Fast Inference for Network Models of Infectious Disease Spread

Models of infectious disease over contact networks offer a versatile means of capturing heterogeneity in populations during an epidemic. Highly connected individuals tend to be infected at a higher rate early during an outbreak than those with fewer connections. A powerful approach based on the probability generating function of the individual degree distribution exists for modelling the mean field dynamics of outbreaks in such a population. We develop the same idea in a stochastic context, by proposing a comprehensive model for 1‐week‐ahead incidence counts. Our focus is inferring contact network (and other epidemic) parameters for some common degree distributions, in the case when the network is non‐homogeneous ‘at random’. Our model is initially set within a susceptible–infectious–removed framework, then extended to the susceptible–infectious–removed–susceptible scenario, and we apply this methodology to influenza A data.     

Bayesian inference for stochastic multitype epidemics in structured populations via random graphs

Summary.  The paper is concerned with new methodology for statistical inference for final outcome infectious disease data using certain structured population stochastic epidemic models. A major obstacle to inference for such models is that the likelihood is both analytically and numerically intractable. The approach that is taken here is to impute missing information in the form of a random graph that describes the potential infectious contacts between individuals. This level of imputation overcomes various constraints of existing methodologies and yields more detailed information about the spread of disease. The methods are illustrated with both real and test data.     

Ultimate Extinction of the Promiscuous Bisexual Galton—Watson Metapopulation

A variant of a sexual Gallon–Watson process is considered. At each generation the population is partitioned among n‘hosts’ (population patches) and individual members mate at random only with others within the same host. This is appropriate for many macroparasite systems, and at low parasite loads it gives rise to a depressed rate of reproduction relative to an asexual system, due to the possibility that females are unmated. It is shown that stochasticity mitigates against this effect, so that for small initial populations the probability of ultimate extinction (the complement of an ‘epidemic’) displays a tradeoff as a function of n between the strength of fluctuations which overcome this ‘mating’ probability, and the probability of the subpopulation in one host being ‘rescued’ by that in another. Complementary approximations are developed for the extinction probability: an asymptotically exact approximation at large n, and for small n a short‐time probability that is exact in the limit where the mean number of offspring per parent is large.     

Estimating likelihoods for spatio-temporal models using importance sampling

This paper describes how importance sampling can be applied to estimate likelihoods for spatio-temporal stochastic models of epidemics in plant populations, where observations consist of the set of diseased individuals at two or more distinct times. Likelihood computation is problematic because of the inherent lack of independence of the status of individuals in the population whenever disease transmission is distance-dependent. The methods of this paper overcome this by partitioning the population into a number of sectors and then attempting to take account of this dependence within each sector, while neglecting that between-sectors. Application to both simulated and real epidemic data sets show that the techniques perform well in comparison with existing approaches. Moreover, the results confirm the validity of likelihood estimates obtained elsewhere using Markov chain Monte Carlo methods.     

A STOCHASTIC LOGISTIC MODEL WITH TIME DELAY

The stochastic version of the logistic model for population growth is generalized to take account of continuously distributed time delay with an exponentially decaying kernel. The theory of diffusion processes is used to analyse the probability density function of the population size. The explicit expression for the stationary distribution is worked out and the effect of time delay on various statistics is discussed.     

Estimation of Balanced Simultaneous Confidence Sets for SIR Models

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we present a method for estimating balanced simultaneous confidence sets for the mean sample path of a stochastic SIR model, thus providing a simple representation of both the typical behavior and the variability of the epidemic. The confidence sets are estimated by a bootstrap procedure, using asymptotic properties of density dependent jump Markov processes. The method is applied to chickenpox epidemics in France and the coverage probability of the confidence sets is estimated in that context.     

ROBUST ESTIMATION FOR EPIDEMIC MODELS

A robust approach to the analysis of epidemic data is suggested. This method is based on a natural extension of M-estimation for i.i.d. observations where the distribution may be asymmetric. It is discussed initially in the context of a general discrete time stochastic process before being applied to previously studied epidemic models. In particular we consider a class of chain binomial models and models based on time dependent branching processes. Robustness and efficiency properties are studied through simulation and some previously analysed data sets are considered.     

ULM,a software for conservation and evolutionary biologists

Matrix models for population dynamics have recently been studied intensively and have many applications to theoretical and applied problems (conservation, management). The computer program ULM (Unified Life Models) collects a good part of the actual knowledge on the subject. It is a powerful tool to study the life cycle of species and meta-populations. In the general framework of discrete dynamical systems and symbolic computation, simple commands and convenient graphics are provided to assist the biologist. The main features of the program are shown through detailed examples: a simple model of a starling population life cycle is first presented leading to basic concepts (growth rates, stable age distribution, sensitivities); the same model is used to study competing strategies in a varying environment (extinction probabilities, stochastic sensitivities); a meta-population model with migrations is then presented; some results on migration strategies and evolutionary stable strategies are eventually proposed.     

Bayesian Inference for a Stochastic Epidemic Model with Uncertain Numbers of Susceptibles of Several Types

A stochastic epidemic model with several kinds of susceptible is used to analyse temporal disease outbreak data from a Bayesian perspective. Prior distributions are used to model uncertainty in the actual numbers of susceptibles initially present. The posterior distribution of the parameters of the model is explored via Markov chain Monte Carlo methods. The methods are illustrated using two datasets, and the results are compared where possible to results obtained by previous analyses.     

On probabilistic parametric inference

This paper formulates a theory of probabilistic parametric inference and explores the limits of its applicability. Unlike Bayesian statistical models, the system does not comprise prior probability distributions. Objectivity is imposed on the theory: a particular direct probability density should always result in the same posterior probability distribution. For calibrated posterior probability distributions it is possible to construct credible regions with posterior-probability content equal to the coverage of the regions, but the calibration is not generally preserved under marginalization. As an application of the theory, the paper also constructs a filter for linear Gauss–Markov stochastic processes with unspecified initial conditions.     

Approximation of transition densities of stochastic differential equations by saddlepoint methods applied to small-time Ito–Taylor sample-path expansions

Likelihood-based inference for parameters of stochastic differential equation (SDE) models is challenging because for most SDEs the transition density is unknown. We propose a method for estimating the transition density that involves expanding the sample path as an Ito–Taylor series, calculating the moment generating function of the retained terms in the Ito–Taylor expansion, then employing a saddlepoint approximation. We perform a numerical comparison with two other methods similarly based on small-time expansions and discuss the pros and cons of our new method relative to other approaches.     

Perfect simulation for Reed-Frost epidemic models

The Reed-Frost epidemic model is a simple stochastic process with parameter q that describes the spread of an infectious disease among a closed population. Given data on the final outcome of an epidemic, it is possible to perform Bayesian inference for q using a simple Gibbs sampler algorithm. In this paper it is illustrated that by choosing latent variables appropriately, certain monotonicity properties hold which facilitate the use of a perfect simulation algorithm. The methods are applied to real data.     

Inference for comparing a multinomial distribution with a known standard

We propose a new type of stochastic ordering which imposes a monotone tendency in differences between one multinomial probability and a known standard one. An estimation procedure is proposed for the constrained maximum likelihood estimate, and then the asymptotic null distribution is derived for the likelihood ratio test statistic for testing equality of two multinomial distributions against the new stochastic ordering. An alternative test is also discussed based on Neyman modified minimum chi-square estimator. These tests are illustrated with a set of heart disease data.     

Likelihood estimation for stochastic compartmental models using Markov chain methods

This paper presents a method for estimating likelihood ratios for stochastic compartment models when only times of removals from a population are observed. The technique operates by embedding the models in a composite model parameterised by an integer k which identifies a switching time when dynamics change from one model to the other. Likelihood ratios can then be estimated from the posterior density of k using Markov chain methods. The techniques are illustrated by a simulation study involving an immigration-death model and validated using analytic results derived for this case. They are also applied to compare the fit of stochastic epidemic models to historical data on a smallpox epidemic. In addition to estimating likelihood ratios, the method can be used for direct estimation of likelihoods by selecting one of the models in the comparison to have a known likelihood for the observations. Some general properties of the likelihoods typically arising in this scenario, and their implications for inference, are illustrated and discussed.     

Bayesian semiparametric modeling for stochastic precedence,with applications in epidemiology and survival analysis

We propose a prior probability model for two distributions that are ordered according to a stochastic precedence constraint, a weaker restriction than the more commonly utilized stochastic order constraint. The modeling approach is based on structured Dirichlet process mixtures of normal distributions. Full inference for functionals of the stochastic precedence constrained mixture distributions is obtained through a Markov chain Monte Carlo posterior simulation method. A motivating application involves study of the discriminatory ability of continuous diagnostic tests in epidemiologic research. Here, stochastic precedence provides a natural restriction for the distributions of test scores corresponding to the non-infected and infected groups. Inference under the model is illustrated with data from a diagnostic test for Johne’s disease in dairy cattle. We also apply the methodology to the comparison of survival distributions associated with two distinct conditions, and illustrate with analysis of data on survival time after bone marrow transplantation for treatment of leukemia.     

Cross-infection in epidemics spread by carriers

A block-structured transient Markov process is introduced to describe an epidemic spreading within two linked populations, of carriers and susceptibles. The epidemic terminates as soon as there are no more carriers or susceptibles present in the population. Our purpose is to determine the distribution of the final susceptible and carrier states, and of any integral path for the susceptible process. The transient epidemic state is also briefly discussed. Then, the model is extended to allow the recovery of infected individuals. Finally, several particular models, some known, are used for illustration.     

Testing the Stochastic Structure of Production: A Flexible Moment-Based Approach

Conventional production function specifications are shown to impose restrictions on the probability distribution of output that cannot be tested with the conventional models. These restrictions have important implications for firm behavior under uncertainty. A flexible representation of a firm's stochastic technology is developed based on the moments of the probability distribution of output. These moments are a unique representation of the technology and are functions of inputs. Large-sample estimators are developed for a linear moment model that is sufficiently flexible to test the restrictions implied by conventional production function specifications. The flexible moment-based approach is applied to milk production data. The first three moments of output are statistically significant functions of inputs. The cross-moment restrictions implied by conventional models are rejected.