Chain Binomial Model Research Articles

Chain Binomial Models and Binomial Autoregressive Processes

We establish a connection between a class of chain-binomial models of use in ecology and epidemiology and binomial autoregressive (AR) processes. New results are obtained for the latter, including expressions for the lag-conditional distribution and related quantities. We focus on two types of chain-binomial model, extinction-colonization and colonization-extinction models, and present two approaches to parameter estimation. The asymptotic distributions of the resulting estimators are studied, as well as their finite-sample performance, and we give an application to real data. A connection is made with standard AR models, which also has implications for parameter estimation.

Analytical methods for a stochastic mainland–island metapopulation model

We study a class of chain-binomial metapopulation models, giving special attention to the ‘mainland–island’ configuration, where patches receive immigrants from an external source. We evaluate the distribution of the number n t of occupied patches at any census time t and establish a law of large numbers that identifies a deterministic trajectory which can be used to approximate the process when the number of patches is large. We also establish a central limit law, which shows that the fluctuations about this trajectory are approximately normally distributed. We describe briefly much finer results that can be used for model calibration.

Limit theorems for discrete-time metapopulation models

We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.

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A chain-binomial model for intra-household spread of Mycobacterium tuberculosis in a low socio-economic setting in Pakistan.

A simulation study using Greenwood's chain-binomial model was carried out to elucidate the spread and control of Mycobacterium tuberculosis among the household contacts of infectious pulmonary tuberculosis (TB) patients. Based on the observed data, the maximum-likelihood estimates (+/-S.E.) of chain-binomial probabilities of intra-household M. tuberculosis transmission from an index case in 3-person and 4-person households were 0.313+/-0.008 and 0.325+/-0.009 respectively. The chi2 goodness-of-fit test of observed and simulated mean expected frequencies of cases revealed good fit for 3-person (P=0.979) and 4-person (P=0.546) households. With the assumption of varying risk of M. tuberculosis transmission across the households under beta-distribution, goodness-of-fit tests of observed and mean simulated expected frequencies revealed the inadequacy of Greenwood's chain-binomial model both for 3-person (P=0.0185) and 4-person (P<0.001) households. Simulated M. tuberculosis control strategy comprising efficient diagnosis, segregation and prompt antibiotic therapy of index pulmonary TB patients showed a substantial reduction of new cases among the household contacts in both household sizes. In conclusion, segregation coupled with prompt antibiotic therapy of the index case, chemoprophylaxis of M. tuberculosis-exposed household contacts, and the assessment of household environmental risks to devise and implement an educational programme may help reduce the TB burden in this and similar settings.

Estimation and inference of R0 of an infectious pathogen by a removal method

The basic reproductive ratio, R 0, is a central quantity in the investigation and management of infectious pathogens. The standard model for describing stochastic epidemics is the continuous time epidemic birth-and-death process. The incidence data used to fit this model tend to be collected in discrete units (days, weeks, etc.), which makes model fitting, and estimation of R 0 difficult. Discrete time epidemic models better match the time scale of data collection but make simplistic assumptions about the stochastic epidemic process. By investigating the nature of the assumptions of a discrete time epidemic model, we derive a bias corrected maximum likelihood estimate of R 0 based on the chain binomial model. The resulting ‘removal’ estimators provide estimates of R 0 and the initial susceptible population size from time series of infectious case counts. We illustrate the performance of the estimators on both simulated data and real epidemics. Lastly, we discuss methods to address data collected with observation error.

A latent variable model for estimating disease transmission rate from data on household outbreaks

A Bayesian latent variable model is proposed for studying household epidemics of infectious diseases in this paper. This model is more general and flexible than the commonly used chain binomial epidemic model. In particular, the model allows for the heterogeneity of the infection transmission rates in related to the sizes and generations of the infectives. Moreover, the model assumes the availability of only the household outbreak sizes, which is more reasonable than assuming the availability of the hardly observed infection chains. The Tanner–Wong's IP algorithm is employed for effective simulations and inferences of this model. Finally, this model was applied to analyzing a real data set on F-4 Asian influenza.

Epidemics Between Bailey and Reed-Frost: A Stochastic Version of the Kermack and Mckendrick Model

A model is developed that describes evolution with respect to time of an infectious disease introduced into a population of susceptibles. The proposed model incorporates at one end Bailey's simple stochastic epidemic and at the other end the Reed-Frost chain-binomial models and is the natural stochastic analogue of Kermack and McKendrick's deterministic model. The epidemic process is characterized by the size of the population and by two infectivity functions. The first one relates to a time dependent outside source of infection. The second one describes the infectivity of an individual as a function of his age-of-infection, that is the time elapsed since his own infection. The proposed model consists of a set of partial differential equations which governs, steered by the given infectivity functions, the evolution with respect to time of a set of density functions. These density functions deliver a complete stochastic description of the infectious-age structure of the population at any moment of time. An expression for the size of the epidemic, that is the probability distribution of the number of infectives, as a function of time follows. Also expressions for the expected arrival times of infectives, useful for the inverse problem, are developed. By letting time tend to infinity earlier results for the final size of the epidemic are confirmed.

Some aspects of inference for chain binomial models

A class of estimators for a chain binomial model, which includes both the maximum likelihood estimator and a previously proposed approximation to the maximum likelihood estimator is obtained from a martingale estimating equation. Extensions of the 1110del are proposed and illustrated on real data. The role of conditional inference is discussed and further examined by a simulation study

A Spatio-temporal Epidemic Model for Influenza

A mathematical model is built for influenza or other similar disease epidemics in a multi-region setting. The model is an extended type of chain-binomial model applied to a large population (Cliff et al., 1981), taking into account interregional infection by interregional contacts of people. If the magnitude of the contact is presented by simple distance-decay spatial interaction or the most primitive gravity model, a conventional gravity-type epidemic model (Murray and Cliff, 1977; Thomas, 1988) is deduced.Given the number of infectives and susceptibles, the chain-binomial model predicts the number of infectives in the next period with binomial probability distribution. Available data are, however, weekly cases per reporting clinic in each prefecture reported by the surveillance project, characterized by continuous variation; the data could be a surrogate index for rates of infection. The author modified the model to use rates of infectives and susceptibles, and used a normal approximation of binomial distribution. With the maximum-likelihood method, this model can be calibrated. The specification of the model is as follows:Li(Yi, t=0, …, Yi, t=T|β°i, δi)=Πt1/√2πVar[Yi, t+1]·exp{-1/2Var[Yi, t+1](Yi, t+1-E[Yi, t+1])}, E[Yi, t+1]=β°i/MiXi, tΣjmijYj, t, Var[Yi, t+1]=β°i/MiXi, tΣjmijYj, t(1-β°i/MiΣjmijYj, t), Xi, t=δi-Σis=0Yi, s, where Mi=Σjmij; Li denotes the likelihood of the model for region i; Xi, t denotes the estimated rate of susceptibles in region i at week t; Yi, t denotes the reported rate of infectives in region i at time t; mij denotes the size of interregional contact with the people in regions j for the people in region i; β°i denotes the infection parameter in region i; δi denotes the parameter concerned with the rate of initial susceptibles in region i.The model posits that the average number of people who come into contact with a susceptible in prefecture i is a constant, and that the average rate of infectives of the people is ΣjmijYj, t/Mi. The probability of a susceptible in region i infected at time t is, therefore, β°iΣjmijYj, t/Mi.This model was applied to a weekly incidence of influenza in each prefecture, from the 41st week, 1988, to the 15th week, 1989, Japan, letting the size of interregional passenger flow Tij correspond to mij as follows: mij=Tij+Tji (i≠j), mii=Tii.Goodness-of-fits (Table 1) of one-week-ahead forecasts were almost satisfactory except for prefectures whose epidemic curves were bi-modal (e.g., Hokkaido) or whose transition speed between epidemic breakout and peak was too high (e.g., Yamagata). The latter might be explained by a cluster of group infection (e.g., school classes) in an earlier phase of the epidemic (see Fig. 4).

ASYMPTOTIC INFERENCE FOR A CLASS OF CHAIN BINOMIAL MODELS

SummaryChain binomial models axe commonly used to model the spread of an epidemic through a population. This paper shows that for a flexible class of chain binomial models an approximate maximum likelihood estimator of the infection rate, derived from a Poisson approximation to the binomial distribution, has an asymptotically normal distribution, as do some other related'estimators.

Fixed accuracy estimation for chain binomial models

In many epidemic models the initial infection rate, suitably defined, plays a major role in determining the probability of an outbreak of a disease becoming a major epidemic. Here we model the epidemic as a chain binomial model and consider an approximate maximum likelihood estimator of the infection rate. It is shown that under mild conditions sampling according to a simple stopping rule yields an asymptotically normally distributed estimator which may be computed during the course of an epidemic. A small simulation study suggests that the asymptotic results applied to small samples yield accurate confidence intervals.

ROBUST ESTIMATION FOR EPIDEMIC MODELS

Summary 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.

A generalized chain binomial model with application to HIV infection

The original Reed-Frost formulation of the chain binomial model is mathematically equivalent to a stochastic model allowing a Poisson number of effective contacts in a time interval. Their formulation cannot accomodate survey data that necessarily correspond to more complex distributions of partners or contacts, or to large populations where complete random mixing is unlikely. This paper generalizes the Reed-Frost model to accomodate these situations in both the one- and two-population settings. The extension to multiple populations is also outlined. Using the model to predict HIV incidence in San Francisco's homosexual population, we show that the total number of contacts over all partners is more important than the distribution of contacts among partners in determining the number of infecteds.

Analysis of Infectious Disease Data.

1. Introduction 2. Chain binomial models 3. Chain models with random effects 4. Latent and infectious periods 5. Heterogeneity of disease spread through a community 6. Generalized Linear Models 7. Martingale methods 8. Methods of inference for large populations

A Markov chain model of extrabinomial variation

This paper studies the properties of a {0, 1}-state Markov chain model of extrabinomial variation. If n is even, the Markov chain binomial model is always overdispersed relative to the binomial model with parameters n and p, while if n is odd, it may be over- or underdispersed relative to the binomial model. Expressions for the likelihood and variance functions are obtained, and for n=3, the Markov chain binomial model is compared with the additive and multiplicative binomial models of Altham (1978), as well as to the beta-binomial model.

Simple two dimensional models for the spread of a disease

A simple two dimensional stochastic model is proposed to model the spread of a disease amongst a fixed population. Two versions of this model are obtained which make assumptions corresponding to the assumptions made in the Greenwood and Reed-Frost Chain Binomial epidemic models. The epidemic is controlled by immunization so as to minimize the expected number of people infected. Some results, conjectures and counterexamples concerning the optimal policy are presented.

A note on chain-binomial models of epidemic spread: What is wrong with the Reed-Frost formulation?

Derivations of the Reed-Frost model are analyzed in terms of the assumptions about the probabilistic process used and in terms of internal consistency. Internally consistent derivations can be exhibited but require assumptions about the basic probabilistic process that are clearly unreasonable. A properly posed, more general model is derivable through the use of En'ko's model and the generating function for the number of contacts per person [6].

Asymptotic final size distribution of the multitype Reed–Frost process

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

Asymptotic final size distribution of the multitype Reed–Frost process

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.