Chain Binomial Model Research Articles

Threshold behaviour for a chain-binomial S-I-S infectious disease.

The present paper is concerned with a chain-binomial deterministic model for an infectious disease of the S-I-S type. The model extends the Cooke et al. (1977) and Longini (1980) model in the sense that it accounts for the distribution of the number of contacts made by each susceptible during an infectious period and for the probabilities of infection at the different contacts with infectives. The aim of the work is to investigate under which conditions the disease becomes endemic or not. Some partial results are first derived, but a complete analysis of the threshold behaviour seems very intricate. A more detailed discussion is then presented for the specific case where at least K + 1 contacts with infectives, K greater than or equal to 1 fixed, are required to make a susceptible infected.

The asymptotic final size distribution of reducible multitype Reed-Frost processes.

The asymptotic final size distribution of a multitype Reed-Frost process, a chain-binomial model for the spread of infection in a finite, closed multitype population, is derived in the case of reducible contact pattern between types. The results are obtained using techniques developed for the irreducible case.

A discrete time model for a S-I-S infectious disease with a random number of contacts between individuals

A chain-binomial deterministic model for the spread of an infectious disease of the S-I-S type is formulated that accounts explicitly for the distribution of the number of contacts made by each susceptible during one time interval. Under certain hypotheses, a threshold theorem for endemicity is derived, bounds for the endemic level are constructed, and the transient behavior of the epidemic process is investigated.

The threshold behaviour of epidemic models

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

The threshold behaviour of epidemic models

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.

Random contact processes, snowball sampling and factorial series distributions

A contact process first posed as a random mapping problem is discussed together with a model for the spread of a rumour through an archaic telephone system. Snowball sampling and the well-known Reed–Frost chain binomial model are also commented on. Our main interest and emphasis is on distributional properties of discrete-time models. Simple combinatorial arguments are used along with a class of discrete distributions originally proposed for catch–recapture sampling.

Random contact processes, snowball sampling and factorial series distributions

A contact process first posed as a random mapping problem is discussed together with a model for the spread of a rumour through an archaic telephone system. Snowball sampling and the well-known Reed–Frost chain binomial model are also commented on. Our main interest and emphasis is on distributional properties of discrete-time models. Simple combinatorial arguments are used along with a class of discrete distributions originally proposed for catch–recapture sampling.

A threshold theorem for the Reed-Frost chain-binomial epidemic

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

A threshold theorem for the Reed-Frost chain-binomial epidemic

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.

A General Chain Binomial Model for Infectious Diseases

A multiparameter chain binomial model is formulated to describe outbreaks of an infectious disease in households. This model can be easily fitted to data as the maximum likelihood estimates and associated standard errors have simple explicit expressions. The model contains the Greenwood and Reed-Frost models as particular cases, so the appropriateness of either of these models may be tested by simple parametric inference. The usefulness of this general chain binomial model is illustrated with an application to household data for the common cold.

AN APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATOR FOR CHAIN BINOMIAL MODELS

SummaryUsing the Poisson approximation to the Binomial distribution, we construct an approximate maximum likelihood estimator (MLE) for a class of chain binomial models. Our estimator proves to have properties which may make it preferable to the exact WLE.

A chain binomial model of endemicity

The classical Reed-Frost chain binomial epidemic model is modified to allow infected individuals to become susceptible again following recovery from their infection. Markov chain methods are applied to obtain probabilities for the duration of the epidemic. The average duration time of the epidemic is found to be infinite for certain population sizes at a fixed level of infectious contact, establishing a critical population size needed to maintain endemicity. The epidemic model is also formulated and analyzed deterministically. The stochastic and deterministic models are compared using a numerical example, the stochastic model being more appropriate for small populations. However, the two models are found to be equivalent for larger populations.

An Epidemic Chain Model

An epidemic chain model is developed by assuming a beta distribution for the probability of being infected by contact with a given infective from the same household. This model includes, as a particular case, the epidemic chain model corresponding to the stochastic Kermack-McKendrick model and, as a limiting case, the Reed-Frost chain binomial model. The advantages of the more general model are illustrated with an application to household data for the common cold. Finally, it is shown how the coefficient of variation of the duration of the infectious period may be estimated without any direct observations on this duration.

An estimator for chain-binomial epidemic models

Using the Poisson approximation to the binomial distribution, we obtain an approximate maximum likelihood estimator for the infection parameter in a chain-binomial epidemic. We examine some properties of this estimator and show that it has certain advantages over the exact MLE.

THE RISK OF STREPTOCOCCAL INFECTIONS IN RHEUMATIC AND NON-RHEUMATIC FAMILIES: AN APPLICATION OF GREENWOOD'S CHAIN-BINOMIAL MODEL1

Infectious transmission of streptococci in families has been demonstrated to follow Greenwood's chain-binomial model in its simplest form which did not need beta transformation to correct for differences in susceptibility between families. Following the application of this model, an estimate of the risk of acquisition of M-typable group A beta-hemolytic streptococci is given (p = 0.05--0.06). The risk is the same among siblings of rheumatic fever patients and children from non-rheumatic families. This emphasizes that the familial aggregation associated with rheumatic fever is not due to increased susceptibility of such families to streptococcal infections.

An estimation procedure for household disease data

SUMMARY A procedure is proposed for estimating a parameter of an epidemic model from household data. The procedure is essentially a method of moments for martingales. It is computationally convenient and is less restrictive in its model assumptions than the maximum likelihood methods used so far. The approach is illustrated with measles data and is also shown to be useful when households contain susceptibles of different types. This paper is concerned with data generated by the spread of an infectious disease within households. Each of the households initially has one or more infectives, and susceptibles may become infected by adequately close contacts with infectives from the same household. A newly infected individual passes through a stage during which he is latent and then through a stage during which he is infectious, before being removed by isolation, by death or by naturally losing his infectiousness and becoming immune for the remaining duration of the epidemic. Such data are usually analysed by maximum likelihood methods using one of two types of epidemic models to describe the spread of the disease within a household. Either a chain binomial model, like those initially introduced by Greenwood and Reed-Frost, is used or the Kermack-McKendrick model, as formulated in its stochastic form by Bartlett (1949), is used. An extensive discussion of these models and methods is given in the book by Bailey (1975). The method proposed in this paper is based on an approach suggested for counting processes in a Berkeley Ph.D. thesis by 0. 0. Aalen and is essentially a method of moments for martingales constructed from the epidemic model. There are two advantages of the resulting estimator and its asymptotic normality. One advantage is its computational convenience and another is that it applies for an epidemic model with more realistic assump- tions about the durations of the latent and infectious periods than those made in the chain binomial and Kermack-McKendrick models. We take for our epidemic model a continuous time model. The duration of the latent period and the duration of the infectious period are taken to be independent random variables which, apart from in ? 4, are taken to have arbitrary distributions. We let v denote the mean dura- tion of the infectious period. Infection occurs, within households, according to an orderly point process with intensity at any time being jointly proportional to the number of suscep- tibles and the number of infectious individuals in the household at that time. The constant of proportionality ,B is the within-household infection rate.

Optimal health programs of immunization and isolation for some approximations to chain-binomial epidemic models

An isolation program and a compound medicare program consisting of immunization and isolation are considered for the Reed-Frost chain-binomial epidemic model. The assumption is made that the probability of contact is either small or large. The principle of optimality is used to derive dynamic-programming equations for the cost of control programs of both deterministic and stochastic versions of these models. Optimal immunization policies together with corresponding optimal costs are presented.

Control for some approximations to chain-binomial epidemic models

A policy of immunization is considered for the Greenwood and Reed-Frost chain-binomial epidemic models. In the case of the Reed-Frost model the assumption is made that the probability of contact between two individuals is either small or large. The principle of optimality is used to derive dynamic programming equations for both deterministic and stochastic versions of these models. Optimal immunization policies together with corresponding optimal costs are presented.

369: An Approximate Chain-Binomial Model for Simple Epidemics

The study of the spread of epidemics is an important feature for the understanding of the growth of infectives in a population. The difficulty in developing a proper mathematical model for epidemics is due to the fact that their intrinsic properties and biological behaviour are not very well known. However, when a large population is considered one can make the assumption that the growth of infectives is Poisson. Using this assumption, Bartlett [1959] has developed theoretical models for the study of epidemics. Studies of simple and general epidemics were also done by Weiss [1965] and Dietz ([1966], [1967]). When one considers an epidemic in a population of small size it is not reasonable to assume that the growth of infectives is Poisson. Epidemics such as measles often occur in groups of small size and it is, therefore, interesting to investigate epidemics of this nature. It is reported in Bailey [1966] that a chain-binomial model was used for the analysis of data from measles epidemics in Rhode Island. In this model, it was assumed that the chance of an infection in a unit interval of time is p (= 1 q) so that the process develops as a chain of binomial distributions. In the present paper, a modification to the Reed-Frost chain-binomial model has been made without allowing the removal of infectives so that the total extinction of susceptibles is inevitable. With this assumption, a relation for the generating function for the number of susceptibles in the rnth time interval is derived. In the special case, when the probability of infection is small, the expected value and the variance of the number of susceptibles in the nth interval are obtained and the results are found to be realistic.

Use of a chain binomial in the epidemiology of caries.

A chain binomial model of the caries process was defined and tested with data that consisted of the total number of carious deciduous molars in 10- and 11-year-old children. The model was shown to be a satisfactory approximation of the caries process.