The Outcome of a Stochastic Epidemic—a Note on Bailey's Paper

Whittle, Peter

doi:10.1093/biomet/42.1-2.116

Cited by 147 publications

(161 citation statements)

References 4 publications

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“…It is of interest both for model-analysis and for statistical inference, the latter applying in the situation where the only data available from an outbreak are the numbers infected. For the special case of the general stochastic epidemic, Whittle (1955) derived a triangular system of linear equations for the probability mass function of the final size. Ball (1986) generalised this result to provide a corresponding system of equations for the GSE; it is the solution of these equations that is our focus in the present paper.…”

Section: Introductionmentioning

confidence: 99%

Computation of final outcome probabilities for the generalised stochastic epidemic

Demiris

O’Neill

2006

Stat Comput

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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.

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Section: Introductionmentioning

confidence: 99%

Computation of final outcome probabilities for the generalised stochastic epidemic

Demiris

O’Neill

2006

Stat Comput

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“…2 for the model-weighted distribution. This phenomenon is well known in stochastic epidemiological models (Bailey 1953;Whittle 1955). Many of the initial infections do not result in a For each value of Q U |U , 100 independent distributions of unvaccinated individuals were generated.…”

Section: Resultsmentioning

confidence: 99%

Multigeneration Reproduction Ratios and the Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

et al. 2011

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An SIR epidemiological community-structured model is constructed to investigate the effects of clustered distributions of unvaccinated individuals and the distribution of the primary case relative to vaccination levels. The communities here represent groups such as neighborhoods within a city or cities within a region. The model contains two levels of mixing, where individuals make more intra-group than inter-group contacts. Stochastic simulations and analytical results are utilized to explore the model. An extension of the effective reproduction ratio that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to characterize the system. Using these methods, we show that both the vaccination coverage and the variation in vaccination levels among communities affect the likelihood and severity of epidemics. The location of the primary infectious case and the degree of mixing between communities are also important factors in determining the dynamics of outbreaks. In some cases, increasing the efficacy of a vaccine can in fact increase the effective reproduction ratio in early generations, due to the effects of population structure on the likely initial location of an infection.

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“…Note that if a infectives are initially present, instead of just one, then these probabilities change to (r/n) a and 1 -(r/n) a . Whittle (1955) strengthens this result by determining the probability that an epidemic of not more than a given intensity takes place, where intensity denotes the total number of susceptibles who eventually contract the disease. This birth-death approximation is clearly a powerful and general device for determining initial population development.…”

Section: The General Epidemicmentioning

confidence: 88%

Stochastic Effects in Population Models

Renshaw

1999

Advanced Ecological Theory

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IntroductionThe remarkable variety of dynamic behaviour exhibited by many species of plants, insects and animals has stimulated great interest in the development of both biological experiments and mathematical models. After a relatively slow start in the 1920s and 1930s, the pace of research quickened dramatically throughout the remainder of the 20th century, though mathematicians and biologists have tended to take widely diverging paths. Theoreticians often profess an interest in biology purely to give them access to an intriguing set of stochastic (i.e. probability) equations, with the occasional biological reference being thrown in merely to perpetuate the mirage of potential applicability. Whilst, supposedly grass-roots 'mathematical biologists' are often tempted to develop vaguely plausible deterministic models which reflect mathematical hope rather than biological reality.Many researchers still use one approach to the total exclusion of the other. The reasons are two-fold. First, pioneering biological studies were greatly influenced by deterministic mathematics and reluctance to accept the importance of stochastic ideas is still deeply ingrained. Second, mathematicians are often taught in a practical vacuum, with the result that instead of using mathematics to interpret and understand biological phenomena they become transfixed by the models themselves. Renshaw (1991) presents a unifying approach between these two extremes, showing that both deterministic and stochastic models have important roles to play and should therefore be considered together. Popular deterministic ideas of even simple systems involving logistic, predator-prey and competition relationships can change markedly when viewed from a stochastic viewpoint. In biology we are often asked to infer the nature of population development from a single data set, yet different realizations of the same process can vary enormously. Even theoretical stochastic solutions are only of limited help here, and simple computer simulation procedures need to be constructed which provide much needed insight into the underlying biological generating mechanisms. Renshaw (1991) also advocates full recognition that the environment has a spatial dimension, since individual population members rarely

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The Outcome of a Stochastic Epidemic—a Note on Bailey's Paper

Cited by 147 publications

References 4 publications

Computation of final outcome probabilities for the generalised stochastic epidemic

Computation of final outcome probabilities for the generalised stochastic epidemic

Multigeneration Reproduction Ratios and the Effects of Clustered Unvaccinated Individuals on Epidemic Outbreak

Stochastic Effects in Population Models

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