Stochastic multi-type SIR epidemics among a population partitioned into households

Ball, Frank; Lyne, Owen D.

doi:10.1017/s000186780001065x

Cited by 53 publications

(55 citation statements)

References 27 publications

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“…We determine the probability that a given individual is infected by considering (the size of) its susceptibility set. This concept has proved a fruitful framework within which to study the final outcome of epidemics where individuals interact with each other in more than one way (see, for example, Ball and Lyne [22] and Ball and Neal [23]). …”

Section: Informal Description Of Methodsmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

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134

148

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This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

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Section: Informal Description Of Methodsmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

Self Cite

134

148

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“…Under the assumption that Probability of containment for epidemics 175 all such contact processes are independent, as N → ∞ we can approximate the epidemic model by a multitype branching process in which individuals correspond to households in the underlying epidemic model. As described in full detail in [3], the basic idea is that, for large N , the probability that a single household receives more than one internal contact converges to 0, and so each infected household then gives rise to independent 'offspring' consisting of other household epidemics. Specifically, the types of individual in the branching process correspond to different initial configurations of household epidemics, i.e.…”

Section: Underlying Epidemic Modelmentioning

confidence: 99%

“…By standard theory, the largest eigenvalue of M, R * say, is a threshold parameter for the branching process in the sense that the process goes extinct almost surely if and only if R * ≤ 1. This parameter can also be viewed as a threshold parameter for the underlying epidemic model, as described in [2] and [3].…”

Section: Reproduction Numbermentioning

confidence: 99%

The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

Spencer

O’Neill

2011

J. Appl. Probab.

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This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

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“…[7,8]. In order to increase heterogeneity, the household structure of the population is also taken into account in [6], and global contacts are made through the edges of a random graph with a given degree distribution in [9], where household structure is also considered.…”

Section: Introductionmentioning

confidence: 99%

Solvability of implicit final size equations for SIR epidemic models

Bidari¹,

Chen²,

Peters³

et al. 2016

Mathematical Biosciences

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Final epidemic size relations play a central role in mathematical epidemiology. These can be written in the form of an implicit equation which is not analytically solvable in most of the cases. While final size relations were derived for several complex models, including multiple infective stages and models in which the durations of stages are arbitrarily distributed, the solvability of those implicit equations have been less studied. In this paper the SIR homogeneous mean-field and pairwise models and the heterogeneous mean-field model are studied. It is proved that the implicit equation for the final epidemic size has a unique solution, and that through writing the implicit equation as a fixed point equation in a suitable form, the iteration of the fixed point equation converges to the unique solution. The Markovian SIR epidemic model on finite networks is also studied by using the generation-based approach. Explicit analytic formulas are derived for the final size distribution for line and star graphs of arbitrary size. Iterative formulas for the final size distribution enable us to study the accuracy of mean-field approximations for the complete graph.

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Stochastic multi-type SIR epidemics among a population partitioned into households

Cited by 53 publications

References 27 publications

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

Solvability of implicit final size equations for SIR epidemic models

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