How big is an outbreak likely to be? Methods for epidemic final-size calculation

House, Thomas; Ross, Joshua V.; Sirl, David

doi:10.1098/rspa.2012.0436

Cited by 56 publications

(64 citation statements)

References 59 publications

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“…The final size analysis has been deepened by Ball for the single population scenario [24] and also extended to include multi-type epidemics [25,30]. For a thorough discussion of numerical methods, and potential pitfalls, for solving the resultant final size distributions see House et al [128]. The major drawback to using result (53) is that ∆T and e next are represented implicitly in terms of a set of exp(1) distributed 'latent' random variables [64] rather than given directly.…”

Section: Non-markovian Arrival Timesmentioning

confidence: 99%

Dynamics of infectious diseases

et al. 2014

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Modern infectious disease epidemiology has a strong history of using mathematics both for prediction and to gain a deeper understanding. However the study of infectious diseases is a highly interdisciplinary subject requiring insights from multiple disciplines, in particular a biological knowledge of the pathogen, a statistical description of the available data and a mathematical framework for prediction. Here we begin with the basic building blocks of infectious disease epidemiologythe SIS and SIR type models -before considering the progress that has been made over the recent decades and the challenges that lie ahead. Throughout we focus on the understanding that can be developed from relatively simple models, although accurate prediction will inevitably require far greater complexity beyond the scope of this review. In particular, we focus on three critical aspects of infectious disease models that we feel fundamentally shape their dynamics: heterogeneously structured populations; stochasticity; and spatial structure. Throughout we relate the mathematical models and their results to a variety of real-world problems.

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Section: Non-markovian Arrival Timesmentioning

confidence: 99%

Dynamics of infectious diseases

et al. 2014

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“…For this variable y = 0 means that the entire population becomes infected and then removed during the epidemic, while y = 1 means that the disease dies out without causing any infection. Equation (12) takes the form yS 0 = y 2/n N τ +γ τ − y 1/n N γ τ for the new unknown y. This form is not suitable to prove the uniqueness and to apply the iteration hence the following transformations are carried out first to obtain a proper fixed point equation.…”

Section: Uniqueness Of the Nontrivial Solution Of The Implicit Equationmentioning

confidence: 99%

“…The final epidemic size was determined for the same values of τ from the homogeneous mean-field given by (4) using the iteration in Proposition 2. Similarly, using the iteration in Proposition 6, the final epidemic size is determined from the homogeneous pairwise model as R ∞ = N − S ∞ with S ∞ being the solution of (12). The comparison is shown in Figure 1 for a complete graph with N = 1000 nodes varying τ from 0 to 0.005.…”

Section: Comparison Of Mean-field and Stochastic Models For The Complmentioning

confidence: 99%

“…The stochastic model with heterogeneous infectivity and susceptibility has also been studied in [17]. A large number of approaches to calculate the final-size distribution are reviewed in [12] with paying particular attention to numerical implementation, including a novel application for iterative methods. The iterative method is used in [11] to develop a new methodology for the efficient computation of epidemic final size distributions for a broad class of Markovian models, applied in particular to SIR epidemic on complete graphs.…”

Section: Introductionmentioning

confidence: 99%

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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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“…This is the probability distribution that a given proportion of the clump will have been infected over the course of the within-clump epidemic. This can be calculated via a number of methods (Ball, 1986;House et al, 2013). Figure 3 shows the mean and variance of the offspring distribution as a function of β and n; the mean is equal to R * .…”

Section: Initial Behaviourmentioning

confidence: 99%

The effect of clumped population structure on the variability of spreading dynamics

Black

House

Keeling

et al. 2014

Journal of Theoretical Biology

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Processes that spread through local contact, including outbreaks of infectious diseases, are inherently noisy, and are frequently observed to be far noisier than predicted by standard stochastic models that assume homogeneous mixing. One way to reproduce the observed levels of noise is to introduce significant individual-level heterogeneity with respect to infection processes, such that some individuals are expected to generate more secondary cases than others. Here we consider a population where individuals can be naturally aggregated into clumps (subpopulations) with stronger interaction within clumps than between them. This clumped structure induces significant increases in the noisiness of a spreading process, such as the transmission of infection, despite complete homogeneity at the individual level. Given the ubiquity of such clumped aggregations (such as homes, schools and workplaces for humans or farms for livestock) we suggest this as a plausible explanation for noisiness of many epidemic time series.

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How big is an outbreak likely to be? Methods for epidemic final-size calculation

Cited by 56 publications

References 59 publications

Dynamics of infectious diseases

Dynamics of infectious diseases

Solvability of implicit final size equations for SIR epidemic models

The effect of clumped population structure on the variability of spreading dynamics

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