A Note on the Derivation of Epidemic Final Sizes

Miller, Joel C.

doi:10.1007/s11538-012-9749-6

Cited by 165 publications

(192 citation statements)

References 26 publications

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“…41,42 Estimating interactions is difficult, however, because studies are rarely powered to estimate subgroup effects and because many interaction variables may be unknown.…”

Section: Discussionmentioning

confidence: 99%

Estimating the Per-Exposure Effect of Infectious Disease Interventions

2014

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The average effect of an infectious disease intervention (e.g., a vaccine) varies across populations with different degrees of exposure to the pathogen. As a result, many investigators favor a per-exposure effect measure that is considered independent of the population level of exposure and that can be used in simulations to estimate the total disease burden averted by an intervention across different populations. However, while per-exposure effects are frequently estimated, the quantity of interest is often poorly defined, and assumptions in its calculation are typically left implicit. In this paper, we build upon work by Halloran and Struchiner (Epidemiology. 1995; 6:172-151) to develop a formal definition of the per-exposure effect and discuss conditions necessary for its unbiased estimation. With greater care paid to the parameterization of transmission models, their results can be better understood and can thereby be of greater value to decision-makers.

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“…41,42 Estimating interactions is difficult, however, because studies are rarely powered to estimate subgroup effects and because many interaction variables may be unknown.…”

Section: Discussionmentioning

confidence: 99%

Estimating the Per-Exposure Effect of Infectious Disease Interventions

2014

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“…ηR ∞ , recalling that η is the disease's mortality fraction. For the standard SIR model this is straightforward to do, resulting with R ∞ being the solution of an implicit equation (Andreasen, 2011;Miller, 2012). A similar approach can be used in the present model, though the details are more complicated.…”

Section: Large-time Analysismentioning

confidence: 99%

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Chen

Ward

Xie

2019

Theoretical Population Biology

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In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human-human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number R is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established.

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“…The final epidemic size can be obtained from an implicit equation that can be derived from the differential equations, see e.g. [18,19], and also from the stochastic formulation of the model, see for example Section 4.3 of [1]. Our goal with exploring this model is to prove the existence and uniqueness of the solution of this implicit equation and to show that simple iteration converges to the unique solution.…”

Section: Homogeneous Mean-field Modelmentioning

confidence: 99%

“…Our goal with exploring this model is to prove the existence and uniqueness of the solution of this implicit equation and to show that simple iteration converges to the unique solution. For the Reader's convenience we also show a simple derivation of the implicit equation following [18].…”

Section: Homogeneous Mean-field Modelmentioning

confidence: 99%

“…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. An alternate approach to deriving final sizes is suggested in [18]. This approach directly considers the underlying stochastic process of the epidemic rather than the approximating deterministic equations and gives insight into why the relations hold.…”

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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A Note on the Derivation of Epidemic Final Sizes

Cited by 165 publications

References 26 publications

Estimating the Per-Exposure Effect of Infectious Disease Interventions

Estimating the Per-Exposure Effect of Infectious Disease Interventions

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain

Solvability of implicit final size equations for SIR epidemic models

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