Strong approximations for epidemic models

Ball, Frank; Donnelly, Peter

doi:10.1016/0304-4149(94)00034-q

Cited by 201 publications

(203 citation statements)

References 29 publications

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“…The branching process discussed in the previous section is still valid and would allow us to calculate the variance in the overall number of infected but the method is problematic to implement (Nerman, 1981;Ball and Donnelly, 1995). Instead, we investigate this phase of the epidemic by deriving a diffusion approximation for the process in the limit of the number of clumps m → ∞ The variance tends to decrease as we increase β past this threshold as large outbreaks become more probable.…”

Section: Early Asymptotic 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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Section: Early Asymptotic 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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“…We use the construction argument of Ball (1983) and Ball & Donnelly (1995) to couple E E E M M M I I I and Z Z Z I I I . They described the construction of a single-host epidemic model from a limiting branching process.…”

Section: Given By E E E M M Mmentioning

confidence: 99%

“…They showed that if the branching process is subcritical, the epidemic and branching processes coincide for N → ∞, where N is the number of susceptible hosts. For that, we need to adapt to our model the independent and identically distributed life histories of the individuals, given as (L, ξ ) in Ball & Donnelly (1995), where L is the time elapsing between an individual's infection and its death, and ξ is a Poisson process of times at which contacts are made. We specify the life histories for dogs as (L 1 , ξ 1 ), where L 1 is exponentially distributed with rate λ 1 and ξ 1 is a point process of rate θ ρ at which sheep make infective contacts with its excreta, and the life histories for sheep with (L 2 , ξ 2 ), where L 2 is exponentially distributed with rate λ 2 and ξ 2 [0, L 2 ) = 0 and ξ 2 {L 2 } = 1, since an infected sheep is connected with exactly one dog and the infection is transmitted at death of the sheep.…”

Section: Given By E E E M M Mmentioning

confidence: 99%

“…Whittle (1955) have shown that initial and final stages of epidemic processes can often be approximated by suitable branching processes. More recently, Ball (1983), Ball & Donnelly (1995), Barbour & Utev (2004) and Barbour (2007) have used different construction arguments to quantify the accuracy of such approximations. We will use the argument of Ball (1983) and Ball & Donnelly (1995) to couple our subcritical epidemic process, which models the transmission dynamics of Echinococcus granulosus, to a suitable branching process.…”

Section: Introductionmentioning

confidence: 99%

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Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus

Heinzmann¹

2010

Workshop on Branching Processes and Their Applications

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Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus Heinzmann, D Heinzmann, D (2010 Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus Abstract An approximation is derived for the time to extinction in a sub-critical epidemic two-host interaction process for the macroparasite Echinococcus granulosus. The argument is based on coupling the epidemic model with a two-type branching process, and then to approximate the time to extinction for the branching process. It is shown that the approximate time is proportional to the logarithm of a weighted sum of the initially infectives in the host populations plus a Gumbel random variable. The accuracy of the approximation is illustrated. Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus Dominik HeinzmannAbstract An approximation is derived for the time to extinction in a sub-critical epidemic two-host interaction process for the macroparasite Echinococcus granulosus. The argument is based on coupling the epidemic model with a two-type branching process, and then to approximate the time to extinction for the branching process. It is shown that the approximate time is proportional to the logarithm of a weighted sum of the initially infectives in the host populations plus a Gumbel random variable. The accuracy of the approximation is illustrated. AMS: 60J80; 92D30

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“…Frank Ball and colleagues [4,5] showed that the initial phase of a certain class of epidemic models can be well approximated by branching processes for which an exact probability of invasion can be calculated. For example, in the large population limit, the probability that a single infectious individual will cause an epidemic in the stochastic Susceptible-Infectious-Susceptible (SIS) and stochastic Susceptible-Infectious-Recovered (SIR) models is 1 -d b , where b is the rate at which an infectious individual infects susceptible members of the population, and d is the rate at which the pathogen is cleared from an infectious individual.…”

mentioning

confidence: 99%

Continuum Approximation of Invasion Probabilities

Borchering¹,

McKinley²

2018

Multiscale Model. Simul.

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Abstract. In the last decade there has been growing criticism of the use of stochastic differential equations to approximate discrete-state-space, continuous-time Markov chain population models. In particular, several authors have demonstrated the failure of Diffusion Approximation, as it is often called, to approximate expected extinction times for populations that start in a quasi-stationary state. In this work we investigate a related, but distinct, population dynamics property for which Diffusion Approximation is unreliable: invasion probabilities. We consider the situation in which a few individuals are introduced into a population and ask whether their collective lineage can successfully invade. Because the population count is so small during the critical period of success or failure, the process is intrinsically stochastic and discrete. In addition to demonstrating how and why the Diffusion Approximation fails in the large population limit, we contrast this analysis with that of a sometimes more successful alternative WKB-like approach. Through numerical investigations, we also study how these approximations perform in an important intermediate regime. Surprisingly, we find that there are times when the Diffusion Approximation performs well, particularly when parameters are near-critical and the population size is small to intermediate.

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Strong approximations for epidemic models

Cited by 201 publications

References 29 publications

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

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

Time to extinction in a two-host interaction model for the macroparasite Echinococcus granulosus

Continuum Approximation of Invasion Probabilities

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