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

Ball, Frank; Sirl, David; Trapman, Pieter

doi:10.1016/j.mbs.2009.12.003

Cited by 133 publications

(156 citation statements)

References 45 publications

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“…In both the standard and improved closures, the now-explicit assumption made about clustering is that each transitive link exists with independent probability φ, and so we would expect networks where transitive links are themselves clustered together into cliques as in [1] or unclustered as in [28] where no triangles overlap not to give good dynamical agreement with the proposed closures.…”

Section: Moment Closuresmentioning

confidence: 99%

From Markovian to pairwise epidemic models and the performance of moment closure approximations

et al. 2011

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Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

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Section: Moment Closuresmentioning

confidence: 99%

From Markovian to pairwise epidemic models and the performance of moment closure approximations

et al. 2011

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“…In these cases the impact of the network structure on the epidemic threshold and final epidemic size is clear. For example, in [2] the final epidemic size is given implicitly by an equation connecting it to moments of the network degree and other factors. The percolation-theory-based approach [17,27] applies even more widely for general transmission and recovery processes.…”

Section: Introductionmentioning

confidence: 99%

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

2018

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We present the generalised mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalised model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma-and uniformly-distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalised pairwise model lies in approximating the time evolution of the epidemic.

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“…More recently, considerable attention has been given to analysing the spread of infection on populations with structure modelled by a random graph (Andersson 1998, Britton et al 2008, including the case where the random graph has a specified degree distribution (Newman 2002, Kenah andRobins 2007). Some of these models have been combined to give, for example, the multitype households model of Ball and Lyne (2001), the model of Ball and Neal (2008) incorporating random networks and homogeneously mixing contacts and the network and households model of Ball et al (2009Ball et al ( , 2010. Other modifications and extensions of random network models are considered by, for example, Trapman (2007), Miller (2009) and Gleeson and Melnik (2009).…”

Section: Introductionmentioning

confidence: 99%

An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

Ball

Sirl

2012

Adv. Appl. Probab.

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1393/1/multiTypeNHMSubmit.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk An SIR epidemic model on a population with random network and household structure and several types of individualsFrank Ball * David Sirl † 22nd July 2010 AbstractWe consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own 'household' and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball et al. (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

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

Cited by 133 publications

References 45 publications

From Markovian to pairwise epidemic models and the performance of moment closure approximations

From Markovian to pairwise epidemic models and the performance of moment closure approximations

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

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