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From Markovian to pairwise epidemic models and the performance of moment closure approximations
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Taylor, Michael, Simon, Péter L., Green, Darren M., House, Thomas A. and Kiss, Istvan Z. (2011) From Markovian to pairwise epidemic models and the performance of moment closure approximations. Journal of Mathematical Biology, Vol.64 (No.6). pp. 10211042. ISSN 03036812

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Official URL: http://dx.doi.org/10.1007/s0028501104433
Abstract
Many if not all models of disease transmission on networks can be linked to the exact statebased 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 individualbased 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 largescale 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 higherorder network structures may compromise their effectiveness.
Item Type:  Submitted Journal Article 

Subjects:  Q Science > QA Mathematics R Medicine > R Medicine (General) 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Epidemics  Mathematical models, System analysis 
Journal or Publication Title:  Journal of Mathematical Biology 
Publisher:  Springer 
ISSN:  03036812 
Official Date:  14 May 2011 
Volume:  Vol.64 
Number:  No.6 
Number of Pages:  22 
Page Range:  pp. 10211042 
Identification Number:  10.1007/s0028501104433 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  Engineering and Physical Sciences Research Council (EPSRC), Országos Tudományos Kutatási Alapprogramok (OTKA) 
Grant number:  EP/H001085/1 (EPSRC), 81403 (OTKA), EP/H016139/1 (EPSRC) 
References:  1. van Baalen M (2000) Pair approximations for different spatial geometries, in: Dieckmann U et al (eds) The 
URI:  http://wrap.warwick.ac.uk/id/eprint/39062 
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