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Exact deterministic representation of Markovian SIR epidemics on networks with and without loops
journal contribution
posted on 2023-06-08, 23:45 authored by Istvan Kiss, Charles G Morris, Fanni Sélley, Péter L Simon, Robert R WilkinsonIn a previous paper Sharkey et al. [14] proved the exactness of closures at the level of triples for Markovian SIR (susceptible-infected-removed) dynamics on tree-like networks. This resulted in a deterministic representation of the epidemic dynamics on the network that can be numerically evaluated. In this paper, we extend this modelling framework to certain classes of networks exhibiting loops. We show that closures where the loops are kept intact are exact, and lead to a simplified and numerically solvable system of ODEs (ordinary-differential-equations). The findings of the paper lead us to a generalisation of closures that are based on partitioning the network around nodes that are cut-vertices (i.e. the removal of such a node leads to the network breaking down into at least two disjointed components or subnetworks). Exploiting this structural property of the network yields some natural closures, where the evolution of a particular state can typically be exactly given in terms of the corresponding or projected states on the subnetworks and the cut-vertex. A byproduct of this analysis is an alternative probabilistic proof of the exactness of the closures for tree-like networks presented in Sharkey et al. [14]. In this paper we also elaborate on how the main result can be applied to more realistic networks, for which we write down the ODEs explicitly and compare output from these to results from simulation. Furthermore, we give a general, recipe-like method of how to apply the reduction by closures technique for arbitrary networks, and give an upper bound on the maximum number of equations needed for an exact representation.
History
Publication status
- Published
File Version
- Accepted version
Journal
Journal of Mathematical BiologyISSN
0303-6812Publisher
Springer VerlagExternal DOI
Issue
3Volume
70Page range
437-464Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2015-12-15First Open Access (FOA) Date
2015-12-15First Compliant Deposit (FCD) Date
2015-12-15Usage metrics
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