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Approximate Master Equations for Dynamical Processes onGraphs

Published online by Cambridge University Press:  24 April 2014

N. Nagy ,
I.Z. Kiss  and
P.L. Simon
N. Nagy
Affiliation:
Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary
I.Z. Kiss
Affiliation:
School of Mathematical and Physical Sciences, Department of Mathematics University of Sussex, Falmer, Brighton BN1 9QH, UK
P.L. Simon*
Affiliation:
Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Hungary
*
⋆Corresponding author. E-mail: [email protected]
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Abstract

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

Keywords

SIS epidemicODE approximationnetwork process
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 2: Epidemics models on networks , 2014 , pp. 43 - 57
Copyright
© EDP Sciences, 2014

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