A many-body field theory approach to stochastic models in population biology.


Dodd, PJ; Ferguson, NM; (2009) A many-body field theory approach to stochastic models in population biology. PLoS One, 4 (9). e6855. ISSN 1932-6203 DOI: https://doi.org/10.1371/journal.pone.0006855

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Abstract

BACKGROUND: Many models used in theoretical ecology, or mathematical epidemiology are stochastic, and may also be spatially-explicit. Techniques from quantum field theory have been used before in reaction-diffusion systems, principally to investigate their critical behavior. Here we argue that they make many calculations easier and are a possible starting point for new approximations. METHODOLOGY: We review the many-body field formalism for Markov processes and illustrate how to apply it to a 'Brownian bug' population model, and to an epidemic model. We show how the master equation and the moment hierarchy can both be written in particularly compact forms. The introduction of functional methods allows the systematic computation of the effective action, which gives the dynamics of mean quantities. We obtain the 1-loop approximation to the effective action for general (space-) translation invariant systems, and thus approximations to the non-equilibrium dynamics of the mean fields. CONCLUSIONS: The master equations for spatial stochastic systems normally take a neater form in the many-body field formalism. One can write down the dynamics for generating functional of physically-relevant moments, equivalent to the whole moment hierarchy. The 1-loop dynamics of the mean fields are the same as those of a particular moment-closure.

Item Type: Article
Faculty and Department: Faculty of Epidemiology and Population Health > Dept of Infectious Disease Epidemiology
PubMed ID: 19730742
Web of Science ID: 269461100001
URI: http://researchonline.lshtm.ac.uk/id/eprint/2270

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