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We argue that the large-dimensional dynamical systems which frequently occur in biological models can sometimes be effectively reduced to much smaller ones. We illustrate this by applying projection operator techniques to a mean-field model of an infectious disease spreading through a population of households. In this way, we are able to accurately approximate the dynamics of the system in terms of a few key quantities greatly reducing the number of equations required. We investigate linear stability in this framework and find a new way of calculating the familiar threshold criterion for household systems.
Reference DOI: 10.1098/rsif.2007.0231
Abstract: Although often studied in their own right, contact networks are generated by social and biological processes. In contrast to static network representations the contact structure arising from such processes will be highly dynamic and connections between individuals will be directed and vary in strength. Although potentially complicating any analyses, an understanding of the impacts of such processes on disease dynamics may provide deeper insights and suggest alternative control strategies. Although in principle one could model behavioural and epidemic processes jointly, here we suggest a simplifying approach in which the underlying (dynamic) network is first derived from a behavioural model. As an applied example we study the network of indirect contacts between individuals induced (on a daily and a weekly basis) by an individual-based spatially explicit model of foraging and avoidance behaviour in grazing animals. The derived contact structures are relevant to faecally mediated infections, but only initial results describing the contact network are presented here: a high turnover in the daily (and weekly) numbers of individual contacts reveals the dynamic nature of the contact network; and there is considerably greater variation in individual exposure than in numbers of individual contacts.
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31st July 2007