Spatial heterogeneity is vitally important to the dynamics and persistence of many biological populations. Often this spatial heterogeneity can be a reflection of variations in the underlying distribution of resources. However, a secondary cause of spatial heterogeneity is the interaction between organisms leading to spatial correlations; a good example of which is the spread of infection through a network.
Traditionally, such spatial structure was only reproduced by complex spatial models, more akin to simulations than mathematical equations. In recent years however a series of tools has been developed to directly model the spatial correlations that can arise when discrete organisms interact with their local environment. One of the simplest type of such models is the pair-wise model, where instead of modelling the number of individuals, the number of interacting pairs is modelled by a set of differential equations.
This framework is particularly useful for understanding the spread of diseases, where a contact network determines who interacts (and therefore can infect) whom. A mixture of analytical and numerical studies have shown that including the correlations that develop within this network structure can have strong quantitative (as well as qualitative) effects.
Future work intends to apply this research to the spread of STDs (sexually transmitted diseases) where the network of contacts is well defined, and the low number of average connections means that there will be large deviations away from standard (well-mixed or mean-field) models. In addition we are interested in applying such techniques to clustered (family-based) networks.
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