Correlations between stochastic epidemics in two interacting populations

It is increasingly apparent that heterogeneity in the interaction between individuals plays an important role in the dynamics, persistence, evolution and control of infectious diseases. In epidemic modelling two main forms of heterogeneity are commonly considered: spatial heterogeneity due to the segregation of populations and heterogeneity in risk at the same location. The transition from random-mixing to heterogeneous-mixing models is made by incorporating the interaction, or coupling, within and between subpopulations. However, such couplings are difficult to measure explicitly; instead, their action through the correlations between subpopulations is often all that can be observed. Here, using moment-closure methodology supported by stochastic simulation, we investigate how the coupling and resulting correlation are related. We focus on the simplest case of interactions, two identical coupled populations, and show that for a wide range of parameters the correlation between the prevalence of infection takes a relatively simple form. In particular, the correlation can be approximated by a logistic function of the between population coupling, with the free parameter determined analytically from the epidemiological parameters. These results suggest that detailed case-reporting data alone may be sufficient to infer the strength of between population interaction and hence lead to more accurate mathematical descriptions of infectious disease behaviour.