Deterministic epidemic models on contact networks: correlations and unbiological terms

The relationship between system-level and subsystem-level master equations is investigated and then utilised for a systematic and potentially automated derivation of the hierarchy of moment equations in a susceptible-infectious-removed (SIR) epidemic model. In the context of epidemics on contact networks we use this to show that the approximate nature of some deterministic models such as mean-field and pair-approximation models can be partly understood by the identification of implicit anomalous terms. These terms describe unbiological processes which can be systematically removed up to and including the nth order by nth order moment closure approximations. These terms lead to a detailed understanding of the correlations in network-based epidemic models and contribute to understanding the connection between individual-level epidemic processes and population-level models. The connection with metapopulation models is also discussed. Our analysis is predominantly made at the individual level where the first and second order moment closure models correspond to what we term the individual-based and pair-based deterministic models, respectively. Matlab code is included as supplementary material for solving these models on transmission networks of arbitrary complexity.