Approximating steady state distributions for household structured epidemic models

Abstract

Household-structured infectious disease models consider the increased transmission potential between individuals of the same household when compared with two individuals in different households. Accounting for these heterogeneities in transmission enables control measures to be more effectively planned. Ideally, pre-control data may be used to fit such a household-structured model at an endemic steady state, before making dynamic forward-predictions under different proposed strategies. However, this requires the accurate calculation of the steady states for the full dynamic model. We observe that steady state SIS dynamics with household structure cannot necessarily be described by the master equation for a single household, instead requiring consideration of the full system. However, solving the full system of equations becomes increasingly computationally intensive, particularly for higher-dimensional models. We compare two approximations to the full system: the single household master equation; and a proposed alternative method, using the Fokker-Planck equation. Moment closure is another commonly used method, but for more complicated systems, the equations quickly become unwieldy and very difficult to derive. In comparison, using the master equation for a single household is easily implementable, however it can be quite inaccurate. In this paper we compare these methods in terms of accuracy and ease of implementation. We find that there are regions of parameter space in which each method outperforms the other, and that these regions of parameter space can be characterised by the infection prevalence, or by the correlation between household states.

Keywords: Approximation; Epidemiology; Fokker–Planck; Master equation; Stochastic household modelling.