Approximate disease dynamics in household-structured populations

Abstract

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.

Figure 1

Comparison of approximate (red) and real (black) epidemic curves for populations of m houses of size n. Cases have (n, δ/β, R_g=βS₀/ν, m) given by: (a) (2,2,3,300); (b) (3,2,3,200); (c) (4,2,3,200); (d) (2,8,1.5,300); (e) (3,8,1.5,200) and (f) (4,8,1.5,200). m is chosen so as to accentuate errors.