Threshold parameters for a model of epidemic spread among households and workplaces

Abstract

The basic reproduction number R0 is one of the most important concepts in modern infectious disease epidemiology. However, for more realistic and more complex models than those assuming homogeneous mixing in the population, other threshold quantities can be defined that are sometimes more useful and easily derived in terms of model parameters. In this paper, we present a model for the spread of a permanently immunizing infection in a population socially structured into households and workplaces/schools, and we propose and discuss a new household-to-household reproduction number RH for it. We show how RH overcomes some of the limitations of a previously proposed threshold parameter, and we highlight its relationship with the effort required to control an epidemic when interventions are targeted at randomly selected households.

Figure 1

Graphical description of the quantities (a) R _GG, (b) R _GL, (c) R _LG and (d) R _LL, used in the main text to construct the household-to-household next-generation matrix, from which the threshold parameter R _H is then computed. In this case, R _g=2, μ _H=2 and μ _W=1. Straight arrows represent local infections within households (rectangles) or workplaces (ellipses), arc arrows represent global infections. The black circle represents the household primary case and the grey circle indicates all the other infected cases in the household (attributed to, although not necessarily caused directly by, the primary case).

Figure 2

Values of , i.e. the household-to-household reproduction number after the vaccination of a proportion 1−1/R _H of randomly selected households. The curves correspond (from bottom to top) to R _g=0.2, 0.5, 1, 2, 4, 8 and 15. Numerical values are obtained by fixing μ _H=2 (other values lead to qualitatively similar results), and assuming that all workplaces have size n=6 and that the epidemic in a workplace spreads according to the standard SIR model defined in Ball (; see also Ball (1986) and Andersson & Britton (2000) and the electronic supplementary material), with infectious period of fixed length and individual-to-individual infection rate β/(n−1). The results are similar for other values of n, thus showing that it can be extended to a non-singular workplace size distribution (the value of μ _H already takes into account variable sizes for households). The model is equivalent to a Reed–Frost model with individual-to-individual escaping probability π=exp(−β/(n−1)) (see Ludwig (1975) and Pellis et al. (2008) and the electronic supplementary material). Other model assumptions (other distributions for the length of the infectious periods in the standard SIR model or other models with time-varying infection rates) lead to qualitatively similar results.