Calculation of Disease Dynamics in a Population of Households

Abstract

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.

The effect of clumped population structure on the variability of spreading dynamics

“…During this phase the epidemic dynamics are well approximated by a branching process between clumps (Ball et al, 1997;Ross et al, 2010). The within-clump dynamics are modelled as a continuous-time Markov chain, X(t), with transition rate matrix Q = (q i j ; i, j ∈ S ).…”

“…(3) can be efficiently evaluated numerically using exponential discounting (Ross et al, 2010). Combined with a basic root finding algorithm, this allows us to easily compute r. All results are derived by finding the value of α which gives r = 1 for a given clump size and within-clump transmission rate β.…”

“…Previous work has detailed how the offspring distribution can be efficiently calculated (Ross et al, 2010). The mean of the offspring distribution is then just R * , the expected number of secondary clumps infected by a primary.…”

“…The mean of the offspring distribution is then just R * , the expected number of secondary clumps infected by a primary. Alternatively, R * can be calculated more efficiently using path integral techniques, (Pollett and Stefanov, 2002;Ross et al, 2010)…”

“…For given values of β and n we scale the between-clump transmission rate, α, to achieve this. There is another quantity which we could instead fix: the clump reproductive ratio R * , which is the expected number of secondary clumps infected by a primary clump (Ball et al, 1997;Ross et al, 2010). We note that these two quantities are linked (Svensson, 2007;Wallinga and Lipsitch, 2007) and we chose to fix r instead of R * because the early growth rate is most easily estimated from data.…”