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In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.

Original publication




Journal article


Journal of mathematical biology

Publication Date





691 - 734


Medical Research Council Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College London, St. Mary's Hospital, Norfolk Place, London, W2 1PG, UK.


Humans, Models, Statistical, Markov Chains, Stochastic Processes, Family Characteristics, Schools, Computer Simulation, Workplace, Basic Reproduction Number, Influenza, Human