Integrating stochasticity and network structure into an epidemic model

Abstract

While the foundations of modern epidemiology are based upon deterministic models with homogeneous mixing, it is being increasingly realized that both spatial structure and stochasticity play major roles in shaping epidemic dynamics. The integration of these two confounding elements is generally ascertained through numerical simulation. Here, for the first time, we develop a more rigorous analytical understanding based on pairwise approximations to incorporate localized spatial structure and diffusion approximations to capture the impact of stochasticity. Our results allow us to quantify, analytically, the impact of network structure on the variability of an epidemic. Using the susceptible–infectious–susceptible framework for the infection dynamics, the pairwise stochastic model is compared with the stochastic homogeneous-mixing (mean-field) model—although to enable a fair comparison the homogeneous-mixing parameters are scaled to give agreement with the pairwise dynamics. At equilibrium, we show that the pairwise model always displays greater variation about the mean, although the differences are generally small unless the prevalence of infection is low. By contrast, during the early epidemic growth phase when the level of infection is increasing exponentially, the pairwise model generally shows less variation.

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

“…Thus the factor multiplying the exponential is linear in n. Setting n = 1, we recover the basic SIR result in Dangerfield et al (2009).…”

Experimental and Modelling Investigation of Monolayer Development with Clustering

“…Great progress has been made in developing continuum models that relax the mean-field assumption for various applications including surface chemistry reactions (Mai et al 1993(Mai et al , 1994, interacting plant communities (Bolker and Pacala 1997;Law and Dieckmann 2000;Law et al 2003), infectious disease progression (Dangerfield et al 2008;Keeling et al 1997;Sharkey 2008Sharkey , 2011 and point particle birth-death processes (Young et al 2001). Our recent work has focused on adapting these techniques to discrete models of cell migration, proliferation and death processes which include finite size effects by allowing, at most, only one agent to occupy a particular location in space.…”

Modelling behavioural contagion

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