Cluster approximations for infection dynamics on random networks

Abstract

In this paper, we consider a simple stochastic epidemic model on large regular random graphs and the stochastic process that corresponds to this dynamics in the standard pair approximation. Using the fact that the nodes of a pair are unlikely to share neighbors, we derive the master equation for this process and obtain from the system size expansion the power spectrum of the fluctuations in the quasistationary state. We show that whenever the pair approximation deterministic equations give an accurate description of the behavior of the system in the thermodynamic limit, the power spectrum of the fluctuations measured in long simulations is well approximated by the analytical power spectrum. If this assumption breaks down, then the cluster approximation must be carried out beyond the level of pairs. We construct an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement. For these parameter values, the power spectrum of the fluctuations in finite systems can be computed analytically from the master equation of the corresponding stochastic process.

Phase lag in epidemics on a network of cities

“…Although we have not explicitly investigated this quantity in the main text, it is also of interest, and by analogy with P (4) we would expect it to be real too. From Eq.…”

“…The theory of the frequency and amplitude of stochastic oscillations in models of epidemics of childhood diseases has been extensively developed over the last few years [1][2][3][4][5][6][7][8][9][10], but the question of the synchrony of these oscillations has received comparatively little attention. This is despite its undoubted importance; whether the oscillations in different locations are in phase or out of phase with each other will clearly have consequences for the duration of an epidemic and for the persistence of a disease.…”

“…In both cases, the frequency and amplitude of these oscillations is very well captured by the van Kampen system-size expansion [1][2][3][4][5][6][7][8]15,16], and it seems reasonable to suppose that the same approach should also be able to quantify the synchrony and phase lag between oscillations of different kinds of individuals. In this paper we apply this method to study possible synchronization between fluctuations in the number of infected individuals in a network of cities between which individuals commute.…”

Outbreak properties of epidemic models: The roles of temporal forcing and stochasticity on pathogen invasion dynamics

“…Although the power of master equations for simulating and deriving analytical insight into the behaviour of stochastic epidemic models has been recognised previously (Chen & Bokka, 2005;Grabowski & Kosinski 2004;Keeling and Ross, 2008;Rozhnova & Nunes, 2009), we illustrate here how insight into the effects of temporal variability in transmission on stochastic infectious disease dynamics may also be incorporated within this framework. This is a key advance in the study of infectious disease dynamics, as much of the insight to date on the effects of seasonality on disease dynamics has resulted largely from analytical or numerical analysis of deterministic epidemic (or endemic) models (Bailey, 1975;Bolker and Grenfell, 1993;Dietz, 1976;Moneim, 2007;Stone et al, 2007).…”

“…The use of master equations, whereby the probability of occurrence of each possible disease state is simultaneously considered, to understand the behaviour of stochastic infectious disease models has been described elsewhere (Keeling & Ross, 2008), along with applications to epidemic processes in homogeneous models (Chen & Bokka 2005) and structured/hierarchical models (Grabowski & Kosinski 2004;Rozhnova and Nunes 2009), yet by comparison to their deterministic counterparts, relatively little infection modelling work has adopted such methods.…”