Exact Equations for SIR Epidemics on Tree Graphs

Abstract

We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this “deterministic” representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

Message passing and moment closure for susceptible-infected-recovered epidemics on finite networks

“…Recently, two practicable and provably exact representations of stochastic epidemic models on finite trees have been proposed. These are the message passing approach of Karrer and Newman [3] and the pair-based moment-closure representation of Sharkey et al [4][5][6]. Here we generalise and compare these two representations and show that the pair-based equations can be derived from the message passing formalism.…”

“…To start to link the message passing method with the pairbased models [4][5][6], we express some relevant probabilities for connected pairs in tree networks. The probability S i S j that neighbouring individuals i and j [(i,j ) ∈ A or (j,i) ∈ A] are susceptible at time t is given by…”

“…We recognise Eq. (8) as a generalisation (to arbitrary recovery processes) of the pair-based moment-closure equations [6] which assume the following approximation for the probability of the state of a connected triple,…”

Special Issue on Spatial Moment Techniques for Modelling Biological Processes

“…More specifically, House (2015) introduces an approximate algebraic method to describe population dynamics on a discrete network that avoids the use of special features (such as the absence of loops), which are often used in similar contexts, and the new results are validated using an SIR model. Sharkey et al (2015) also consider an SIR model on a contact network and prove that a new pair-based moment closure representation is consistent with the infectious time series for networks with no cycles in the associated graph. Hiebeler and co-workers consider a moment dynamics description of an SIS model, and they specifically focus on the influence of population size, where the population is partitioned into groups or communities (Hiebeler et al 2015).…”

“…In particular, the special issue covers incorporate a broad review paper by Plank and Law (2015), which documents a general framework that can be used to model movement, birth, and death of multiple types of interacting agents in a non-homogeneous setting, which is relevant to a range of biological conditions, such as moving population fronts. The studies presented by Sharkey (2015) and House (2015) focus on describing dynamics on discrete structures, such as a graph, since these kinds of structures are thought of as being representative of the heterogeneities that exist in contacts between individuals. More specifically, House (2015) introduces an approximate algebraic method to describe population dynamics on a discrete network that avoids the use of special features (such as the absence of loops), which are often used in similar contexts, and the new results are validated using an SIR model.…”