From Markovian to pairwise epidemic models and the performance of moment closure approximations

Abstract

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

Outbreak analysis of an SIS epidemic model with rewiring

“…In particular, Keeling [14] introduced the susceptible-infected correlation function C SI in regular lattices, which is defined as the ratio of the current number of susceptible-infected pairs (denoted by [SI]) to the expected number from the mean-field assumption, namely, k[S] [I]/N where k is the (constant) number of neighbours per node, [S] the number of susceptible nodes, [I] the number of infected nodes, and N the size of the lattice (total number of nodes). The same correlation measure was also used for regular lattices, for instance in [2] and, more recently, in [30].…”

“…is the greater vulnerability of nodes with higher degrees. This implies that the pair approximation traditionally considered in regular lattices ( [21,30]) must be generalized in order to incorporate more information about the degree distribution. In addition, it also must be taken into account that the disease states have different degree distributions in the network [33] which implies additional sources of variability in the model [13].…”

Approximate Master Equations for Dynamical Processes on Graphs

“…Concerted efforts on the analysis of different ODE (ordinary differential equation) models of various dynamics on networks has led to a better understanding of how these models relate to each other [7,15,16], what the assumptions that these rely on are, and whether these models can serve as the limiting case of stochastic or exact models in some well defined limit [1,3,14]. The specific limits may typically depend on the size and type of the network or the time horizon over which agreement is sought.…”

On the early epidemic dynamics for pairwise models

“…The factor 2 comes from the fact that the triples are not oriented (cf. [3,39] for the oriented case).…”