Abstract
Network models of disease spread play an important role in elucidating the impact of long-lasting infectious contacts on the dynamics of epidemics. Moment-closure approximation is a common method of generating low-dimensional deterministic models of epidemics on networks, which has found particular success for diseases with susceptible-infected-recovered (SIR) dynamics. However, the effect of network structure is arguably more important for sexually transmitted infections, where epidemiologically relevant contacts are comparatively rare and longstanding, and which are in general modelled via the susceptible-infected-susceptible (SIS)-paradigm. In this paper, we introduce an improvement to the standard pairwise approximation for network models with SIS-dynamics for two different network structures: the isolated open triple (three connected individuals in a line) and the k-regular network. This improvement is achieved by tracking the rate of change of errors between triple values and their standard pairwise approximation. For the isolated open triple, this improved pairwise model is exact, while for k-regular networks a closure is made at the level of triples to obtain a closed set of equations. This improved pairwise approximation provides an insight into the errors introduced by the standard pairwise approximation, and more closely matches both higher-order moment-closure approximations and explicit stochastic simulations with only a modest increase in dimensionality to the standard pairwise approximation.
Highlights
The spread of any epidemic can be conceptualised as a process on a network, where individuals are represented as nodes and epidemiologically relevant contacts as edges between nodes
3. k-regular networks In Section 2, we considered the accuracy of the standard pairwise approximation on the isolated open triple, and derived a closed exact set of equations describing the errors such an approximation makes
The following results hold true whether referring to proportions or numbers in Appendix C, we provide a conversion table to transform the results from this section to numbers, and provide the model derived in terms of numbers
Summary
The spread of any epidemic can be conceptualised as a process on a network, where individuals are represented as nodes and epidemiologically relevant contacts as edges between nodes. An abundance of different network-based approaches to disease spread have been developed over the years, varying in scope, application, and sophistication These range from, at one extreme, Markovian state-based models, where the probability of a system being in a certain state is given exactly by its master equations (see Kiss et al, 2017 for an introduction to such methods), to explicit stochastic simulations of epidemics on networks (see Goodreau et al, 2017 and Whittles et al, 2019 for recent examples) at the other. We consider both higher-order moment-closure approximations and explicit stochastic simulations for this type of network, to act as benchmarks for our improved pairwise approximation.
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