162 research outputs found
A motif-based approach to network epidemics
Networks have become an indispensable tool in modelling infectious diseases, with the structure of epidemiologically relevant contacts known to affect both the dynamics of the infection process and the efficacy of intervention strategies. One of the key reasons for this is the presence of clustering in contact networks, which is typically analysed in terms of prevalence of triangles in the network. We present a more general approach, based on the prevalence of different four-motifs, in the context of ODE approximations to network dynamics. This is shown to outperform existing models for a range of small world networks
Exact and approximate moment closures for non-Markovian network epidemics
Moment-closure techniques are commonly used to generate low-dimensional
deterministic models to approximate the average dynamics of stochastic systems
on networks. The quality of such closures is usually difficult to asses and the
relationship between model assumptions and closure accuracy are often
difficult, if not impossible, to quantify. Here we carefully examine some
commonly used moment closures, in particular a new one based on the concept of
maximum entropy, for approximating the spread of epidemics on networks by
reconstructing the probability distributions over triplets based on those over
pairs. We consider various models (SI, SIR, SEIR and Reed-Frost-type) under
Markovian and non-Markovian assumption characterising the latent and infectious
periods. We initially study two special networks, namely the open triplet and
closed triangle, for which we can obtain analytical results. We then explore
numerically the exactness of moment closures for a wide range of larger motifs,
thus gaining understanding of the factors that introduce errors in the
approximations, in particular the presence of a random duration of the
infectious period and the presence of overlapping triangles in a network. We
also derive a simpler and more intuitive proof than previously available
concerning the known result that pair-based moment closure is exact for the
Markovian SIR model on tree-like networks under pure initial conditions. We
also extend such a result to all infectious models, Markovian and
non-Markovian, in which susceptibles escape infection independently from each
infected neighbour and for which infectives cannot regain susceptible status,
provided the network is tree-like and initial conditions are pure. This works
represent a valuable step in deepening understanding of the assumptions behind
moment closure approximations and for putting them on a more rigorous
mathematical footing.Comment: Main text (45 pages, 11 figures and 3 tables) + supplementary
material (12 pages, 10 figures and 1 table). Accepted for publication in
Journal of Theoretical Biology on 27th April 201
The end time of SIS epidemics driven by random walks on edge-transitive graphs
Network epidemics is a ubiquitous model that can represent different
phenomena and finds applications in various domains. Among its various
characteristics, a fundamental question concerns the time when an epidemic
stops propagating. We investigate this characteristic on a SIS epidemic induced
by agents that move according to independent continuous time random walks on a
finite graph: Agents can either be infected (I) or susceptible (S), and
infection occurs when two agents with different epidemic states meet in a node.
After a random recovery time, an infected agent returns to state S and can be
infected again. The End of Epidemic (EoE) denotes the first time where all
agents are in state S, since after this moment no further infections can occur
and the epidemic stops.
For the case of two agents on edge-transitive graphs, we characterize EoE as
a function of the network structure by relating the Laplace transform of EoE to
the Laplace transform of the meeting time of two random walks. Interestingly,
this analysis shows a separation between the effect of network structure and
epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically
in the size of the graph) under different parameter scalings, identifying
regimes where EoE converges in distribution to a proper random variable or to
infinity. We also highlight the impact of different graph structures on EoE,
characterizing it under complete graphs, complete bipartite graphs, and rings
Insights from unifying modern approximations to infections on networks
Networks are increasingly central to modern science owing to their ability to conceptualize multiple interacting components of a complex system. As a specific example of this, understanding the implications of contact network structure for the transmission of infectious diseases remains a key issue in epidemiology. Three broad approaches to this problem exist: explicit simulation; derivation of exact results for special networks; and dynamical approximations. This paper focuses on the last of these approaches, and makes two main contributions.
Firstly, formal mathematical links are demonstrated between several prima facie unrelated dynamical approximations. And secondly, these links are used to derive two novel dynamical models for network epidemiology, which are compared against explicit stochastic simulation. The success of these new models provides improved understanding about the interaction of network structure and transmission dynamics
Pairwise approximation for SIR-type network epidemics with non-Markovian recovery
We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic
Heterogeneous network epidemics: real-time growth, variance and extinction of infection
Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multitype branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction by time t that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distributionâin particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual
A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling
For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks
- âŠ