SIR epidemics on random graphs with a fixed degree sequence

Bohman, Tom; Picollelli, Michael E.

doi:10.1002/rsa.20401

Cited by 31 publications

(54 citation statements)

References 38 publications

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“…We establish the upper bound on V (i) in (8) by showing that Z V < 0 for all i ≤ T with high probability. Similarly, we establish the upper bounds on d ± (v) in (9) by showing that Z ± (v) < 0 for all i ≤ T with high probability.…”

Section: Dynamic Concentrationmentioning

confidence: 83%

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A note on the random greedy independent set algorithm

Bennett

Bohman

2016

Random Struct Algorithms

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Let r be a fixed constant and let scriptH be an r‐uniform, D‐regular hypergraph on N vertices. Assume further that D>Nϵ for some ϵ>0. Consider the random greedy algorithm for forming an independent set in scriptH. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and I∪{v} contains no edge in scriptH). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of scriptH; that is, the process terminates at a maximal independent set. We prove that if scriptH satisfies certain degree and codegree conditions then there are Ωtrue(N·((logN)/D)1r−1true) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H‐free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016

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Section: Dynamic Concentrationmentioning

confidence: 83%

“…where λ = /4r. Throughout this section we use the bound |V (i)| > r + ND −λ , which follows from (7), (8) and the fact that we may set ζ > 0 sufficiently small relative to .…”

Section: Definition 2 (Codegrees) For a Pair Of Vertices V V Let mentioning

confidence: 99%

A note on the random greedy independent set algorithm

Bennett

Bohman

2016

Random Struct Algorithms

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“…PGF models [52,36] provide low-dimensional representations of epidemic dynamics on configuration models and have been proved to be asymptotically exact [7,5,4,20].…”

Section: Probability Generating Function (Pgf) Methodsmentioning

confidence: 99%

Consistent Approximation of Epidemic Dynamics on Degree-Heterogeneous Clustered Networks

Bishop

Kiss

House

2018

Studies in Computational Intelligence

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Realistic human contact networks capable of spreading infectious disease, for example studied in social contact surveys, exhibit both significant degree heterogeneity and clustering, both of which greatly affect epidemic dynamics. To understand the joint effects of these two network properties on epidemic dynamics, the effective degree model of Lindquist et al. [30] is reformulated with a new moment closure to apply to highly clustered networks. A simulation study comparing alternative ODE models and stochastic simulations is performed for SIR (Susceptible-Infected-Removed) epidemic dynamics, including a test for the conjectured error behaviour in [42], providing evidence that this novel model can be a more accurate approximation to epidemic dynamics on complex networks than existing approaches.

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“…Then 14) wherex N (τ ) andỹ N E (τ ) are the deterministic 'number' of susceptible individuals and infectious halfedges, given by (5.26) and (5.11), respectively, but with (random) initial conditions induced by the NSW random graph on N vertices.…”

Section: )mentioning

confidence: 99%

A stochastic SIR network epidemic model with preventive dropping of edges

et al. 2019

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A Markovian Susceptible Infectious Recovered (SIR) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size , assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy–Reed (in which the degrees of individuals are deterministic) and Newman–Strogatz–Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman–Strogatz–Watts version. The basic reproduction number and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when , the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the size of the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N .

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SIR epidemics on random graphs with a fixed degree sequence

Cited by 31 publications

References 38 publications

A note on the random greedy independent set algorithm

A note on the random greedy independent set algorithm

Consistent Approximation of Epidemic Dynamics on Degree-Heterogeneous Clustered Networks

A stochastic SIR network epidemic model with preventive dropping of edges

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