A note on a paper by Erik Volz: SIR dynamics in random networks

Miller, Joel C.

doi:10.1007/s00285-010-0337-9

Cited by 168 publications

(201 citation statements)

References 10 publications

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“…More rigorous approaches avoid these approximations, but are more difficult [24][25][26][27]. Of these, only Volz [24] yields a closed ODE system, (see also [28,29]). This model lacks an illustration like figure 1, hampering communication and further development.…”

Section: Introductionmentioning

confidence: 99%

Edge-based compartmental modelling for infectious disease spread

Miller

Slim²,

Volz

2011

J. R. Soc. Interface.

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227

302

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The primary tool for predicting infectious disease spread and intervention effectiveness is the mass action susceptible -infected -recovered model of Kermack & McKendrick. Its usefulness derives largely from its conceptual and mathematical simplicity; however, it incorrectly assumes that all individuals have the same contact rate and partnerships are fleeting. In this study, we introduce edge-based compartmental modelling, a technique eliminating these assumptions. We derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of partnership duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model and focus on applying the technique under different assumptions about how contact rates are distributed and how long partnerships last.

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Section: Introductionmentioning

confidence: 99%

Edge-based compartmental modelling for infectious disease spread

Miller

Slim²,

Volz

2011

J. R. Soc. Interface.

Self Cite

227

302

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“…Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1)(2)(3)(4)(5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7)(8)(9)(10), opinions (11)(12)(13), information cascades (14)(15)(16), and evolutionary games (17,18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa.…”

mentioning

confidence: 99%

Graph fission in an evolving voter model

Durrett

Gleeson

Lloyd

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

153

251

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We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 − α, one imitates the opinion of the other; otherwise (i.e., with probability α), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting ρ be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how ρ depends on α and the initial fraction u of voters with opinion 1. In case (i), there is a critical value α c which does not depend on u, with ρ ≈ u for α > α c and ρ ≈ 0 for α < α c . In case (ii), the transition point α c ðuÞ depends on the initial density u. For α > α c ðuÞ, ρ ≈ u, but for α < α c ðuÞ, we have ρðα,uÞ ¼ ρðα,1∕2Þ. Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.I n recent years, a variety of research efforts from different disciplines have combined with established studies in social network analysis and random graph models to fundamentally change the way we think about networks. Significant attention has focused on the implications of dynamics in establishing network structure, including preferential attachment, rewiring, and other mechanisms (1-5). At the same time, the impact of structural properties on dynamics on those networks has been studied, (6), including the spread of epidemics (7-10), opinions (11-13), information cascades (14-16), and evolutionary games (17,18). Of course, in many real-world networks the evolution of the edges in the network is tied to the states of the vertices and vice versa. Networks that exhibit such a feedback are called adaptive or coevolutionary networks (19,20). As in the case of static networks, significant attention has been paid to evolutionary games (21)(22)(23)(24) and to the spread of epidemics (25-29) and opinions (30-35), including the polarization of a network of opinions into two groups (36,37). In this paper, we examine two closely related variants of a simple, abstract model for coevolution of a network and the opinions of its members. Holme-Newman ModelOur starting point is the model of Holme and Newman (38-41). They begin with a network of N vertices and M edges, where each vertex x has an opinion ξðxÞ from a set of G possible opinions and the number of people per opinion γ N ¼ N∕G stays bounded as N gets large. On each step of the process, a vertex x is picked at random. If its degree dðxÞ ¼ 0, nothing happens. For dðxÞ > 0, (i) with probability α an edge attached to vertex x is selected and the other end of that edge is moved to a vertex chosen at random from those with o...

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“…7). This model may be solved analytically for the mean value (see ESM [7] §III) and the results are in agreement with [19,20]. Moreover, ESM [7] §IV shows how (10) may be rewritten with a state vector two thirds the size of (10a).…”

Section: First Neighbourhood On-the-fly Sir Modelmentioning

confidence: 59%

“…Knowing (in Y ) that a system behaves in this manner greatly simplifies the inference process, and this is the main reason for the success of the SIR pair-based model for the evolution of mean values on CM networks that is presented in [19,20]. Whether or not the same approach may be used to obtain stochastic results is an open question.…”

Section: Pair-based Modelsmentioning

confidence: 99%

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Spreading dynamics on complex networks: a general stochastic approach

et al. 2013

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Dynamics on networks is considered from the perspective of Markov stochastic processes. We partially describe the state of the system through network motifs and infer any missing data using the available information. This versatile approach is especially well adapted for modelling spreading processes and/or population dynamics. In particular, the generality of our systematic framework and the fact that its assumptions are explicitly stated suggests that it could be used as a common ground for comparing existing epidemics models too complex for direct comparison, such as agentbased computer simulations. We provide many examples for the special cases of susceptible-infectious-susceptible (SIS) and susceptible-infectious-removed (SIR) dynamics (e.g., epidemics propagation) and we observe multiple situations where accurate results may be obtained at low computational cost. Our perspective reveals a subtle balance between the complex requirements of a realistic model and its basic assumptions.

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A note on a paper by Erik Volz: SIR dynamics in random networks

Cited by 168 publications

References 10 publications

Edge-based compartmental modelling for infectious disease spread

Edge-based compartmental modelling for infectious disease spread

Graph fission in an evolving voter model

Spreading dynamics on complex networks: a general stochastic approach

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