Large graph limit for an SIR process in random network with heterogeneous connectivity

Decreusefond, Laurent; Dhersin, Jean-Stéphane; Moyal, Pascal; Tran, Viet Chi

doi:10.1214/11-aap773

Cited by 91 publications

(168 citation statements)

References 23 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Approximate Master Equations for Dynamical Processes on Graphs

Nagy

Kiss

Šimon

2014

Math. Model. Nat. Phenom.

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Abstract. We extrapolate from the exact master equations of epidemic dynamics on fully connected graphs to non-fully connected by keeping the size of the state space N + 1, where N is the number of nodes in the graph. This gives rise to a system of approximate ODEs (ordinary differential equations) where the challenge is to compute/approximate analytically the transmission rates. We show that this is possible for graphs with arbitrary degree distributions built according to the configuration model. Numerical tests confirm that: (a) the agreement of the approximate ODEs system with simulation is excellent and (b) that the approach remains valid for clustered graphs with the analytical calculations of the transmission rates still pending. The marked reduction in state space gives good results, and where the transmission rates can be analytically approximated, the model provides a strong alternative approximate model that agrees well with simulation. Given that the transmission rates encompass information both about the dynamics and graph properties, the specific shape of the curve, defined by the transmission rate versus the number of infected nodes, can provide a new and different measure of network structure, and the model could serve as a link between inferring network structure from prevalence or incidence data.

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Section: Introductionmentioning

confidence: 99%

Approximate Master Equations for Dynamical Processes on Graphs

Nagy

Kiss

Šimon

2014

Math. Model. Nat. Phenom.

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“…Perhaps related but more widely used in the mathematical-biology community are the formal proofs by Decreusefond et al [9], Barbour and Reinert [3] and Janson et al [11] which are all concerned with the limiting mean-field equations of the SIR stochastic epidemic on configuration networks. Decreusefond et al [9] study a measure-valued process capturing the degrees of susceptible individuals and the number of edges between different types of nodes. They prove that, as N → ∞, the measure-valued process converges to a deterministic limit, from which the Volz [43] equations may be derived as a corollary.…”

Section: Model Derivationmentioning

confidence: 99%

“…Edge-based compartmental models [29] have also been successful in capturing SIR dynamics on configuration-like networks and for Markovian models, giving perhaps the most compact ODE models that provide an excellent approximation of the exact stochastic network epidemic. They become exact in some appropriate limits and conditions on the underlying network [3,9,11]. The overwhelming view is that such models complement each other and offer a different or complimentary perspective of the exact stochastic process.…”

Section: Introductionmentioning

confidence: 99%

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

2018

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We present the generalised 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 generalised 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 generalised pairwise model lies in approximating the time evolution of the epidemic.

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“…The single-disease, small initial condition limit of these equations has been proved exact [38]. These equations capture the fact that disconnected components are safe from outside introduction.…”

Section: B the Basic Modelmentioning

confidence: 99%

Cocirculation of infectious diseases on networks

Miller

2013

Phys. Rev. E

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We consider multiple diseases spreading in a static configuration model network. We make standard assumptions that infection transmits from neighbor to neighbor at a disease-specific rate and infected individuals recover at a disease-specific rate. Infection by one disease confers immediate and permanent immunity to infection by any disease. Under these assumptions, we find a simple, low-dimensional ordinary differential equations model which captures the global dynamics of the infection. The dynamics depend strongly on initial conditions. Although we motivate this Rapid Communication with infectious disease, the model may be adapted to the spread of other infectious agents such as competing political beliefs, or adoption of new technologies if these are influenced by contacts. As an example, we demonstrate how to model an infectious disease which can be prevented by a behavior change.

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Large graph limit for an SIR process in random network with heterogeneous connectivity

Cited by 91 publications

References 23 publications

Approximate Master Equations for Dynamical Processes on Graphs

Approximate Master Equations for Dynamical Processes on Graphs

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

Cocirculation of infectious diseases on networks

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