Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5102565 | Physica A: Statistical Mechanics and its Applications | 2017 | 12 Pages |
Abstract
In this paper we are concerned with the SIR model with random vertex weights on ErdÅs-Rényi graph G(n,p). The ErdÅs-Rényi graph G(n,p) is generated from the complete graph Cn with n vertices through independently deleting each edge with probability (1âp). We assign i. i. d. copies of a positive r. v. Ï on each vertex as the vertex weights. For the SIR model, each vertex is in one of the three states 'susceptible', 'infective' and 'removed'. An infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices. An infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at t=0 there is no removed vertex and the number of infective vertices follows a Bernoulli distribution B(n,θ). Our main result is a law of large numbers of the model. We give two deterministic functions HS(Ït),HV(Ït) for tâ¥0 and show that for any tâ¥0, HS(Ït) is the limit proportion of susceptible vertices and HV(Ït) is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment t as n grows to infinity.
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Physical Sciences and Engineering
Mathematics
Mathematical Physics
Authors
Xiaofeng Xue,