Law of large numbers for the SIR model with random vertex weights on Erdős–Rényi graph

In this paper we are concerned with the SIR model with random vertex weights on Erdős–Rényi graph G(n,p). The Erdős–Rényi graph G(n,p) is generated from the complete graph Cn with n vertices through independently deleting each edge with probability (1−p). We assign i. i. d. copies of a positive r. v. ρ 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(ψt),HV(ψt) for t≥0 and show that for any t≥0, HS(ψt) is the limit proportion of susceptible vertices and HV(ψt) 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.