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.