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

Abstract

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 $C_n$ with $n$ vertices through independently deleting each edge with probability $(1-p)$. We assign i. i. d. copies of a positive r. v. $\rho$ 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 $B(n,\theta )$. Our main result is a law of large numbers of the model. We give two deterministic functions $H_S({\psi }_t),H_V({\psi }_t)$ for $t\ge 0$ and show that for any $t\ge 0$, $H_S({\psi }_t)$ is the limit proportion of susceptible vertices and $H_V({\psi }_t)$ 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.

13 pages

References in corpus (1)

• Phase transition for the SIR model with random transition rates on complete graphs

Cited by in corpus (3)

• Asymptotic for critical value of the large-dimensional SIR epidemic on clusters
• Statistical inference for unknown parameters of stochastic SIS epidemics on complete graphs
• Mean field limit of local density-dependent Markov processes on weighted Erdos-Renyi graphs