  In this paper we are concerned with the SIR model with random vertex weights
on Erd\H{o}s-R\'{e}nyi graph $G(n,p)$. The Erd\H{o}s-R\'{e}nyi 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 $t\geq 0$ and show
that for any $t\geq 0$, $H_S(\psi_t)$ 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.
