Hydrodynamics of $N$-urn susceptible-infected-removed epidemics

In this paper we are concerned with $N$-urn susceptible-infected-removed epidemics, where each urn is in one of three states, namely `susceptible', `infected' and `removed'. We assume that recovery rates of infected urns and infection rates between infected and susceptible urns are all coordinate-dependent. We show that the hydrodynamic limit of our model is driven by a deterministic $(C[0, 1])^\prime$-valued process with density which is the solution to a nonlinear $C[0, 1]$-valued ordinary differential equation consistent with a mean-field analysis. We further show that the fluctuation of our process is driven by a generalized Ornstein-Uhlenbeck process. A key step in proofs of above main results is to show that sates of different urns are approximately independent as $N\rightarrow+\infty$.