In this paper, we develop a robust economic model predictive controller for the containment of stochastic susceptible-exposed-infected-vigilant (SEIV) epidemic processes, which drives the process to extinction quickly, while minimizing the rate at which control resources are used. The study we present here is significant in that it addresses the problem of efficiently controlling general stochastic epidemic systems without relying on mean-field approximation, which is an important issue in the theory of stochastic epidemic processes. This enables us to provide rigorous convergence guarantees on the stochastic epidemic model itself, improving over the mean-field type convergence results of most prior work. There are two primary technical difficulties addressed in treating this problem: 1) constructing a means of tractably approximating the evolution of the process so that the designed approximation is robust to the modeling error introduced by the applied moment closure and 2) guaranteeing that the designed controller causes the closed-loop system to drive the SEIV process to extinction quickly. As an application, we use the developed framework for optimizing the use of quarantines in containing an SEIV epidemic outbreak.