The study and analysis of epidemic spreading models help in mitigating the propagation of a disease and its eradication. The model studied in this article is Susceptible-Infected-Susceptible-Unaware-Aware-Unaware (SIS-UAU) on a multiplex network, which captures the simultaneous spreading of epidemic and awareness, and their interplay in a population, under the assumption that both the networks can be represented by connected, undirected graphs. The treatment and campaigning efforts with bounds arising from practical limitations are considered as admissible control inputs to each node in the network. We formulate and solve an optimal control problem where the objective is to bring down the prevalence of disease in the population in a prescribed time. As a precursor to solving this problem, we compute the reachable set of the model, which paves the way to guarantee the existence of a solution to the optimal control problem. The solution to the optimal control problem on an example network is numerically obtained using direct discretization method. © 2019 IEEE.