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In this paper, we present a three-stage algorithm for finding numerical solutions for optimal control problems. The algorithm first performs an exhaustive search through a discrete set of widely dispersed solutions which are representative of large subregions of the search space; then, it uses the search results to initialize a Monte Carlo process that searches quasi-randomly for a best solution; then, it finally uses a Newton-type iteration to converge to a solution that satisfies mathematical conditions of local optimality. We demonstrate our methodology on an epidemiological model of the coronavirus disease with testing and distancing controls applied over a period of 180 days to two different subpopulations (low-risk and high-risk), where model parameters are chosen to fit the city of Houston, Texas, USA. In order to enable the user to select his/her preferred trade-off between (number of deaths) and (herd immunity) outcomes, the objective function includes costs for deaths and non-immunity. Optimal strategies are estimated for a grid of (death cost) × (non-immunity cost) combinations, in order to obtain a Pareto curve that represents optimum trade-offs. The levels of the four controls for the different Pareto-optimal solutions over the 180-day period are visually represented and their characteristics discussed. Three different variants of the algorithm are run in order to determine the relative importance of the three stages in the optimization. Results from the three algorithm variants are fairly consistent, indicating that solutions are robust. Results also show that the Monte Carlo stage plays an especially prominent role in the optimization, but that all three stages of the process make significant contributions towards finding lower-cost, more effective control strategies. [ABSTRACT FROM AUTHOR]