The first two terms represent the benefit of the susceptible and vaccinated populations. The parameters B 1 and B 2 represent the weight constraints for the infected population and the control, respectively. They can also represent balancing coefficients transforming the integral into dollars expended over a finite time period of T days [34] . The goal is to maximize the populations of susceptible and vaccinated individuals, minimize the population of infectives, maximize the benefits of media coverage and vaccination, while minimizing the systemic costs of both media coverage and vaccination. The value u v (t) = u m (t) = 1 represents the maximal control due to vaccination and media coverage, respectively. The terms B u t v 2 2 ( ) and B u t m 2 2 ( ) represent the maximal cost of education, implementation and campaigns on both vaccination and media coverage. S(t) and V(t) account for the fitness of the susceptible and the vaccinated groups as a result of a reduction in the rate at which the vaccine wanes, and vaccination and treatment efforts are implemented [35] . We thus seek optimal controls u t v * ( ) and u t m * ( ) such that