Using control variables such as the level of medicare programme effort and the level of inoculation programme effort, three simple mathematical models of epidemics are transformed into optimal control problems. These are multi-state problems with the state variables as numbers of infectives and susceptibles and with the objective function being minimization of the total present value of the social or economic costs of infectives and medical controls. The problems are analysed with the framework of the maximum principle to obtain or, at least, partially characterize the optimal policies over time.