This paper deals with the study and control of the transmission of an infectious disease described by a Susceptible–Vaccinated–Infected (SVI) epidemiological model. This model takes into account the age structure of the infection, imperfect vaccination and therapeutic treatment of infected people who become susceptible again. We show that the rate of therapeutic treatment can lead to bistability of disease–free and an endemic equilibria. This phenomenon is known as backward bifurcation and can allow the disease to persist in a population even if the basic reproduction number is less than one. The existence of this bifurcation means that controlling this infectious disease remains extremely difficult. This is why we are developing an optimal control strategy combining vaccination and treatment in order to minimize the disease spread in the population. We then consider vaccination and treatment rates as time–dependent functions. This allows us to formulate an optimal control problem whose objective is to minimize the number of infected individuals as well as the cost of applying controls. We establish first–order necessary conditions for optimality and characterize optimal controls. Numerical simulations illustrate the fact that a combination of intervention measures can significantly reduce the transmission dynamics of infectious diseases by keeping the population of infected individuals relatively low.