The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease in
a population of constant size is considered. In order to control the spread of infection,
we propose the model with four bounded controls which describe vaccination of newborns,
vaccination of the susceptible, treatment of infected, and indirect strategies aimed at a
reduction of the incidence rate (e. g. quarantine). The optimal control problem of
minimizing the total number of the infected individuals on a given time interval is stated
and solved. The optimal solutions are obtained with the use of the Pontryagin Maximum
Principle and investigated analytically. Numerical results are presented to illustrate the
optimal solutions.