We consider a deterministic model of Methicillin-resistant Staphylococcus aureus infections in hospitals with seasonal oscillations of the antibiotic prescription rate. The model compartments consist of uncolonized patients with or without antibiotic exposure, colonized patients with or without antibiotic exposure, uncontaminated or contaminated healthcare workers, and free-living bacteria in the environment. We apply optimal control theory to this seven-compartment periodic system of ordinary differential equations to reduce the number of colonized patients and density of bacteria in the environment while minimizing the cost associated with environmental cleaning and antibiotic use in a particular time period. Characterizations of optimal control strategies are formulated and the ways hospitals should adjust these strategies for different scenarios are discussed. Numerical simulations strongly suggest that environmental cleaning is essential in the control of MRSA infections and antibiotic usage is suggested to be maintained at the least possible level. Screening, isolating, and shortening the extremely lengthened stays of colonized patients with antibiotic use history are all effective intervention strategies.