Transmission dynamics and optimal control of measles epidemics

Based on the mechanism and characteristics of measles transmission, we propose a susceptible-exposed-infectious-recovered (SEIR) measles epidemic model with vaccination and investigate the effect of vaccination in controlling the spread of measles. We obtain two critical threshold values, μc1 and μc2, of the vaccine coverage ratio. Measles will be extinct when the vaccination ratio μ>μc1, endemic when μc2<μ<μc1, and outbreak periodically when μ<μc2. In addition, we apply the optimal control theory to obtain an optimal vaccination strategy μ∗(t) and give some numerical simulations for those theoretical findings. Finally, we use our model to simulate the data of measles cases in the U.S. from 1951 to 1962 and design a control strategy.