Transmission dynamics and optimal control of measles epidemics. (English) Zbl 1338.92135
Summary: 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, \(\mu_{c 1}\) and \(\mu_{c 2}\), of the vaccine coverage ratio. Measles will be extinct when the vaccination ratio \(\mu > \mu_{c 1}\), endemic when \(\mu_{c 2} < \mu < \mu_{c 1}\), and outbreak periodically when \(\mu < \mu_{c 2}\). In addition, we apply the optimal control theory to obtain an optimal vaccination strategy \(\mu^\ast(t)\) 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.
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