Abstract
In this chapter, we consider SIR, SIS, and SEIR epidemic models under discontinuous treatment. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on nonsmooth analysis, the basic reproductive number R0 is proved to be a sharp threshold value that completely determines the dynamics of the model. We discuss that the disease will die out in a finite time, which is impossible for the corresponding model with continuous treatment. Furthermore, the numerical simulations indicate that strengthening a treatment measure after infective individuals reach some level is beneficial to disease control.
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