  We study the minimum eradication time problem for controlled
Susceptible-Infected-Recovered (SIR) epidemic models that incorporate
vaccination control and time-varying infected and recovery rates. Unlike the
SIR model with constant rates, the time-varying model is more delicate as the
number of infectious individuals can oscillate, which causes ambiguity for the
definition of the eradication time. We accordingly introduce two definitions
that describe the minimum eradication time, and we prove that for a suitable
choice of the threshold, the two definitions coincide. We also study the
well-posedness of time-dependent Hamilton-Jacobi equation that the minimum
eradication time satisfies in the viscosity sense and verify that the value
function is locally semiconcave under certain conditions.
