  This paper demonstrates input-to-state stability (ISS) of the SIR model of
infectious diseases with respect to the disease-free equilibrium and the
endemic equilibrium. Lyapunov functions are constructed to verify that both
equilibria are individually robust with respect to perturbation of
newborn/immigration rate which determines the eventual state of populations in
epidemics. The construction and analysis are geometric and global in the space
of the populations. In addition to the establishment of ISS, this paper shows
how explicitly the constructed level sets reflect the flow of trajectories.
Essential obstacles and keys for the construction of Lyapunov functions are
elucidated. The proposed Lyapunov functions which have strictly negative
derivative allow us to not only establish ISS, but also get rid of the use of
LaSalle's invariance principle and popular simplifying assumptions.
