The homogenization of the Maxwell equations at fixed frequency is
addressed in this paper. The bulk (homogenized) electric and magnetic
properties of a material with a periodic microstructure are found from the
solution of a local problem on the unit cell by suitable averages. The material
can be anisotropic and satisfies a coercivity
condition. The exciting field is generated by an incident field from sources
outside the material under investigation. A suitable sesquilinear
form is defined for the interior problem, and the exterior Calderón
operator is used to solve the exterior radiating fields. The concept of
two-scale convergence is employed to solve the homogenization problem. A new a
priori estimate is proved as well as a new result on the correctors.Vaccination
of both newborns and susceptibles is included in a
transmission model for a disease that confers immunity. The interplay of the
vaccination strategy together with the vaccine efficacy and waning is studied.
In particular, it is shown that a backward bifurcation leading to bistability can occur. Under mild parameter constraints,
compound matrices are used to show that each orbit limits to an equilibrium. In
the case of bistability, this global result requires
a novel approach since there is no compact absorbing set.