  In this work we use Floquet theory to theoretically study the influence of
circularly polarized light on disordered two-dimensional models exhibiting
topological transitions. We find circularly polarized light can induce a
topological transition in extended Kane-Mele models that include additional
hopping terms and on-site disorder. The topological transitions are understood
from the Floquet-Bloch band structure of the clean system at high symmetry
points in the first Brillouin zone. The light modifies the equilibrium band
structure of the clean system in such a way that the smallest gap in the
Brillouin zone can be shifted from the $M$ points to the $K(K')$ points, the
$\Gamma$ point, or even other lower symmetry points. The movement of the
minimal gap point through the Brillouin zone as a function of laser parameters
is explained in the high frequency regime through the Magnus expansion. In the
disordered model, we compute the Bott index to reveal topological phases and
transitions. The disorder can induce transitions from topologically non-trivial
states to trivial states or vice versa, both examples of Floquet topological
Anderson transitions. As a result of the movement of the minimal gap point
through the Brillouin zone as a function of laser parameters, the nature of the
topological phases and transitions is laser-parameter dependent--a contrasting
behavior to the Kane-Mele model.
