  Schematic su(2)+h3 interaction Hamiltonians, where su(2) plays the role of
the pseudo-spin algebra of fermion operators and h3 is the Heisenberg algebra
for bosons, are shown to be closely related to certain nonlinear models defined
on a single quantum algebra q-su(2) of quasifermions. In particular, q-su(2)
analogues of the Da Providencia-Schutte and extended Lipkin models are
presented. The connection between q and the physical parameters of the
fermion-boson system is analysed, and the integrability properties of the
interaction Hamiltonians are discussed by using polynomial algebras.
