  The notion of quantum algebras is merged with that of Lie systems in order to
establish a new formalism called Poisson-Hopf algebra deformations of Lie
systems. The procedure can be naturally applied to Lie systems endowed with a
symplectic structure, the so-called Lie-Hamilton systems. This is quite a
general approach, as it can be applied to any quantum deformation and any
underlying manifold. One of its main features is that, Lie systems are extended
to generalized systems described by involutive distributions. In this way, we
obtain their new generalized (deformed) counterparts that cover, in particular,
a new oscillator system with a time-dependent frequency and a
position-dependent mass. Based on a recently developed procedure to construct
Poisson-Hopf deformations of Lie-Hamilton systems \cite{Ballesteros5}, a novel
unified approach to deformations of Lie-Hamilton systems on the real plane with
a Vessiot-Guldberg Lie algebra isomorphic to $\mathfrak{sl}(2)$ is proposed. In
addition, we study the deformed systems obtained from Lie-Hamilton systems
associated to the oscillator algebra $\mathfrak{h}_4$, seen as a subalgebra of
the 2-photon algebra $\mathfrak{h}_6$. As a particular application, we propose
an epidemiological model of SISf type that uses the solvable Lie algebra
$\mathfrak{b}_2$ as subalgebra of $\mathfrak{sl}(2)$, by restriction of the
corresponding quantum deformed systems.
