Using the theory of Lie-Hamilton systems, formal generalized stochastic
Hamiltonian systems that enlarge a recently proposed stochastic SIS epidemic
model with a variable infection rate are considered. It is shown that,
independently on the particular interpretation of the time-dependent
coefficients, these systems generally admit an exact solution, up to the case
of the maximal extension within the classification of Lie-Hamilton systems, for
which a superposition rule is constructed. The method provides the algebraic
frame to which any SIS epidemic model that preserves the above mentioned
properties is subjected. In particular, we obtain exact solutions for
generalized SIS Hamitonian models based on the book and oscillator algebras,
denoted respectively by b2β and h4β. The last
generalization corresponds to a SIS system possessing the so-called two-photon
algebra symmetry h6β, according to the embedding chain
b2ββh4ββh6β, for which an
exact solution cannot generally be found, but a nonlinear superposition rule is
explicitly given.Comment: 24 page