Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations

Using the theory of Lie-Hamilton systems, formal generalized time-dependent Hamiltonian systems that enlarge a recently proposed SIS epidemic model with variable infection rate are considered. It is shown that, independently on particular interpretation coefficients, these generally admit an exact solution, up to case maximal extension within classification for which superposition rule constructed. The method provides algebraic frame any preserves above mentioned properties subjected. In particular, we obtain solutions models based book and oscillator algebras, denoted by $\mathfrak{b}_2$ $\mathfrak{h}_4$, respectively. last generalization corresponds system possessing so-called two-photon algebra symmetry $\mathfrak{h}_6$, according embedding chain $\mathfrak{b}_2\subset \mathfrak{h}_4\subset \mathfrak{h}_6$, solution cannot be found, but nonlinear explicitly given.