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 extend a recently proposed 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 Hamiltonian models based on the book and oscillator algebras, denoted by ${\mathfrak{b}}_2$ and ${\mathfrak{h}}_4$, respectively. The last generalization corresponds to an SIS system possessing the so-called two-photon algebra symmetry ${\mathfrak{h}}_6$, according to the embedding chain ${\mathfrak{b}}_2\subset {\mathfrak{h}}_4\subset {\mathfrak{h}}_6$, for which an exact solution cannot generally be found but a nonlinear superposition rule is explicitly given.