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Inspired by the paper Greenhalgh et al. [5] we investigate a class of twodimensional stochastic differential equations related tosusceptible-infected-susceptible epidemic models with demographicstochasticity. While preserving the key features of the model considered in[5], where an ad hoc approach has been utilized to prove existence, uniquenessand non explosivity of the solution, we consider an encompassing family ofmodels described by a stochastic differential equation with random and H\"oldercontinuous coefficients. We prove the existence of a unique strong solution bymeans of a Cauchy-Euler-Peano approximation scheme which is shown to convergein the proper topologies to the unique solution

On a class of stochastic differential equations with random and H\"older continuous coefficients arising in biological modeling