Inspired by the paper Greenhalgh et al. [5] we investigate a class of two dimensional stochastic differential equations related to susceptible-infected-susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in [5], where an ad hoc approach has been utilized to prove existence, uniqueness and non explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and Hölder continuous coefficients. We prove the existence of a unique strong solution by means of a Cauchy-Euler-Peano approximation scheme which is shown to converge in the proper topologies to the unique solution.

No abstract

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie–Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379–406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two-dimensional framework and propose a direct proof of the Duffie–Kan theorem which does not passes through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one-dimensional square root process is assigned.