On a Class of Stochastic Differential Equations with Random and Hölder Continuous Coefficients Arising in Biological Modeling

Inspired by the paper Greenhalgh et al. (Appl Math Comput 276:218–238, 2016), 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 Greenhalgh et al. (2016), 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.