Solving Continuous Models with Dependent Uncertainty: A Computational Approach

Abstract

This paper presents a computational study on a quasi-Galerkin projection-based method to deal with a class of systems of random ordinary differential equations (r.o.d.e.’s) which is assumed to depend on a finite number of random variables (r.v.’s). This class of systems of r.o.d.e.’s appears in different areas, particularly in epidemiology modelling. In contrast with the other available Galerkin-based techniques, such as the generalized Polynomial Chaos, the proposed method expands the solution directly in terms of the random inputs rather than auxiliary r.v.’s. Theoretically, Galerkin projection-based methods take advantage of orthogonality with the aim of simplifying the involved computations when solving r.o.d.e.’s, which means to compute both the solution and its main statistical functions such as the expectation and the standard deviation. This approach requires the previous determination of an orthonormal basis which, in practice, could become computationally burden and, as a consequence, could ruin the method. Motivated by this fact, we present a technique to deal with r.o.d.e.’s that avoids constructing an orthogonal basis and keeps computationally competitive even assuming statistical dependence among the random input parameters. Through a wide range of examples, including a classical epidemiologic model, we show the ability of the method to solve r.o.d.e.’s.