  We present a method to quantify uncertainty in the predictions made by
simulations of mathematical models that can be applied to a broad class of
stochastic, discrete, and differential equation models. Quantifying uncertainty
is crucial for determining how accurate the model predictions are and
identifying which input parameters affect the outputs of interest. Most of the
existing methods for uncertainty quantification require many samples to
generate accurate results, are unable to differentiate where the uncertainty is
coming from (e.g., parameters or model assumptions), or require a lot of
computational resources. Our approach addresses these challenges and
opportunities by allowing different types of uncertainty, that is, uncertainty
in input parameters as well as uncertainty created through stochastic model
components. This is done by combining the Karhunen-Loeve decomposition,
polynomial chaos expansion, and Bayesian Gaussian process regression to create
a statistical surrogate for the stochastic model. The surrogate separates the
analysis of variation arising through stochastic simulation and variation
arising through uncertainty in the model parameterization. We illustrate our
approach by quantifying the uncertainty in a stochastic ordinary differential
equation epidemic model. Specifically, we estimate four quantities of interest
for the epidemic model and show agreement between the surrogate and the actual
model results.
