Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models

Julia Calatayud Gregori,Marc Jornet Sanz,Juan Carlos Cortés López,Benito M Chen-Charpentier

doi:10.3390/sym11010043

Julia Calatayud Gregori, Marc Jornet Sanz + Show 2 more

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https://doi.org/10.3390/sym11010043

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Abstract

In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.

Highlights

Mathematical models based on random differential and difference equations consider input parameters as random variables or stochastic processes [1,2,3]
The aim of this kind of models is to capture the uncertainty often met in the analysis of complex phenomena. This randomness comes from errors in measurements or estimations, a lack of information, or ignorance of the phenomenon under analysis, etc. These are typical features in dealing with the study of epidemiological models, where input data, such as the initial proportion of infected individuals, the rates of contagious and recovery caused by an infectious disease, etc., are not deterministically known
It is important to point out that the largest part of random variables that are used in practice, specially in epidemiological modeling, such as Beta, Gaussian, Gamma, Uniform, Triangular, etc., satisfy these hypotheses

Summary

Introduction

Mathematical models based on random differential and difference equations consider input parameters as random variables or stochastic processes [1,2,3]. This randomness comes from errors in measurements or estimations, a lack of information, or ignorance of the phenomenon under analysis, etc These are typical features in dealing with the study of epidemiological models, where input data, such as the initial proportion of infected individuals, the rates of contagious and recovery caused by an infectious disease, etc., are not deterministically known. In such a situation, it is more realistic to model all this initial information via random variables or stochastic processes rather than by using deterministic constants or functions, respectively. These facts have motivated the development of powerful mathematical techniques capable to model and quantify uncertainty in dealing with epidemiological problems to better understand and describe the dynamics of a disease

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