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

[EN] 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.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Chen-Charpentier, BM.; Cortés, J.; Jornet-Sanz, M. (2019). Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models. Symmetry (Basel). 11(1):1-28. https://doi.org/10.3390/sym11010043S128111Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Bharucha-Reid, A. T. (1964). 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Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC

[EN] Population dynamics models consisting of nonlinear difference equations allow us to get a better understanding of the processes involved in epidemiology. Usually, these mathematical models are studied under a deterministic approach. However, in order to take into account the uncertainties associated with the measurements of the model input parameters, a more realistic approach would be to consider these inputs as random variables. In this paper, we study the random time-discrete epidemiological models SIS, SIR, SIRS, and SEIR using a powerful unified approach based upon the so-called adaptive generalized polynomial chaos (gPC) technique. The solution to these random difference equations is a stochastic process in discrete time, which represents the number of susceptible, infected, recovered, etc individuals at each time step. We show, via numerical experiments, how adaptive gPC permits quantifying the uncertainty for the solution stochastic process of the aforementioned random time-discrete epidemiological model and obtaining accurate results at a cheap computational expense. We also highlight how adaptive gPC can be applied in practice, by means of an example using real data.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the helpful and valuable reviewers' comments that have considerably improved the final form of this manuscript.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M.; Villanueva Micó, RJ. (2018). Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC. Mathematical Methods in the Applied Sciences. 41(18):9618-9627. https://doi.org/10.1002/mma.5315S96189627411

Modelling soil-transmitted helminths using generalised polynomial chaos.

Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.Soil-transmitted helminths are the group of neglected tropical diseases for humans caused by parasitic worms; Ascaris lumbricoides, Trichuris trichiura and hookworm species. It occurs mostly in tropical and subtropical regions as it survives in warm and moist temperature. The disease is a severe public health problem as it is prevalent to school-aged children. In this study, a non-linear mathematical model is formulated to model the transmission dynamics and the spread of soiltransmitted Helminths. Firstly, the deterministic model is formulated, and the stability analysis of the model was performed. The disease-free equilibrium is globally asymptotically stable for the basic reproduction number R0 1, a unique endemic equilibrium exists and is globally asymptotically stable. Secondly, we apply the polynomial chaos approach to the system of differential equations with random coefficients. This approach takes into account the randomness in the model parameters. The polynomial chaos is applied resulting in a system of coupled ordinary differential equations. The resulting system is solved numerically to obtain the first-order and second-order moments of the stochastic output processes. Sensitivity analysis based on Sobol indices is also employed to determine parameters with the most significant influence on the output. Finally, both deterministic and polynomial chaos simulations reveal that the reduction of the contact rate can reduce the size of the epidemic. The polynomial chaos numerical simulations show low volatility in the number of susceptible, exposed and infectious human population compared to the egg and larva density which is chaotic

Uncertainty quantification for nonlinear difference equations with dependent random inputs via a stochastic Galerkin projection technique

[EN] Discrete stochastic systems model discrete response data on some phenomenon with inherent uncertainty. The main goal of uncertainty quantification is to derive the probabilistic features of the stochastic system. This paper deals with theoretical and computational aspects of uncertainty quantification for nonlinear difference equations with dependent random inputs. When the random inputs are independent random variables, a generalized Polynomial Chaos (gPC) approach has been usually used to computationally quantify the uncertainty of stochastic systems. In the gPC technique, the stochastic Galerkin projections are done onto linear spans of orthogonal polynomials from the Askey-Wiener scheme or from Gram-Schmidt orthonormalization procedures. In this regard, recent results have established the algebraic or exponential convergence of these Galerkin projections to the solution process. In this paper, as the random inputs of the difference equation may be dependent, we perform Galerkin projections directly onto linear spans of canonical polynomials. The main contribution of this paper is to study the spectral convergence of these Galerkin projections for the solution process of general random difference equations. Spectral convergence is important to derive the main statistics of the response process at a cheap computational expense. In this regard, the numerical experiments bring to light the theoretical discussion of the paper.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. The co-author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Uncertainty quantification for nonlinear difference equations with dependent random inputs via a stochastic Galerkin projection technique. Communications in Nonlinear Science and Numerical Simulation. 72:108-120. https://doi.org/10.1016/j.cnsns.2018.12.011S1081207

Efficient Uncertainty Quantification and Variance-Based Sensitivity Analysis in Epidemic Modelling Using Polynomial Chaos

The use of epidemic modelling in connection with spread of diseases plays an important role in understanding dynamics and providing forecasts for informed analysis and decision-making. In this regard, it is crucial to quantify the effects of uncertainty in the modelling and in model-based predictions to trustfully communicate results and limitations. We propose to do efficient uncertainty quantification in compartmental epidemic models using the generalized Polynomial Chaos (gPC) framework. This framework uses a suitable polynomial basis that can be tailored to the underlying distribution for the parameter uncertainty to do forward propagation through efficient sampling via a mathematical model to quantify the effect on the output. By evaluating the model in a small number of selected points, gPC provides illuminating statistics and sensitivity analysis at a low computational cost. Through two particular case studies based on Danish data for the spread of Covid-19, we demonstrate the applicability of the technique. The test cases consider epidemic peak time estimation and the dynamics between superspreading and partial lockdown measures. The computational results show the efficiency and feasibility of the uncertainty quantification techniques based on gPC, and highlight the relevance of computational uncertainty quantification in epidemic modelling.Peer reviewe