Computational uncertainty quantification for random time‐discrete epidemiological models using adaptive gPC

Abstract

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. KEYWORDSadaptive gPC, computational methods for stochastic equations, computational uncertainty quantification, random nonlinear difference equations model, random population dynamics model, random time-discrete epidemiological model, stochastic difference equations INTRODUCTIONDiscrete models, usually expressed via finite difference equations, and continuous models, often expressed by means of ordinary and partial differential equations, allow us to get a better understanding of the processes involved in epidemiology. 1-5 These models for population dynamics have been usually studied in a deterministic sense, treating the involved input parameters (initial conditions, forcing term, and/or coefficients) as constants. Recent examples in this regard, dealing with important epidemiological models by applying new deterministic approaches, like nonstandard finite difference schemes, modal infinite series expansions, analysis of bifurcations, for example, include previous works. [6][7][8] However, due to the inherent uncertainty associated with epidemiological phenomena, it would be better to treat the input parameters in a random sense. For instance, the coefficient that describes the proportion of individuals that recover from a disease and become susceptible again should take into account the uncertainties involved in the measurements, due to errors in the collection of data, missed individuals, lack of information, etc.The most well-known method to deal computationally with stochastic systems is Monte Carlo simulation. 9 Although it is an effective and easy to implement approach to quantify the uncertainty, the slowness to get accurately the digits in the computations makes this technique computationally expensive.Only recently, some random continuous-time epidemic models have been studied using generalized polynomial chaos (gPC). [10][11][12]...

On the Legendre differential equation with uncertainties at the regular‐singular point 1: L p ( Ω ) random power series solution and approximation of its statistical moments

“…A variation of gPC expansions for continuous stochastic systems with dependent and jointly absolutely continuous random inputs, which is a method described in the work of Cortés et al and applied in the work of Calatayud et al We consider the canonical bases and the vector of random input coefficients, ζ =( A , Y 0 , Y 1 ). We consider a simple tensor product to construct a basis of monomials of degree less than or equal to m where where i 1 + i 2 + i 3 ≤ m and i ↔( i 1 , i 2 , i 3 ) in a bijective manner.…”

“…Monte Carlo simulations require many realizations or simulations of X(t) to get accurately its statistics (the error convergence rate is inversely proportional to the square root of the number m of realizations); therefore, the computational cost of Monte Carlo simulations is higher than our method based on random series. • A variation of gPC expansions for continuous stochastic systems with dependent and jointly absolutely continuous random inputs, which is a method described in the work of Cortés et al 19 and applied in the work of Calatayud et al 20 We consider the canonical bases…”

Computational methods for random differential equations: probability density function and estimation of the parameters

“…The stochastic Galerkin projection technique and gPC expansions have been profusely used in the literature for uncertainty quantification, see for example [24,35,90,117,126,138,144].…”

“…Then we will introduce a small perturbation into one of the parameters (with mean value being the deterministic estimate calculated) and we will apply the methodology previously exposed. Introducing only a small amount of randomness into the model from deterministic estimates allows a more faithful representation of the time evolution of the population, see for example [33,Example 5], [13,24,117,126,127]. In this article, as we are interested in testing our methodology on approximating the density function of the model output, but not in estimating probability distributions for the input parameters, we do not deal with inverse parameter estimation, see, for example, references [143] (pp.…”

“…It is easy to implement and effective, but may be computationally expensive. A quite latest alternative is based on generalized polynomial chaos (gPC) expansions and the stochastic Galerkin projection technique [24,33,35,46,90,117,126,138,142,143,144]. The stochastic solution is approximated as a linear combination of orthogonal polynomials with respect to the distributions of the input coefficients.…”