In this paper we study random non-autonomous second order linear differential equations by taking advantage of the powerful theory of random difference equations. The coefficients are assumed to be stochastic processes, and the initial conditions are random variables both defined in a common underlying complete probability space. Under appropriate assumptions established on the data stochastic processes and on the random initial conditions, and using key results on difference equations, we prove the existence of an analytic stochastic process solution in the random mean square sense. Truncating the random series that defines the solution process, we are able to approximate the main statistical properties of the solution, such as the expectation and the variance. We also obtain error a priori bounds to construct reliable approximations of both statistical moments. We include a set of numerical examples to illustrate the main theoretical results established throughout the paper. We finish with an example where our findings are combined with Monte Carlo simulations to model uncertainty using real data.

The frailty syndrome increases the morbidity/mortality in older adults, and several studies have shown a higher prevalence of this syndrome in patients with Chronic Obstructive Pulmonary Disease (COPD). The aim of this study was to identify the characteristics of frail patients with COPD to define a new phenotype called "COPD-frail." We conducted a cross-sectional study in a cohort of patients with stable COPD, classified as either frail, pre-frail, or non-frail. Sociodemographic, clinical, and biochemical variables were compared between the three groups of patients. The study included 127 patients, of which 31 were frail, 64 were pre-frail, and 32 non-frail. All subjects had FEV1/FVC below the lower limit of normal (range Z-score:-1.66 and -5.32). Patients in the frail group showed significantly higher scores in the mMRC (modified Medical Research Council) scale, the CAT (COPD Assessment Test), and the BODE (Body mass index, airflow Obstruction, Dyspnea, and Exercise capacity) index. They also showed differences in symptoms according to GOLD (Global Initiative for Chronic Obstructive Lung Disease), as well as more COPD exacerbations, less physical activity, more anxiety and depression symptoms based on HADS (Hospital Anxiety and Depression Scale), and lower hemoglobin, hematocrit, and 25-hydroxycholecalciferol levels. Variables with independent association with frailty included the mMRC score, the HAD index for depression and age. In summary, differential characteristics of frail patients with COPD encourage the definition of a "COPD-frail" phenotype that-if identified early-would allow performing interventions to prevent a negative impact on the morbidity/mortality of these patients.

This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a symbolic Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper.

This paper deals with the damped pendulum random differential equation:Ẍ(t) + 2ω 0 ξẊ(t) + ω 2 0 X(t) = Y (t), t ∈ [0, T ], with initial conditions X(0) = X 0 andẊ(0) = X 1 . The forcing term Y (t) is a stochastic process and X 0 and X 1 are random variables in a common underlying complete probability space (Ω, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L p senses. To understand the probabilistic behaviour of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function f X(t) (x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y (t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Itô type; and Y (t) can be approximated by a sequence {Y N (t)} ∞ N =1 in L 2 ([0, T ] × Ω), which occurs with Karhunen-Loève expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X 0 and X 1 and a specific stochastic process Y (t), and then, we find the probability density function of X(t).

In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X 0 and X 1 . In a previous study [Calbo G. et al, Comput. Math. Appl., 61(9), 2782-2792], a mean square convergent power series solution on (−1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X 0 and X 1 , independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L p solution (1 ≤ p ≤ ∞) to the random Legendre differential equation on the whole domain (−1, 1), as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of X 0 and X 1 . Moreover, the growth condition on the moments of A is characterized by the boundedness of A, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.

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]...

In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay τ > 0: x (t) = ax(t) + bx(t − τ), t ≥ 0, with initial condition x(t) = g(t), −τ ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. By using L p-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an L p-solution too. An analysis of L p-convergence when the delay τ tends to 0 is also performed in detail.

This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.