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The Random Variable Transformation (RVT) technique has been applied in recent years to analyze a wide variety of dynamic models formulated via random differential equations. The applicability of this technique has usually been focused on problems where an explicit solution of the underlying deterministic problem is available. This fact limits the usefulness of the RVT method. This note aims to point out that the RVT technique can be successfully applied without this requirement by showing a wider range of potential applications including very general classes of single-species models. [ABSTRACT FROM AUTHOR]

In this work, we consider control problems represented by a linear differential equation assuming that all the coefficients are random variables and with an additive control that is a stochastic process. Specifically, we will work with controllable problems in which the initial condition and the final target are random variables. The probability density function of the solution and the control has been calculated. The theoretical results have been applied to study, from a probabilistic standpoint, a damped oscillator. [ABSTRACT FROM AUTHOR]

In this work we consider a particular randomized kinetic model for reaction–deactivation of hydrogen peroxide decomposition. We apply the random variable transformation technique to obtain the first probability density function of the solution stochastic process under general conditions. From the first probability density function, we can obtain fundamental statistical information, such as the mean and the variance of the solution, at every instant time. The transformation considered in the application of the random variable transformation technique is not unique. Then, the first probability density function can take different expressions, although essentially equivalent in terms of computing probabilistic information. To motivate this fact, we consider in our analysis two different mappings. Several numerical examples show the capability of our approach and of the obtained results as well. We show, through simulations, that the choice of the transformation, that permits computing the first probability density function, is a crucial issue regarding the computational time. [ABSTRACT FROM AUTHOR]

In this work, we study the full randomized versions of Airy, Hermite and Laguerre differential equations, which depend on a random variable appearing as an equation coefficient as well as two random initial conditions. In previous contributions, the mean square stochastic solutions to the aforementioned random differential equations were constructed via the Fröbenius method, under the assumption of exponential growth of the absolute moments of the equation coefficient, which is equivalent to its essential boundedness. In this paper we aim at relaxing the boundedness hypothesis to allow more general probability distributions for the equation coefficient. We prove that the equations are solvable in the mean square sense when the equation coefficient has finite moment-generating function in a neighborhood of the origin. A thorough discussion of the new hypotheses is included. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P′(t,ω)=A(t,ω)(1−P(t,ω))P(t,ω), t∈[t0,T], P(t0,ω)=P0(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P0 to approximate the density function f1(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P0, and the density of P0, fP0, is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L2([t0,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P0, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For fP0, only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence {f1N(p,t)}N=1∞ of density functions in terms of expectations that tends to f1(p,t) pointwise. Numerical examples illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However, discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state, and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples: The first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data. [ABSTRACT FROM AUTHOR]

Candida Auris is a pathogen that can express multiple virulence factors and is considered a significant menace to the Intensive Care Unit environment. In this work, we propose to model its growth with a computationally randomized Fisher Kolmogorov–Petrovsky–Piskunov partial differential equation. We calibrate the model using in vitro growth data and provide statistical features. We also evaluate the efficacy of a regular and uniform cleaning strategy with vaporized hydrogen peroxide to control Candida Auris' population in ICUs. • Candida Auris is a threat to the healthcare environment. • Random Differential Equations are used to capture the spread of a Candida Auris. • Regular Cleaning with vaporized hydrogen peroxide controls outbreaks of Candida Auris. [ABSTRACT FROM AUTHOR]

The bomb attacks in Madrid three days before the general elections of March 14, 2004, and their possible influence on the victory of PSOE (Spanish Workers Socialist Party), defeating PP (Popular Party), have been a matter of study from several points of view (i.e., sociological, political, or statistical). In this paper, we present a dynamic model based on a system of differential equations such that it, using data from Spanish CIS (National Center of Sociological Research), describes the evolution of voting intention of the Spanish people over time. Using this model, we conclude that the probability is very low that the PSOE would have won had the attack not happened. Moreover, after the attack, the PSOE increased an average of 5.6% in voting on March 14 and an average of 11.2% of the Spanish people changed their vote between March 11 and March 14. These figures are in accordance with other studies. [ABSTRACT FROM AUTHOR]

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable. [ABSTRACT FROM AUTHOR]

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation. [ABSTRACT FROM AUTHOR]