Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series

[EN] The aim of this paper is to solve a class of non-autonomous linear fractional differential equations with random inputs. A mean square convergent series solution is constructed in the case that the fractional order a of that Caputo derivative lies in ]0,1] using a random Frobenius approach. The analysis is conducted by using the so-called mean square random calculus. The mean square convergence of the series solution is established assuming mild conditions on random inputs (diffusion coefficient and initial condition). We show that these conditions are satisfied for a variety of unbounded random variables. In addition, explicit expressions to approximate the mean, the variance and the covariance functions of the random series solution are given. Two full illustrative examples are shown. (C) 2017 Elsevier Ltd. All rights reserved.Authors gratefully acknowledge the comments made by reviewers, which have greatly enriched the manuscript. This work has been partially supported by Ministerio de Economia y Competitividad grant MTM2013-41765-P.Burgos-Simon, C.; Calatayud-Gregori, J.; Cortés, J.; Villafuerte, L. (2018). Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Applied Mathematics Letters. 78:95-104. https://doi.org/10.1016/j.aml.2017.11.009S951047

A full probabilistic solution of the random linear fractional differential equation via the Random Variable Transformation technique

[EN] This paper provides a full probabilistic solution of the randomized fractional linear nonhomogeneous differential equation with a random initial condition via the computation of the first probability density function of the solution stochastic process. To account for most generality in our analysis, we assume that uncertainty appears in all input parameters (diffusion coefficient, source term, and initial condition) and that a wide range of probabilistic distributions can be assigned to these parameters. Throughout our study, we will consider that the fractional order of Caputo derivative lies in] 0,1], that corresponds to the main standard case. To conduct our analysis, we take advantage of the random variable transformation technique to construct approximations of the first probability density function of the solution process from a suitable infinite series representation. We then prove these approximations do converge to the exact density assuming mild conditions on random input parameters. Our theoretical findings are illustrated through 2 numerical examples.Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Programa de Ayudas de Investigacion y Desarrollo, Grant/Award Number: PAID-2014; UNiversitat Politecncia de ValenciaBurgos-Simon, C.; Calatayud-Gregori, J.; Cortés, J.; Navarro-Quiles, A. (2018). A full probabilistic solution of the random linear fractional differential equation via the Random Variable Transformation technique. Mathematical Methods in the Applied Sciences. 41(18):9037-9047. https://doi.org/10.1002/mma.4881S90379047411

A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations

[EN] First, we propose a deterministic age-structured epidemiological model to study the diffusion of e-commerce in Spain. Afterwards, we determine the parameters (death, birth and growth rates) of the underlying demographic model as well as the parameters (transmission of the use of e-commerce rates) of the proposed epidemiological model that best fit real data retrieved from the Spanish National Statistical Institute. Motivated by the two following facts: first the dynamics of acquiring the use of a new technology as e-commerce is mainly driven by the feedback after interacting with our peers (family, friends, mates, mass media, etc.), hence having a certain delay, and second the inherent uncertainty of sampled real data and the social complexity of the phenomena under analysis, we introduce aftereffect and stochastic perturbations in the initial deterministic model. This leads to a delayed stochastic model for e-commerce. We then investigate sufficient conditions in order to guarantee the stability in probability of the equilibrium point of the dynamic e-commerce delayed stochastic model. Our theoretical findings are numerically illustrated using real data. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P.Burgos-Simon, C.; Cortés, J.; Shaikhet, L.; Villanueva Micó, RJ. (2018). A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations. Communications in Nonlinear Science and Numerical Simulation. 64:149-158. https://doi.org/10.1016/j.cnsns.2018.04.022S1491586

A computational technique to predict the level of glucose of a diabetic patient with uncertainty in the short term

[EN] On advanced stages of the disease, diabetic patients have to inject insulin doses to maintain blood glucose levels inside of a healthy range. The decision of how much insulin is injected implies somehow to predict the level of glucose they will have after a certain time. Due to the sudden changes in the glucose levels, their estimation is a very difficult task. If we were able to give reliable estimations in advance, it would facilitate the process of taking therapeutic decisions to control the disease and improve the health of the patient. In this work, we present a technique to estimate the glucose level of a diabetic patient, capturing the measurement errors produced by continuous glucose monitoring systems (CGMSs), smart devices that measure glucose levels. To do that, we will use a model of glucose dynamics and we calibrate it with the aim to capture the glucose level data of the patient in a time interval of 30 minutes and the uncertainty given by the glucose measurement. Then, we use the calibrated parameters to predict the levels of glucose over the next 15 minutes. Repeating this procedure every 15 minutes, we are able to give short¿term accurate predictions.This work has been partially supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2017-89664-P and RTI2018-095180-B-I00 and by Fundación Eugenio Rodriguez Pascual 2019 -GLENO ProjectBurgos Simon, C.; Cervigón, C.; Hidalgo, J.; Villanueva Micó, RJ. (2019). A computational technique to predict the level of glucose of a diabetic patient with uncertainty in the short term. Computational and Mathematical Methods. 2(2):1-11. https://doi.org/10.1002/cmm4.1064S11122(2004). Third-Party Reimbursement for Diabetes Care, Self-Management Education, and Supplies. Diabetes Care, 28(Supplement 1), S62-S63. doi:10.2337/diacare.28.suppl_1.s62Bloomgarden, Z. T. (2004). Consequences of Diabetes: Cardiovascular disease. Diabetes Care, 27(7), 1825-1831. doi:10.2337/diacare.27.7.1825BrownA.Time‐in‐range: what's an achievable goal with diabetes?2017.https://diatribe.org/time-range-whats-achievable-goal-diabetesFonseca, V. A., Grunberger, G., Anhalt, H., Bailey, T. S., Blevins, T., … Garg, S. K. (2016). CONTINUOUS GLUCOSE MONITORING: A CONSENSUS CONFERENCE OF THE AMERICAN ASSOCIATION OF CLINICAL ENDOCRINOLOGISTS AND AMERICAN COLLEGE OF ENDOCRINOLOGY. Endocrine Practice, 22(8), 1008-1021. doi:10.4158/ep161392.csChristiansen, M. P., Klaff, L. J., Brazg, R., Chang, A. R., Levy, C. J., Lam, D., … Bailey, T. S. (2018). A Prospective Multicenter Evaluation of the Accuracy of a Novel Implanted Continuous Glucose Sensor: PRECISE II. Diabetes Technology & Therapeutics, 20(3), 197-206. doi:10.1089/dia.2017.0142Bock, A., François, G., & Gillet, D. (2015). A therapy parameter-based model for predicting blood glucose concentrations in patients with type 1 diabetes. Computer Methods and Programs in Biomedicine, 118(2), 107-123. doi:10.1016/j.cmpb.2014.12.002Acedo, L., Botella, M., Cortés, J. C., Hidalgo, J. I., Maqueda, E., & Villanueva, R. J. (2018). Swarm hybrid optimization for a piecewise model fitting applied to a glucose model. Journal of Systems and Information Technology, 20(4), 404-416. doi:10.1108/jsit-10-2017-0103Alegre-Sanahuja, J., Cortés, J.-C., Villanueva, R.-J., & Santonja, F.-J. (2017). Predicting mobile apps spread: An epidemiological random network modeling approach. SIMULATION, 94(2), 123-130. doi:10.1177/003754971771260

Probabilistic prediction of outbreaks of meningococcus W-135 infections over the next few years in Spain

[EN] The genogroups of meningococcal and other bacteria are in competition in the ecosystem they form with the human hosts. Changes in vaccination strategies, prophylactic measures or usual habits, may also change the distribution of the genogroups in the ecosystem but, usually, this competition is ignored in most epidemiological models, despite it can be highly influential in the evolution of infection diseases and outbreaks. Our goal is to propose a susceptible carrier susceptible (SCS) epidemiological model to determine the percentage of carriers in the population, and introduce a fractional Lotka Volterra competition model to describe the evolution of the meningococcal genogroups in Spain among the carriers. Using data from the distribution of the genogroups in Spain in 2011 and 2012, we find the model parameters and their uncertainties according to a probabilistic fitting approach. On this basis, we predict the evolution of the carriers of the different genogroups over the next few years and, in particular, the percentage of carriers of meningococcus W-135 with a 95% confidence interval. Then, we estimate the probability of having a possible outbreak of meningococcus W-135 in Spain over the next few years. According to our model and, under the present conditions, the risk of a serious outbreak of W-135 in Spain in the next 3 years is below 0.3%.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P and the FIS grant PI13/01459. We also acknowledge Dr. Julio Vazquez from the Carlos III Institute of Health for providing the epidemiological data used in this work.Acedo Rodríguez, L.; Burgos-Simon, C.; Cortés, J.; Villanueva Micó, RJ. (2017). Probabilistic prediction of outbreaks of meningococcus W-135 infections over the next few years in Spain. Physica A Statistical Mechanics and its Applications. 486:106-117. https://doi.org/10.1016/j.physa.2017.05.043S10611748

Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions

[EN] The Baranyi-Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville-Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi-Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Bevia-Escrig, V.; Burgos-Simon, C.; Cortés, J.; Villanueva Micó, RJ. (2021). Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions. Fractal and Fractional. 5(2):1-18. https://doi.org/10.3390/fractalfract5020026S1185

Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

[EN] The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Burgos-Simon, C.; Cortés, J.; Debbouche, A.; Villafuerte, L.; Villanueva Micó, RJ. (2019). Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus. Applied Mathematics and Computation. 352:15-29. https://doi.org/10.1016/j.amc.2019.01.039S152935

Probabilistic Fitting of Glucose Models with Real-Coded Genetic Algorithms

[EN] Type 1 Diabetes patients have to control their blood glucose levels using insulin therapy. Numerous factors (such as carbohydrate intake, physical activity, time of day, etc.) greatly complicate this task. In this article we propose a modeling method that will allow us to make predictions of blood glucose level evolution with a time horizon of 24 hours. This may allow the adjustment of insulin doses in advance and could help to improve the living conditions of diabetes patients. Our approach starts from a system of finite difference equations that characterizes the interaction between insulin and glucose (in the field, this is known as a minimal model). This model has several parameters whose values vary widely depending on patient characteristics and time. Thus, in the first phase of our strategy, We will enrich the patient¿s historical data by adding white Gaussian noise, which will allow us to perform a probabilistic fitting with a 95% confidence interval. Then, the model¿s parameters are adjusted based on the history of each patient using a genetic algorithm and dividing the day into 12 time intervals. In the final stage, we will perform a whole-day forecast from an ensemble of the models fitted in the previous phase. Th e validity of our strategy will be tested using the Parkers¿ error grid analysis. Our experimental results based on data from real diabetic patients show that this technique is capable of robust predictions that take into account all the uncertainty associated with the interaction between insulin and glucose.We acknowledge support from Spanish Ministry of Economy and Competitiveness under project RTI2018-095180- B-I00 and Madrid Regional Goverment - FEDER grants B2017/BMD3773 (GenObIA-CM) and Y2018/NMT-4668 (Micro-Stress- MAP-CM). Devices for adquiring data from patients were adquired with the support of Fundacion Eugenio Rodriguez Pascual 2019 grant - Desarrollo de sistemas adaptativos y bioinspirados para el control glucemico con infusores subcutaneos continuos de insulina y monitores continuos de glucosa (Development of adaptive and bioinspired systems for glycaemic control with continuous subcutaneous insulin infusors and continuous glucose monitors).Cervigón, C.; Velasco, JM.; Burgos-Simon, C.; Villanueva Micó, RJ.; Hidalgo, JI. (2021). Probabilistic Fitting of Glucose Models with Real-Coded Genetic Algorithms. IEEE. 736-743. https://doi.org/10.1109/CEC45853.2021.9504836S73674

Modelling the dynamics of frequent users of electronic commenrce in Spain using optimization techniques for inverse problems with uncertainty

[EN] In this paper, we retrieve data about the frequent users of electronic commerce during the period 2011-2016 from the Spanish National Institute of Statistics. These data,coming from surveys, have intrinsic uncertainty that we describe using appropriate random variables. Then, we propose a stochastic model to study the dynamics of frequent users of electronic commerce. The goal of this paper is to solve the inverse problem that consists of determining themodel parameters as suitable parametric random variables, in such a way the model output be capable of capturing the data uncertainty, at the time instantswhere sample data are available, via adequate probability density functions. To achieve the aforementioned goal, we propose a computational procedure that involves building a nonlinear objective function, based on statistical moment measures, to be minimized using a variation of the particle swarm optimization algorithm.This work has been supported by the Ministerio de Economía, Industria y Competitividad Grant MTM2017-89664-P.Burgos-Simon, C.; Cortés, J.; Lombana, IC.; Martínez Rodríguez, D.; Villanueva Micó, RJ. (2018). Modelling the dynamics of frequent users of electronic commenrce in Spain using optimization techniques for inverse problems with uncertainty. Journal of Optimization Theory and Applications. 158(3):1-12. https://doi.org/10.1007/s10957-018-1382-6S1121583Dorini, F.A., Cecconello, M.S., Dorini, M.B.: On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Commun. Nonlinear Sci. Numer. Simul. 33, 160–173 (2016). https://doi.org/10.1016/j.cnsns.2015.09.009Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012). https://doi.org/10.1016/j.amc.2011.12.088Dorini, F., Cunha, M.: Statistical moments of the random linear transport equation. J. Comput. Phys. 227(19), 8541–8550 (2008). https://doi.org/10.1016/j.jcp.2008.06.002Xu, Z., Tipireddy, R., Lin, G.: Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Appl. Math. Model. 40(9–10), 5542–5559 (2016). https://doi.org/10.1016/j.apm.2015.12.04Mourad, K., Debbouche, A.: Complete controllability of nonlocal fractional stochastic differential evolution equations with Poisson jumps in Hilbert spaces. Int. J. Adv. Appl. Math. Mech. 3(1), 41–48 (2015)Casabán, M.C., Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Villanueva, R.J.: A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 32, 199–210 (2016). https://doi.org/10.1016/j.cnsns.2015.08.009Casabán, M.C., Cortés, J.C., Navarro-Quiles, A., Romero, J.V., Roselló, M.D., Villanueva, R.J.: Computing probabilistic solutions of the Bernoulli random differential equation. J. Comput. Appl. Math. 309, 396–407 (2017). https://doi.org/10.1016//j.cam.2016.02.034Cortés, J.C., Santonja, F.J., Tarazona, A.C., Villanueva, R.J., Villanueva-Oller, J.: A probabilistic estimation and prediction technique for dynamic continuous social science models: the evolution of the attitude of the basque country population towards ETA as a case study. Appl. Math. Comput. 264, 13–20 (2015). https://doi.org/10.1016/j.amc.2015.03.128Spanish INE: Encuesta sobre equipamiento y uso de tecnologías de información y comunicación en los hogares (Survey on equipment and use of the information technologies and communication in the household. http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736176741&menu=resultados&idp=1254735576692 . Accessed 20 June 2018Spanish INE: Indicadores demográficos básicos (Basic demographic indicators). http://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica_C&cid=1254736177003&menu=resultados&idp=1254735573002 . Accessed 20 June 2018Christakis, N.A., Fowler, J.H.: Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives. Little, Brown and Company, Boston (2009)Brauer, F., Castillo-Chávez, C.: Mathematical Models in Population Biology and Epidemiology. Springer, New York (2001). https://doi.org/10.1007/978-1-4757-3516-1Norman, L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions. Wiley, London (1994)Khemka, N., Jacob, C.: Exploratory toolkit for evolutionary and swarm-based optimization. Math. J. 11(3), 376–391 (2010). https://doi.org/10.3888/tmj.11.3-5Acedo, L., Burgos, C., Hidalgo, J.I., Sánchez-Alonso, V., Villanueva, R.J., Villanueva-Oller, J.: Calibrating a large network model describing the transmission dynamics of the human papillomavirus using a particle swarm optimization algorithm in a distributed computing environment. Int. J. High Perf. Comput. Appl. (2017). https://doi.org/10.1177/109434201769786

Estrategias Financieras Sintéticas con Opciones de Venta y Futuros

En este trabajo se muestra una relación fundamental entre el precio de un contrato de futuro y las primas de una opción de compra europea y una opción de venta europea, todos ellos sobre un mismo subyacente y un mismo vencimiento. La deducción se realiza de dos formas, primero a través de una posición inversora creada con un futuro comprado y con una put europea vendida y, en segundo lugar, con una posición inversora consistente en la venta de un futuro y la compra de una opción put europea.Burgos Simon, C.; Cortés López, JC.; Navarro Quiles, A. (2017). Estrategias Financieras Sintéticas con Opciones de Venta y Futuros. http://hdl.handle.net/10251/82960DE