3,700 research outputs found

    Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

    Full text link
    [EN] This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.This work has been partially supported by the Ministerio de Economia y Competitividad Grants MTM2013-41765-P and TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 24(1):86-97. https://doi.org/10.1016/j.cnsns.2014.12.016S869724

    Solving random homogeneous linear second-order differential equations: a full probabilistic description

    Full text link
    [EN] In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Second, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on random variable transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.This work has been partially supported by the Spanish M. C. Y. T. Grant MTM2013-41765-P.Casabán, M.; Cortés, J.; Romero, J.; Roselló, M. (2016). Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterranean Journal of Mathematics. 13(6):3817-3836. https://doi.org/10.1007/s00009-016-0716-6S38173836136Øksendal B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2007)Soong T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Neckel, T., Rupp, F.: Random Differential Equations in Scientific Computing. Versita, London (2013)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 1–18 (2014). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Chen-Charpentier, B.M.: A random differential transform method: theory and applications. Appl. Math. Lett. 25(10), 1490–1494 (2012). doi: 10.1016/j.aml.2011.12.033Licea, J.A., Villafuerte, L., Chen-Charpentier, B.M.: Analytic and numerical solutions of a Riccati differential equation with random coefficients. J. Comput. Appl. Math. 239, 208–219 (2013). doi: 10.1016/j.cam.2012.09.040Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random homogeneous linear second-order difference equations. Appl. Math. Lett. 34, 27–32 (2014). doi: 10.1016/j.aml.2014.03.010Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216 (5), 1524–1530 (2010). doi: 10.16/j.amc.2010.03.001El-Tawil, M., El-Tahan, W., Hussein, A.: Using FEM-RVT technique for solving a randomly excited ordinary differential equation with a random operator. Appl. Math. Comput. 187(2), 856–867 (2007). doi: 10.1016/j.amc.2006.08.164Hussein, 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). doi: 10.1016/j.amc.2011.12.088Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Probabilistic solution of random SI-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 24(1–3), 86–97 (2015). doi: 10.1016/j.cnsns.2014.12.016Casabán, M.C., Cortés, J.C., Romero, J.V., Roselló, M.D.: Determining the first probability density function of linear random initial value problems by the random variable transformation (RVT) technique: a comprehensive study. In: Abstract and Applied Analysis 2014-ID248512, pp. 1–25 (2014). doi: 10.1155/2013/248512Casabán, 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). doi: 10.1016/j.cnsns.2015.08.009El-Wakil, S.A., Sallah, M., El-Hanbaly, A.M.: Random variable transformation for generalized stochastic radiative transfer in finite participating slab media. Phys. A 435 66–79 (2015). doi: 10.1016/j.physa.2015.04.033Dorini, F.A., Cunha, M.C.C.: On the linear advection equation subject to random fields velocity. Math. Comput. Simul. 82, 679–690 (2011). doi: 10.16/j.matcom.2011.10.008Dhople, S.V., Domínguez-García, D.: A parametric uncertainty analysis method for Markov reliability and reward models. IEEE Trans. Reliab. 61(3), 634–648 (2012). doi: 10.1109/TR.2012.2208299Williams, M.M.R.: Polynomial chaos functions and stochastic differential equations. Ann. Nucl. Energy 33(9), 774–785 (2006). doi: 10.1016/j.anucene.2006.04.005Chen-Charpentier, B.M., Stanescu, D.: Epidemic models with random coefficients. Math. Comput. Model. 52(7/8), 1004–1010 (2009). doi: 10.1016/j.mcm.2010.01.014Papoulis A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1991

    A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications

    Full text link
    [EN] This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.Spanish Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-P; Generalitat Valenciana, Grant/Award Number: APOSTD/2019/128; Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCortés, J.; El-Labany, S.; Navarro-Quiles, A.; Selim, MM.; Slama, H. (2020). A comprehensive probabilistic analysis of SIR-type epidemiological models based on full randomized Discrete-Time Markov Chain formulation with applications. Mathematical Methods in the Applied Sciences. 43(14):8204-8222. https://doi.org/10.1002/mma.6482S820482224314Hamra, G., MacLehose, R., & Richardson, D. (2013). Markov Chain Monte Carlo: an introduction for epidemiologists. International Journal of Epidemiology, 42(2), 627-634. doi:10.1093/ije/dyt043Becker, N. (1981). A General Chain Binomial Model for Infectious Diseases. Biometrics, 37(2), 251. doi:10.2307/2530415Allen, L. J. S. (2010). An Introduction to Stochastic Processes with Applications to Biology. doi:10.1201/b12537Hethcote, H. W. (2000). The Mathematics of Infectious Diseases. SIAM Review, 42(4), 599-653. doi:10.1137/s0036144500371907Brauer, F., & Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics. doi:10.1007/978-1-4757-3516-1Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Some results about randomized binary Markov chains: theory, computing and applications. International Journal of Computer Mathematics, 97(1-2), 141-156. doi:10.1080/00207160.2018.1440290Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Randomizing the parameters of a Markov chain to model the stroke disease: A technical generalization of established computational methodologies towards improving real applications. Journal of Computational and Applied Mathematics, 324, 225-240. doi:10.1016/j.cam.2017.04.040Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Slama, H., Hussein, A., El-Bedwhey, N. A., & Selim, M. M. (2019). An approximate probabilistic solution of a random SIR-type epidemiological model using RVT technique. Applied Mathematics and Computation, 361, 144-156. doi:10.1016/j.amc.2019.05.019Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 195(2), 179-193. doi:10.1016/j.mbs.2005.02.004Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Van den Driessche, P. (2017). Reproduction numbers of infectious disease models. Infectious Disease Modelling, 2(3), 288-303. doi:10.1016/j.idm.2017.06.002Heffernan, J. ., Smith, R. ., & Wahl, L. . (2005). Perspectives on the basic reproductive ratio. Journal of The Royal Society Interface, 2(4), 281-293. doi:10.1098/rsif.2005.0042Khalil, K. M., Abdel-Aziz, M., Nazmy, T. T., & Salem, A.-B. M. (2012). An Agent-Based Modeling for Pandemic Influenza in Egypt. Intelligent Systems Reference Library, 205-218. doi:10.1007/978-3-642-25755-1_1

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

    Full text link
    [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). On the theory of random equations. Proceedings of Symposia in Applied Mathematics, 40-69. doi:10.1090/psapm/016/0189071Xiu, D., & Karniadakis, G. E. (2002). The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 24(2), 619-644. doi:10.1137/s1064827501387826Chen-Charpentier, B.-M., Cortés, J.-C., Licea, J.-A., Romero, J.-V., Roselló, M.-D., Santonja, F.-J., & Villanueva, R.-J. (2015). Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach. Mathematics and Computers in Simulation, 109, 113-129. doi:10.1016/j.matcom.2014.09.002Cortés, J.-C., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 50, 1-15. doi:10.1016/j.cnsns.2017.02.011Chen-Charpentier, B. M., & Stanescu, D. (2010). Epidemic models with random coefficients. Mathematical and Computer Modelling, 52(7-8), 1004-1010. doi:10.1016/j.mcm.2010.01.014Lucor, D., Su, C.-H., & Karniadakis, G. E. (2004). Generalized polynomial chaos and random oscillators. International Journal for Numerical Methods in Engineering, 60(3), 571-596. doi:10.1002/nme.976Santonja, F., & Chen-Charpentier, B. (2012). Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos. Computational and Mathematical Methods in Medicine, 2012, 1-8. doi:10.1155/2012/742086Stanescu, D., & Chen-Charpentier, B. M. (2009). Random coefficient differential equation models for bacterial growth. Mathematical and Computer Modelling, 50(5-6), 885-895. doi:10.1016/j.mcm.2009.05.017Calatayud, J., Cortés, J. C., Jornet, M., & Villanueva, R. J. (2018). Computational uncertainty quantification for random time-discrete epidemiological models using adaptive gPC. Mathematical Methods in the Applied Sciences, 41(18), 9618-9627. doi:10.1002/mma.5315Villegas, M., Augustin, F., Gilg, A., Hmaidi, A., & Wever, U. (2012). Application of the Polynomial Chaos Expansion to the simulation of chemical reactors with uncertainties. Mathematics and Computers in Simulation, 82(5), 805-817. doi:10.1016/j.matcom.2011.12.001Xiu, D., & Em Karniadakis, G. (2002). Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Computer Methods in Applied Mechanics and Engineering, 191(43), 4927-4948. doi:10.1016/s0045-7825(02)00421-8Shi, W., & Zhang, C. (2012). Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations. Applied Numerical Mathematics, 62(12), 1954-1964. doi:10.1016/j.apnum.2012.08.007Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations. Journal of Nonlinear Sciences and Applications, 11(09), 1077-1084. doi:10.22436/jnsa.011.09.06Casabán, M.-C., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 86-97. doi:10.1016/j.cnsns.2014.12.016Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Dorini, F. A., & Cunha, M. C. C. (2008). Statistical moments of the random linear transport equation. Journal of Computational Physics, 227(19), 8541-8550. doi:10.1016/j.jcp.2008.06.002Hussein, A., & Selim, M. M. (2012). Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Applied Mathematics and Computation, 218(13), 7193-7203. doi:10.1016/j.amc.2011.12.088Hussein, A., & Selim, M. M. (2015). Solution of the stochastic generalized shallow-water wave equation using RVT technique. The European Physical Journal Plus, 130(12). doi:10.1140/epjp/i2015-15249-3Hussein, A., & Selim, M. M. (2013). A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion. Journal of Quantitative Spectroscopy and Radiative Transfer, 125, 84-92. doi:10.1016/j.jqsrt.2013.03.018Xu, Z., Tipireddy, R., & Lin, G. (2016). Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Applied Mathematical Modelling, 40(9-10), 5542-5559. doi:10.1016/j.apm.2015.12.041Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2017). Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Applied Mathematics Letters, 68, 150-156. doi:10.1016/j.aml.2016.12.015El-Tawil, M. A. (2005). The approximate solutions of some stochastic differential equations using transformations. Applied Mathematics and Computation, 164(1), 167-178. doi:10.1016/j.amc.2004.04.062Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Calatayud, J., Cortés, J. C., & Jornet, M. (2018). Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences, 42(17), 5649-5667. doi:10.1002/mma.5333Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Solving second-order linear differential equations with random analytic coefficients about ordinary points: A full probabilistic solution by the first probability density function. Applied Mathematics and Computation, 331, 33-45. doi:10.1016/j.amc.2018.02.051Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation, 32, 199-210. doi:10.1016/j.cnsns.2015.08.009Kegan, B., & West, R. W. (2005). Modeling the simple epidemic with deterministic differential equations and random initial conditions. Mathematical Biosciences, 194(2), 217-231. doi:10.1016/j.mbs.2005.02.002Crestaux, T., Le Maıˆtre, O., & Martinez, J.-M. (2009). Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 94(7), 1161-1172. doi:10.1016/j.ress.2008.10.008Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 93(7), 964-979. doi:10.1016/j.ress.2007.04.002Chen-Charpentier, B. M., Cortés, J.-C., Romero, J.-V., & Roselló, M.-D. (2013). Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation, 219(9), 4208-4218. doi:10.1016/j.amc.2012.11.007Ernst, O. G., Mugler, A., Starkloff, H.-J., & Ullmann, E. (2011). On the convergence of generalized polynomial chaos expansions. ESAIM: Mathematical Modelling and Numerical Analysis, 46(2), 317-339. doi:10.1051/m2an/2011045Giraud, L., Langou, J., & Rozloznik, M. (2005). The loss of orthogonality in the Gram-Schmidt orthogonalization process. Computers & Mathematics with Applications, 50(7), 1069-1075. doi:10.1016/j.camwa.2005.08.009Marzouk, Y. M., Najm, H. N., & Rahn, L. A. (2007). Stochastic spectral methods for efficient Bayesian solution of inverse problems. Journal of Computational Physics, 224(2), 560-586. doi:10.1016/j.jcp.2006.10.010Marzouk, Y., & Xiu, D. (2009). A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems. Communications in Computational Physics, 6(4), 826-847. doi:10.4208/cicp.2009.v6.p826SCOTT, D. W. (1979). On optimal and data-based histograms. Biometrika, 66(3), 605-610. doi:10.1093/biomet/66.3.605National Spanish Health Survey (Encuesta Nacional de Salud de España, ENSE)http://pestadistico.inteligenciadegestion.msssi.es/publicoSNS/comun/ArbolNodos.asp

    A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique

    Full text link
    [EN] This paper provides a complete probabilistic description of SIS-type epidemiological models where all the input parameters (contagion rate, recovery rate and initial conditions) are assumed to be random variables. By applying the Random Variable Transformation technique, the first probability density function, the mean and the variance functions as well as confidence intervals associated to the solution of SIS-type epidemiological models are determined under the general hypothesis that the random inputs have any joint probability density function. The distributions to describe the time until a given proportion of the population remains susceptible and infected are also determined. Lastly, a probabilistic description of the so-called basic reproductive number is included. The theoretical results are applied to an illustrative example showing good fitting.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P and TRA2012-36932. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2016). A comprehensive probabilistic solution of random SIS-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 32:199-210. https://doi.org/10.1016/j.cnsns.2015.08.009S1992103

    COMPUTATIONAL METHODS FOR RANDOM DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS

    Full text link
    Desde las contribuciones de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob y Johann Bernoulli en el siglo XVII hasta ahora, las ecuaciones en diferencias y las diferenciales han demostrado su capacidad para modelar satisfactoriamente problemas complejos de gran interés en Ingeniería, Física, Epidemiología, etc. Pero, desde un punto de vista práctico, los parámetros o inputs (condiciones iniciales/frontera, término fuente y/o coeficientes), que aparecen en dichos problemas, son fijados a partir de ciertos datos, los cuales pueden contener un error de medida. Además, pueden existir factores externos que afecten al sistema objeto de estudio, de modo que su complejidad haga que no se conozcan de forma cierta los parámetros de la ecuación que modeliza el problema. Todo ello justifica considerar los parámetros de la ecuación en diferencias o de la ecuación diferencial como variables aleatorias o procesos estocásticos, y no como constantes o funciones deterministas, respectivamente. Bajo esta consideración aparecen las ecuaciones en diferencias y las ecuaciones diferenciales aleatorias. Esta tesis hace un recorrido resolviendo, desde un punto de vista probabilístico, distintos tipos de ecuaciones en diferencias y diferenciales aleatorias, aplicando fundamentalmente el método de Transformación de Variables Aleatorias. Esta técnica es una herramienta útil para la obtención de la función de densidad de probabilidad de un vector aleatorio, que es una transformación de otro vector aleatorio cuya función de densidad de probabilidad es conocida. En definitiva, el objetivo de este trabajo es el cálculo de la primera función de densidad de probabilidad del proceso estocástico solución en diversos problemas basados en ecuaciones en diferencias y diferenciales aleatorias. El interés por determinar la primera función de densidad de probabilidad se justifica porque dicha función determinista caracteriza la información probabilística unidimensional, como media, varianza, asimetría, curtosis, etc., de la solución de la ecuación en diferencias o diferencial correspondiente. También permite determinar la probabilidad de que acontezca un determinado suceso de interés que involucre a la solución. Además, en algunos casos, el estudio teórico realizado se completa mostrando su aplicación a problemas de modelización con datos reales, donde se aborda el problema de la estimación de distribuciones estadísticas paramétricas de los inputs en el contexto de las ecuaciones en diferencias y diferenciales aleatorias.Ever since the early contributions by Isaac Newton, Gottfried Wilhelm Leibniz, Jacob and Johann Bernoulli in the XVII century until now, difference and differential equations have uninterruptedly demonstrated their capability to model successfully interesting complex problems in Engineering, Physics, Chemistry, Epidemiology, Economics, etc. But, from a practical standpoint, the application of difference or differential equations requires setting their inputs (coefficients, source term, initial and boundary conditions) using sampled data, thus containing uncertainty stemming from measurement errors. In addition, there are some random external factors which can affect to the system under study. Then, it is more advisable to consider input data as random variables or stochastic processes rather than deterministic constants or functions, respectively. Under this consideration random difference and differential equations appear. This thesis makes a trail by solving, from a probabilistic point of view, different types of random difference and differential equations, applying fundamentally the Random Variable Transformation method. This technique is an useful tool to obtain the probability density function of a random vector that results from mapping another random vector whose probability density function is known. Definitely, the goal of this dissertation is the computation of the first probability density function of the solution stochastic process in different problems, which are based on random difference or differential equations. The interest in determining the first probability density function is justified because this deterministic function characterizes the one-dimensional probabilistic information, as mean, variance, asymmetry, kurtosis, etc. of corresponding solution of a random difference or differential equation. It also allows to determine the probability of a certain event of interest that involves the solution. In addition, in some cases, the theoretical study carried out is completed, showing its application to modelling problems with real data, where the problem of parametric statistics distribution estimation is addressed in the context of random difference and differential equations.Des de les contribucions de Isaac Newton, Gottfried Wilhelm Leibniz, Jacob i Johann Bernoulli al segle XVII fins a l'actualitat, les equacions en diferències i les diferencials han demostrat la seua capacitat per a modelar satisfactòriament problemes complexos de gran interés en Enginyeria, Física, Epidemiologia, etc. Però, des d'un punt de vista pràctic, els paràmetres o inputs (condicions inicials/frontera, terme font i/o coeficients), que apareixen en aquests problemes, són fixats a partir de certes dades, les quals poden contenir errors de mesura. A més, poden existir factors externs que afecten el sistema objecte d'estudi, de manera que, la seua complexitat faça que no es conega de forma certa els inputs de l'equació que modelitza el problema. Tot aço justifica la necessitat de considerar els paràmetres de l'equació en diferències o de la equació diferencial com a variables aleatòries o processos estocàstics, i no com constants o funcions deterministes. Sota aquesta consideració apareixen les equacions en diferències i les equacions diferencials aleatòries. Aquesta tesi fa un recorregut resolent, des d'un punt de vista probabilístic, diferents tipus d'equacions en diferències i diferencials aleatòries, aplicant fonamentalment el mètode de Transformació de Variables Aleatòries. Aquesta tècnica és una eina útil per a l'obtenció de la funció de densitat de probabilitat d'un vector aleatori, que és una transformació d'un altre vector aleatori i la funció de densitat de probabilitat és del qual és coneguda. En definitiva, l'objectiu d'aquesta tesi és el càlcul de la primera funció de densitat de probabilitat del procés estocàstic solució en diversos problemes basats en equacions en diferències i diferencials. L'interés per determinar la primera funció de densitat es justifica perquè aquesta funció determinista caracteritza la informació probabilística unidimensional, com la mitjana, variància, asimetria, curtosis, etc., de la solució de l'equació en diferències o l'equació diferencial aleatòria corresponent. També permet determinar la probabilitat que esdevinga un determinat succés d'interés que involucre la solució. A més, en alguns casos, l'estudi teòric realitzat es completa mostrant la seua aplicació a problemes de modelització amb dades reals, on s'aborda el problema de l'estimació de distribucions estadístiques paramètriques dels inputs en el context de les equacions en diferències i diferencials aleatòries.Navarro Quiles, A. (2018). COMPUTATIONAL METHODS FOR RANDOM DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/98703TESI

    Computing probabilistic solutions of the Bernoulli random differential equation

    Full text link
    [EN] The random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process. In this paper, that technique is applied to construct a closed form expression of the solution for the Bernoulli random differential equation. In order to account for the general scenario, all the input parameters (coefficients and initial condition) are assumed to be absolutely continuous random variables with an arbitrary joint probability density function. The analysis is split into two cases for which an illustrative example is provided. Finally, a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data.This work has been partially supported by the Ministerio de Economía y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. Contratos Predoctorales UPV 2014- Subprograma 1.Casabán, M.; Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics. 309:396-407. https://doi.org/10.1016/j.cam.2016.02.034S39640730

    Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique

    Full text link
    [EN] Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P.Cortés, J.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation. 50:1-15. https://doi.org/10.1016/j.cnsns.2017.02.011S1155

    Early warning signs for saddle-escape transitions in complex networks

    Get PDF
    Many real world systems are at risk of undergoing critical transitions, leading to sudden qualitative and sometimes irreversible regime shifts. The development of early warning signals is recognized as a major challenge. Recent progress builds on a mathematical framework in which a real-world system is described by a low-dimensional equation system with a small number of key variables, where the critical transition often corresponds to a bifurcation. Here we show that in high-dimensional systems, containing many variables, we frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions. This generic class of transitions has been missed in the search for early-warnings up to now. In fact, the saddle-type mechanism also applies to low-dimensional systems with saddle-dynamics. Near a saddle a system moves slowly and the state may be perceived as stable over substantial time periods. We develop an early warning sign for the saddle-type transition. We illustrate our results in two network models and epidemiological data. This work thus establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition. In complex models and big data we anticipate that saddle-transitions will be encountered frequently in the future.Comment: revised versio

    Clinical microbiology with multi-view deep probabilistic models

    Get PDF
    Clinical microbiology is one of the critical topics of this century. Identification and discrimination of microorganisms is considered a global public health threat by the main international health organisations, such as World Health Organisation (WHO) or the European Centre for Disease Prevention and Control (ECDC). Rapid spread, high morbidity and mortality, as well as the economic burden associated with their treatment and control are the main causes of their impact. Discrimination of microorganisms is crucial for clinical applications, for instance, Clostridium difficile (C. diff ) increases the mortality and morbidity of healthcare-related infections. Furthermore, in the past two decades, other bacteria, including Klebsiella pneumoniae (K. pneumonia), have demonstrated a significant propensity to acquire antibiotic resistance mechanisms. Consequently, the use of an ineffective antibiotic may result in mortality. Machine Learning (ML) has the potential to be applied in the clinical microbiology field to automatise current methodologies and provide more efficient guided personalised treatments. However, microbiological data are challenging to exploit owing to the presence of a heterogeneous mix of data types, such as real-valued high-dimensional data, categorical indicators, multilabel epidemiological data, binary targets, or even time-series data representations. This problem, which in the field of ML is known as multi-view or multi-modal representation learning, has been studied in other application fields such as mental health monitoring or haematology. Multi-view learning combines different modalities or views representing the same data to extract richer insights and improve understanding. Each modality or view corresponds to a distinct encoding mechanism for the data, and this dissertation specifically addresses the issue of heterogeneity across multiple views. In the probabilistic ML field, the exploitation of multi-view learning is also known as Bayesian Factor Analysis (FA). Current solutions face limitations when handling high-dimensional data and non-linear associations. Recent research proposes deep probabilistic methods to learn hierarchical representations of the data, which can capture intricate non-linear relationships between features. However, some Deep Learning (DL) techniques rely on complicated representations, which can hinder the interpretation of the outcomes. In addition, some inference methods used in DL approaches can be computationally burdensome, which can hinder their practical application in real-world situations. Therefore, there is a demand for more interpretable, explainable, and computationally efficient techniques for highdimensional data. By combining multiple views representing the same information, such as genomic, proteomic, and epidemiologic data, multi-modal representation learning could provide a better understanding of the microbial world. Hence, in this dissertation, the development of two deep probabilistic models, that can handle current limitations in state-of-the-art of clinical microbiology, are proposed. Moreover, both models are also tested in two real scenarios regarding antibiotic resistance prediction in K. pneumoniae and automatic ribotyping of C. diff in collaboration with the Instituto de Investigación Sanitaria Gregorio Marañón (IISGM) and the Instituto Ramón y Cajal de Investigación Sanitaria (IRyCIS). The first presented algorithm is the Kernelised Sparse Semi-supervised Heterogeneous Interbattery Bayesian Analysis (SSHIBA). This algorithm uses a kernelised formulation to handle non-linear data relationships while providing compact representations through the automatic selection of relevant vectors. Additionally, it uses an Automatic Relevance Determination (ARD) over the kernel to determine the input feature relevance functionality. Then, it is tailored and applied to the microbiological laboratories of the IISGM and IRyCIS to predict antibiotic resistance in K. pneumoniae. To do so, specific kernels that handle Matrix-Assisted Laser Desorption Ionization (MALDI)-Time-Of-Flight (TOF) mass spectrometry of bacteria are used. Moreover, by exploiting the multi-modal learning between the spectra and epidemiological information, it outperforms other state-of-the-art algorithms. Presented results demonstrate the importance of heterogeneous models that can analyse epidemiological information and can automatically be adjusted for different data distributions. The implementation of this method in microbiological laboratories could significantly reduce the time required to obtain resistance results in 24-72 hours and, moreover, improve patient outcomes. The second algorithm is a hierarchical Variational AutoEncoder (VAE) for heterogeneous data using an explainable FA latent space, called FA-VAE. The FA-VAE model is built on the foundation of the successful KSSHIBA approach for dealing with semi-supervised heterogeneous multi-view problems. This approach further expands the range of data domains it can handle. With the ability to work with a wide range of data types, including multilabel, continuous, binary, categorical, and even image data, the FA-VAE model offers a versatile and powerful solution for real-world data sets, depending on the VAE architecture. Additionally, this model is adapted and used in the microbiological laboratory of IISGM, resulting in an innovative technique for automatic ribotyping of C. diff, using MALDI-TOF data. To the best of our knowledge, this is the first demonstration of using any kind of ML for C. diff ribotyping. Experiments have been conducted on strains of Hospital General Universitario Gregorio Marañón (HGUGM) to evaluate the viability of the proposed approach. The results have demonstrated high accuracy rates where KSSHIBA even achieved perfect accuracy in the first data collection. These models have also been tested in a real-life outbreak scenario at the HGUGM, where successful classification of all outbreak samples has been achieved by FAVAE. The presented results have not only shown high accuracy in predicting each strain’s ribotype but also revealed an explainable latent space. Furthermore, traditional ribotyping methods, which rely on PCR, required 7 days while FA-VAE has predicted equal results on the same day. This improvement has significantly reduced the time response by helping in the decision-making of isolating patients with hyper-virulent ribotypes of C. diff on the same day of infection. The promising results, obtained in a real outbreak, have provided a solid foundation for further advancements in the field. This study has been a crucial stepping stone towards realising the full potential of MALDI-TOF for bacterial ribotyping and advancing our ability to tackle bacterial outbreaks. In conclusion, this doctoral thesis has significantly contributed to the field of Bayesian FA by addressing its drawbacks in handling various data types through the creation of novel models, namely KSSHIBA and FA-VAE. Additionally, a comprehensive analysis of the limitations of automating laboratory procedures in the microbiology field has been carried out. The shown effectiveness of the newly developed models has been demonstrated through their successful implementation in critical problems, such as predicting antibiotic resistance and automating ribotyping. As a result, KSSHIBA and FA-VAE, both in terms of their technical and practical contributions, signify noteworthy progress both in the clinical and the Bayesian statistics fields. This dissertation opens up possibilities for future advancements in automating microbiological laboratories.La microbiología clínica es uno de los temas críticos de este siglo. La identificación y discriminación de microorganismos se considera una amenaza mundial para la salud pública por parte de las principales organizaciones internacionales de salud, como la Organización Mundial de la Salud (OMS) o el Centro Europeo para la Prevención y Control de Enfermedades (ECDC). La rápida propagación, alta morbilidad y mortalidad, así como la carga económica asociada con su tratamiento y control, son las principales causas de su impacto. La discriminación de microorganismos es crucial para aplicaciones clínicas, como el caso de Clostridium difficile (C. diff ), el cual aumenta la mortalidad y morbilidad de las infecciones relacionadas con la atención médica. Además, en las últimas dos décadas, otros tipos de bacterias, incluyendo Klebsiella pneumoniae (K. pneumonia), han demostrado una propensión significativa a adquirir mecanismos de resistencia a los antibióticos. En consecuencia, el uso de un antibiótico ineficaz puede resultar en un aumento de la mortalidad. El aprendizaje automático (ML) tiene el potencial de ser aplicado en el campo de la microbiología clínica para automatizar las metodologías actuales y proporcionar tratamientos personalizados más eficientes y guiados. Sin embargo, los datos microbiológicos son difíciles de explotar debido a la presencia de una mezcla heterogénea de tipos de datos, tales como datos reales de alta dimensionalidad, indicadores categóricos, datos epidemiológicos multietiqueta, objetivos binarios o incluso series temporales. Este problema, conocido en el campo del aprendizaje automático (ML) como aprendizaje multimodal o multivista, ha sido estudiado en otras áreas de aplicación, como en el monitoreo de la salud mental o la hematología. El aprendizaje multivista combina diferentes modalidades o vistas que representan los mismos datos para extraer conocimientos más ricos y mejorar la comprensión. Cada vista corresponde a un mecanismo de codificación distinto para los datos, y esta tesis aborda particularmente el problema de la heterogeneidad multivista. En el campo del aprendizaje automático probabilístico, la explotación del aprendizaje multivista también se conoce como Análisis de Factores (FA) Bayesianos. Las soluciones actuales enfrentan limitaciones al manejar datos de alta dimensionalidad y correlaciones no lineales. Investigaciones recientes proponen métodos probabilísticos profundos para aprender representaciones jerárquicas de los datos, que pueden capturar relaciones no lineales intrincadas entre características. Sin embargo, algunas técnicas de aprendizaje profundo (DL) se basan en representaciones complejas, dificultando así la interpretación de los resultados. Además, algunos métodos de inferencia utilizados en DL pueden ser computacionalmente costosos, obstaculizando su aplicación práctica. Por lo tanto, existe una demanda de técnicas más interpretables, explicables y computacionalmente eficientes para datos de alta dimensionalidad. Al combinar múltiples vistas que representan la misma información, como datos genómicos, proteómicos y epidemiológicos, el aprendizaje multimodal podría proporcionar una mejor comprensión del mundo microbiano. Dicho lo cual, en esta tesis se proponen el desarrollo de dos modelos probabilísticos profundos que pueden manejar las limitaciones actuales en el estado del arte de la microbiología clínica. Además, ambos modelos también se someten a prueba en dos escenarios reales relacionados con la predicción de resistencia a los antibióticos en K. pneumoniae y el ribotipado automático de C. diff en colaboración con el IISGM y el IRyCIS. El primer algoritmo presentado es Kernelised Sparse Semi-supervised Heterogeneous Interbattery Bayesian Analysis (SSHIBA). Este algoritmo utiliza una formulación kernelizada para manejar correlaciones no lineales proporcionando representaciones compactas a través de la selección automática de vectores relevantes. Además, utiliza un Automatic Relevance Determination (ARD) sobre el kernel para determinar la relevancia de las características de entrada. Luego, se adapta y aplica a los laboratorios microbiológicos del IISGM y IRyCIS para predecir la resistencia a antibióticos en K. pneumoniae. Para ello, se utilizan kernels específicos que manejan la espectrometría de masas Matrix-Assisted Laser Desorption Ionization (MALDI)-Time-Of-Flight (TOF) de bacterias. Además, al aprovechar el aprendizaje multimodal entre los espectros y la información epidemiológica, supera a otros algoritmos de última generación. Los resultados presentados demuestran la importancia de los modelos heterogéneos ya que pueden analizar la información epidemiológica y ajustarse automáticamente para diferentes distribuciones de datos. La implementación de este método en laboratorios microbiológicos podría reducir significativamente el tiempo requerido para obtener resultados de resistencia en 24-72 horas y, además, mejorar los resultados para los pacientes. El segundo algoritmo es un modelo jerárquico de Variational AutoEncoder (VAE) para datos heterogéneos que utiliza un espacio latente con un FA explicativo, llamado FA-VAE. El modelo FA-VAE se construye sobre la base del enfoque de KSSHIBA para tratar problemas semi-supervisados multivista. Esta propuesta amplía aún más el rango de dominios que puede manejar incluyendo multietiqueta, continuos, binarios, categóricos e incluso imágenes. De esta forma, el modelo FA-VAE ofrece una solución versátil y potente para conjuntos de datos realistas, dependiendo de la arquitectura del VAE. Además, este modelo es adaptado y utilizado en el laboratorio microbiológico del IISGM, lo que resulta en una técnica innovadora para el ribotipado automático de C. diff utilizando datos MALDI-TOF. Hasta donde sabemos, esta es la primera demostración del uso de cualquier tipo de ML para el ribotipado de C. diff. Se han realizado experimentos en cepas del Hospital General Universitario Gregorio Marañón (HGUGM) para evaluar la viabilidad de la técnica propuesta. Los resultados han demostrado altas tasas de precisión donde KSSHIBA incluso logró una clasificación perfecta en la primera colección de datos. Estos modelos también se han probado en un brote real en el HGUGM, donde FA-VAE logró clasificar con éxito todas las muestras del mismo. Los resultados presentados no solo han demostrado una alta precisión en la predicción del ribotipo de cada cepa, sino que también han revelado un espacio latente explicativo. Además, los métodos tradicionales de ribotipado, que dependen de PCR, requieren 7 días para obtener resultados mientras que FA-VAE ha predicho resultados correctos el mismo día del brote. Esta mejora ha reducido significativamente el tiempo de respuesta ayudando así en la toma de decisiones para aislar a los pacientes con ribotipos hipervirulentos de C. diff el mismo día de la infección. Los resultados prometedores, obtenidos en un brote real, han sentado las bases para nuevos avances en el campo. Este estudio ha sido un paso crucial hacia el despliegue del pleno potencial de MALDI-TOF para el ribotipado bacteriana avanzado así nuestra capacidad para abordar brotes bacterianos. En conclusión, esta tesis doctoral ha contribuido significativamente al campo del FA Bayesiano al abordar sus limitaciones en el manejo de tipos de datos heterogéneos a través de la creación de modelos noveles, concretamente, KSSHIBA y FA-VAE. Además, se ha llevado a cabo un análisis exhaustivo de las limitaciones de la automatización de procedimientos de laboratorio en el campo de la microbiología. La efectividad de los nuevos modelos, en este campo, se ha demostrado a través de su implementación exitosa en problemas críticos, como la predicción de resistencia a los antibióticos y la automatización del ribotipado. Como resultado, KSSHIBA y FAVAE, tanto en términos de sus contribuciones técnicas como prácticas, representan un progreso notable tanto en los campos clínicos como en la estadística Bayesiana. Esta disertación abre posibilidades para futuros avances en la automatización de laboratorios microbiológicos.Programa de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Juan José Murillo Fuentes.- Secretario: Jerónimo Arenas García.- Vocal: María de las Mercedes Marín Arriaz
    corecore