Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems

[EN] This paper deals with the explicit determination of the first probability density function of the solution stochastic process to random autonomous first-order linear systems of difference equations under very general hypotheses. This finding is applied to extend the classical stability classification of the zero-equilibrium point based on phase portrait to the random scenario. An example illustrates the potentiality of the theoretical results established and their connection with their deterministic counterpart.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. Ana Navarro Quiles acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigation y Desarrollo (PAID), Universitat Politecnica de Valencia.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (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. https://doi.org/10.1016/j.aml.2016.12.0151501566

A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis

"This is the peer reviewed version of the following article: Cortés, J-C, Navarro-Quiles, A, Romero, J-V, Roselló, M-D. A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Comp and Math Methods. 2019; 1:e1013, which has been published in final form at https://doi.org/10.1002/cmm4.1013. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] In this paper a randomized version of the Beverton-Holt type discrete model is proposed. Its solution stochastic process and the random steady state are determined. Its first probability density function and second probability density function are obtained by means of the random variable transformation method, providing a full probabilistic description of the solution. Finally, several numerical examples are shown.This work has been partially supported by the Ministerio de Economía, Industria y Competitividad under grant MTM2017-89664-P. The authors express their deepest thanks and respect to the editors and reviewers for their valuable comments.Cortés, J.; Navarro-Quiles, A.; Romero, J.; Roselló, M. (2019). A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis. Computational and Mathematical Methods. 1(1):1-12. https://doi.org/10.1002/cmm4.1013S11211Kwasnicki, W. (2013). Logistic growth of the global economy and competitiveness of nations. Technological Forecasting and Social Change, 80(1), 50-76. doi:10.1016/j.techfore.2012.07.007De la Sen, M. (2008). The generalized Beverton–Holt equation and the control of populations. Applied Mathematical Modelling, 32(11), 2312-2328. doi:10.1016/j.apm.2007.09.007Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2018). Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty. Journal of Computational and Applied Mathematics, 337, 190-208. doi:10.1016/j.cam.2018.01.015Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Cortés, 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.2017.04.040Cortés, 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.2016.12.01

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

[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

Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters

[EN] In spite of its simple formulation via a nonlinear differential equation, the Gompertz model has been widely applied to describe the dynamics of biological and biophysical parts of complex systems (growth of living organisms, number of bacteria, volume of infected cells, etc.). Its parameters or coefficients and the initial condition represent biological quantities (usually, rates and number of individual/particles, respectively) whose nature is random rather than deterministic. In this paper, we present a complete uncertainty quantification analysis of the randomized Gomperz model via the computation of an explicit expression to the first probability density function of its solution stochastic process taking advantage of the Liouville-Gibbs theorem for dynamical systems. The stochastic analysis is completed by computing other important probabilistic information of the model like the distribution of the time until the solution reaches an arbitrary value of specific interest and the stationary distribution of the solution. Finally, we apply all our theoretical findings to two examples, the first of numerical nature and the second to model the dynamics of weight of a species using real data.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, V.; Burgos, C.; Cortés, J.; Navarro-Quiles, A.; Villanueva Micó, RJ. (2020). Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters. Chaos, Solitons and Fractals. 138:1-12. https://doi.org/10.1016/j.chaos.2020.109908S112138Golec, J., & Sathananthan, S. (2003). Stability analysis of a stochastic logistic model. Mathematical and Computer Modelling, 38(5-6), 585-593. doi:10.1016/s0895-7177(03)90029-xCortés, J. C., Jódar, L., & Villafuerte, L. (2009). Random linear-quadratic mathematical models: Computing explicit solutions and applications. Mathematics and Computers in Simulation, 79(7), 2076-2090. doi:10.1016/j.matcom.2008.11.008Dorini, 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., Bobko, N., & Dorini, L. B. (2016). A note on the logistic equation subject to uncertainties in parameters. Computational and Applied Mathematics, 37(2), 1496-1506. doi:10.1007/s40314-016-0409-6Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., & Roselló, M.-D. (2019). Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation, 72, 121-138. doi:10.1016/j.cnsns.2018.12.013Calatayud, J., Cortés, J. C., & Jornet, M. (2019). Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations. Mathematical Methods in the Applied Sciences, 42(18), 7259-7267. doi:10.1002/mma.5834Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2016). Probabilistic solution of the homogeneous Riccati differential equation: A case-study by using linearization and transformation techniques. Journal of Computational and Applied Mathematics, 291, 20-35. doi:10.1016/j.cam.2014.11.028Hesam, S., Nazemi, A. R., & Haghbin, A. (2012). Analytical solution for the Fokker–Planck equation by differential transform method. Scientia Iranica, 19(4), 1140-1145. doi:10.1016/j.scient.2012.06.018Lakestani, M., & Dehghan, M. (2009). Numerical solution of Fokker-Planck equation using the cubic B-spline scaling functions. Numerical Methods for Partial Differential Equations, 25(2), 418-429. doi:10.1002/num.20352Mao, X., Yuan, C., & Yin, G. (2005). Numerical method for stationary distribution of stochastic differential equations with Markovian switching. Journal of Computational and Applied Mathematics, 174(1), 1-27. doi:10.1016/j.cam.2004.03.016Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., & Villanueva, R.-J. (2017). Computing probabilistic solutions of the Bernoulli random differential equation. Journal of Computational and Applied Mathematics, 309, 396-407. doi:10.1016/j.cam.2016.02.034Kegan, 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.2005.02.004Cortés, 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.2016.12.015Cortés, J. C., Navarro‐Quiles, A., Romero, J., & Roselló, M. (2019). (CMMSE2018 paper) Solving the random Pielou logistic equation with the random variable transformation technique: Theory and applications. Mathematical Methods in the Applied Sciences, 42(17), 5708-5717. doi:10.1002/mma.5440Dorini, F. A., & Cunha, M. C. C. (2011). On the linear advection equation subject to random velocity fields. Mathematics and Computers in Simulation, 82(4), 679-690. doi:10.1016/j.matcom.2011.10.008Slama, 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/i2017-11763-6Hussein, 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.2013.03.018Hussein, A., & Selim, M. M. (2019). A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique. Journal of Quantitative Spectroscopy and Radiative Transfer, 232, 54-65. doi:10.1016/j.jqsrt.2019.04.034Cortés, J.-C., Jódar, L., Camacho, F., & Villafuerte, L. (2010). Random Airy type differential equations: Mean square exact and numerical solutions. Computers & Mathematics with Applications, 60(5), 1237-1244. doi:10.1016/j.camwa.2010.05.046Bekiryazici, Z., Merdan, M., & Kesemen, T. (2020). Modification of the random differential transformation method and its applications to compartmental models. Communications in Statistics - Theory and Methods, 50(18), 4271-4292. doi:10.1080/03610926.2020.1713372Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Laird, A. K. (1965). Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell. British Journal of Cancer, 19(2), 278-291. doi:10.1038/bjc.1965.32Nahashon, S. N., Aggrey, S. E., Adefope, N. A., Amenyenu, A., & Wright, D. (2006). Growth Characteristics of Pearl Gray Guinea Fowl as Predicted by the Richards, Gompertz, and Logistic Models. Poultry Science, 85(2), 359-363. doi:10.1093/ps/85.2.35