Stabilization of positive linear continuous-time systems by using a Brauer´s theorem

[EN] In this paper we study the stability property of positive linear continuous-time systems. This property is useful to study the asymptotic behavior of a dynamical system and specifically, in positive systems. Stabilization of linear systems using feedbacks has been deeply studied during the last decades. Motivated by some results, in this paper we find conditions on the system such that the eigenvalues of the closed loop system are in the open left half plane of the complex plane C. We do this by applying a Brauer s theorem.The authors would like to thank the referees and the editor for their comments and useful suggestions for improvement of the manuscript. This research was partially supported by Spanish Grant MTM2013-43678-P.Cantó Colomina, B.; Cantó Colomina, R.; Urbano Salvador, AM. (2016). Stabilization of positive linear continuous-time systems by using a Brauer´s theorem. International Journal of Complex Systems in Science. 6(1):23-28. http://hdl.handle.net/10251/81742S23286

Stabilization of positive linear discrete-time systems by using a Brauer's theorem

The stabilization problem of positive linear discrete-time systems (PLDS) by linear state feedback is considered. A method based on a Brauer s theorem is proposed for solving the problem. It allows us to modify some eigenvalues of the system without hanging the rest of them. The problem is studied for the single-input single-output (SISO) and for multi-input multioutput (MIMO) cases and sufficient conditions for stability and positivity of the closed-loop system are proved.The results are illustrated by numerical examples and the proposed method is used in stochastic systems.This work is supported by the Spanish DGI Grant MTM2010-18228.Cantó Colomina, B.; Cantó Colomina, R.; Kostova, S. (2014). Stabilization of positive linear discrete-time systems by using a Brauer's theorem. Scientific World Journal. 2014:1-6. https://doi.org/10.1155/2014/856356S162014Caccetta, L., & Rumchev, V. G. (2000). Annals of Operations Research, 98(1/4), 101-122. doi:10.1023/a:1019244121533Allen, L. J. S., & van den Driessche, P. (2008). The basic reproduction number in some discrete-time epidemic models. Journal of Difference Equations and Applications, 14(10-11), 1127-1147. doi:10.1080/10236190802332308Delchamps, D. F. (1988). State Space and Input-Output Linear Systems. doi:10.1007/978-1-4612-3816-4Méndez-Barrios, C.-F., Niculescu, S.-I., Chen, J., & Maya-Méndez, M. (2013). Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach. International Journal of Control, 87(2), 346-362. doi:10.1080/00207179.2013.834075Anderson, B. D. O., Ilchmann, A., & Wirth, F. R. (2013). Stabilizability of linear time-varying systems. Systems & Control Letters, 62(9), 747-755. doi:10.1016/j.sysconle.2013.05.003De Leenheer, P., & Aeyels, D. (2001). Stabilization of positive linear systems. Systems & Control Letters, 44(4), 259-271. doi:10.1016/s0167-6911(01)00146-3Fornasini, E., & Valcher, M. E. (2012). Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems. IEEE Transactions on Automatic Control, 57(5), 1208-1221. doi:10.1109/tac.2011.2173416Bru, R., Cantó, R., Soto, R. L., & Urbano, A. M. (2011). A Brauer’s theorem and related results. Central European Journal of Mathematics, 10(1), 312-321. doi:10.2478/s11533-011-0113-0Soto, R. L., & Rojo, O. (2006). Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra and its Applications, 416(2-3), 844-856. doi:10.1016/j.laa.2005.12.026Silva, M. S., & de Lima, T. P. (2003). Looking for nonnegative solutions of a Leontief dynamic model. Linear Algebra and its Applications, 364, 281-316. doi:10.1016/s0024-3795(02)00569-4Mourad, B. (2013). Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear and Multilinear Algebra, 61(9), 1234-1243. doi:10.1080/03081087.2012.746330Pakshin, P. V., & Ugrinovskii, V. A. (2006). Stochastic problems of absolute stability. Automation and Remote Control, 67(11), 1811-1846. doi:10.1134/s0005117906110051Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19(1), 75-91. doi:10.1215/s0012-7094-52-01910-8Perfect, H. (1955). Methods of constructing certain stochastic matrices. II. Duke Mathematical Journal, 22(2), 305-311. doi:10.1215/s0012-7094-55-02232-8Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. doi:10.1137/1.9781611971262Cantó, B., Cardona, S. C., Coll, C., Navarro-Laboulais, J., & Sánchez, E. (2011). Dynamic optimization of a gas-liquid reactor. Journal of Mathematical Chemistry, 50(2), 381-393. doi:10.1007/s10910-011-9941-1Fieberg, J., & Ellner, S. P. (2001). Stochastic matrix models for conservation and management: a comparative review of methods. Ecology Letters, 4(3), 244-266. doi:10.1046/j.1461-0248.2001.00202.

Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images

Accurate image segmentation is used in medical diagnosis since this technique is a noninvasive pre-processing step for biomedical treatment. In this work we present an efficient segmentation method for medical image analysis. In particular, with this method blood cells can be segmented. For that, we combine the wavelet transform with morphological operations. Moreover, the wavelet thresholding technique is used to eliminate the noise and prepare the image for suitable segmentation. In wavelet denoising we determine the best wavelet that shows a segmentation with the largest area in the cell. We study different wavelet families and we conclude that the wavelet db1 is the best and it can serve for posterior works on blood pathologies. The proposed method generates goods results when it is applied on several images. Finally, the proposed algorithm made in MatLab environment is verified for a selected blood cells.Boix García, M.; Cantó Colomina, B. (2013). Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images. Mathematical Biosciences and Engineering. 10(2):279-294. doi:10.3934/mbe.2013.10.279S27929410

Sobre las Matrices Totalmente No Positivas

[ES] En este trabajo presentamos un procedimiento para construir un tipo de matrices llamadas totalmente no positivas, estudiar sus propiedades y obtener las relaciones que tienen con otra clase de matrices llamadas totalmente no negativas.Este trabajo ha sido financiado por el proyecto MTM2017-85669-P-AR.Cantó Colomina, B.; Cantó Colomina, R.; Urbano Salvador, AM. (2021). Sobre las Matrices Totalmente No Positivas. Compobell. 49-52. http://hdl.handle.net/10251/191304495

Structured parametric epidemic models

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton-Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number Script capital R-0 and the stability of the model in the equilibrium is studied using Script capital R-0 in terms of the model parameters.This work has been partially supported by MTM2010-18228. The authors wish to express their thanks to the reviewers for helpful comments and suggestions.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2014). Structured parametric epidemic models. International Journal of Computer Mathematics. 91(2):188-197. https://doi.org/10.1080/00207160.2013.800864188197912Allen, L. J. S., & Thrasher, D. B. (1998). The effects of vaccination in an age-dependent model for varicella and herpes zoster. IEEE Transactions on Automatic Control, 43(6), 779-789. doi:10.1109/9.679018Ben-Zvi, A., McLellan, P. J., & McAuley, K. B. (2004). Identifiability of Linear Time-Invariant Differential-Algebraic Systems. 2. The Differential-Algebraic Approach. Industrial & Engineering Chemistry Research, 43(5), 1251-1259. doi:10.1021/ie030534jBoyadjiev, C., & Dimitrova, E. (2005). An iterative method for model parameter identification. Computers & Chemical Engineering, 29(5), 941-948. doi:10.1016/j.compchemeng.2004.08.036Cantó, B., Coll, C., & Sánchez, E. (2011). Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations. Mathematical Problems in Engineering, 2011, 1-12. doi:10.1155/2011/510519Cao, H., & Zhou, Y. (2012). The discrete age-structured SEIT model with application to tuberculosis transmission in China. Mathematical and Computer Modelling, 55(3-4), 385-395. doi:10.1016/j.mcm.2011.08.017Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28(4). doi:10.1007/bf00178324Dion, J.-M., Commault, C., & van der Woude, J. (2003). Generic properties and control of linear structured systems: a survey. Automatica, 39(7), 1125-1144. doi:10.1016/s0005-1098(03)00104-3Emmert, K. E., & Allen, L. J. S. (2004). Population Persistence and Extinction in a Discrete-time, Stage-structured Epidemic Model. Journal of Difference Equations and Applications, 10(13-15), 1177-1199. doi:10.1080/10236190410001654151Farina, L., & Rinaldi, S. (2000). Positive Linear Systems. doi:10.1002/9781118033029Van den Hof, J. M. (1998). Structural identifiability of linear compartmental systems. IEEE Transactions on Automatic Control, 43(6), 800-818. doi:10.1109/9.679020T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.Li, C.-K., & Schneider, H. (2002). Applications of Perron-Frobenius theory to population dynamics. Journal of Mathematical Biology, 44(5), 450-462. doi:10.1007/s002850100132Li, X., & Wang, W. (2005). A discrete epidemic model with stage structure☆. Chaos, Solitons & Fractals, 26(3), 947-958. doi:10.1016/j.chaos.2005.01.063Ma, J., & Earn, D. J. D. (2006). Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease. Bulletin of Mathematical Biology, 68(3), 679-702. doi:10.1007/s11538-005-9047-7Wang, W., & Zhao, X.-Q. (2004). An epidemic model in a patchy environment. Mathematical Biosciences, 190(1), 97-112. doi:10.1016/j.mbs.2002.11.00

Epidemic dynamics of an infection through the pathogen density in the environment

[EN] We propose a mathematical model for the indirect transmission via a contaminated environment by a disease agent and we analyze the possibility that the initial infection can spread in the population. We have used the significance of the basic reproduction number, since the disease free state is either stable or unstable depending on its value, to obtain an explicit expression for $R_0$. This expression is found to give threshold conditions for the stability of the disease- free equilibrium or the existence of an endemic equilibrium in a population.This research was partially supported by Ministerio de Economia y Competitividad under grant MTM2013-43678-P.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2016). Epidemic dynamics of an infection through the pathogen density in the environment. Proceeding of the Bulgarian Academy of Sciences. 69(7):835-844. http://hdl.handle.net/10251/99651S83584469

Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations

This paper presents the use of an iteration method to solve the identifiability problem for a class of discretized linear partial differential algebraic equations. This technique consists in replacing the partial derivatives in the PDAE by differences and analyzing the difference algebraic equations obtained. For that, the theory of discrete singular systems, which involves Drazin inverse matrix, is used. This technique can also be applied to other differential equations in mathematical physics. © 2011 Begoa Cant et al.The authors are very grateful to the referees for their comments and suggestions. The paper is supported by Grant PAID-05-10-003-295 and Grant MTM2010-18228.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2011). Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations. Mathematical Problems in Engineering. 1-12. https://doi.org/10.1155/2011/510519S112LUCHT, W. (2002). On quasi-linear PDAEs with convection: Applications, indices, numerical solution. Applied Numerical Mathematics, 42(1-3), 297-314. doi:10.1016/s0168-9274(01)00157-xDebrabant, K., & Strehmel, K. (2005). Convergence of Runge–Kutta methods applied to linear partial differential-algebraic equations. Applied Numerical Mathematics, 53(2-4), 213-229. doi:10.1016/j.apnum.2004.08.023Ben-Zvi, A., McLellan, P. J., & McAuley, K. B. (2003). Identifiability of Linear Time-Invariant Differential-Algebraic Systems. I. The Generalized Markov Parameter Approach. Industrial & Engineering Chemistry Research, 42(25), 6607-6618. doi:10.1021/ie030317iBen-Zvi, A., McLellan, P. J., & McAuley, K. B. (2004). Identifiability of Linear Time-Invariant Differential-Algebraic Systems. 2. The Differential-Algebraic Approach. Industrial & Engineering Chemistry Research, 43(5), 1251-1259. doi:10.1021/ie030534jSchittkowski, K. (2007). Parameter Identification in One-Dimensional Partial Differential Algebraic Equations. GAMM-Mitteilungen, 30(2), 352-375. doi:10.1002/gamm.200790023Dai, L. (Ed.). (1989). Singular Control Systems. Lecture Notes in Control and Information Sciences. doi:10.1007/bfb0002475Boyadjiev, C., & Dimitrova, E. (2005). An iterative method for model parameter identification. Computers & Chemical Engineering, 29(5), 941-948. doi:10.1016/j.compchemeng.2004.08.036Dion, J.-M., Commault, C., & van der Woude, J. (2003). Generic properties and control of linear structured systems: a survey. Automatica, 39(7), 1125-1144. doi:10.1016/s0005-1098(03)00104-3Usmani, R. A. (1994). Inversion of a tridiagonal jacobi matrix. Linear Algebra and its Applications, 212-213, 413-414. doi:10.1016/0024-3795(94)90414-6Lewis, J. W. (1982). Inversion of tridiagonal matrices. Numerische Mathematik, 38(3), 333-345. doi:10.1007/bf0139643

Modelos matemáticos en epidemiología. Parámetros significativos

En este trabajo se presenta un modelo epidemiológico en el que los individuos están en las categorías de susceptibles, infectados y recuperados, con todos los parámetros no negativos. Por las características del modelo este es no lineal. Tras linealizar el modelo se obtiene un sistema lineal discreto. El comportamiento del modelo se analiza a partir del número reproductivo básico del sistema obtenido, determinando así cuando la epidemia constituye sólo un brote y cuando no. Con el fin de determinar una posible política de vacunación y estudiar cómo afectaría esta política al comportamiento de la epidemia, se ha modificado el modelo añadiendo la categoría de vacunados. Por esta vía, se ha obtenido el número reproductivo básico del sistema cuando la vacunación está presente y con los dos números reproductivos básicos se ha obtenido el número de vacunas necesarias para este caso de epidemiología.Este trabajo está subvencionado por MTM2010-18228.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2013). Modelos matemáticos en epidemiología. Parámetros significativos. Compobell, S.L. http://hdl.handle.net/10251/73779

Una caracterización para la alcanzabilidad de sistemas periódicos generalizados con retardos de estados

Los sistemas con retardo aparecen frecuentemente en cualquier tipo de proceso que involucre transmisión de datos a largas distancias y en procesos químicos, biológicos, eléctricos, etc. El modelo matemático que permite analizar este tipo de procesos corresponde a un sistema dinámico de control. En este trabajo se estudia el caso en que los coeficientes del sistema varían de forma periódica y se analiza la influencia del retardo cuando este afecta al estado del sistema. Para ello, se obtiene la solución del sistema y se construyen las matrices de alcanzabilidad del mismo. Estas matrices se utilizan para caracterizar la propiedad de alcanzabilidad de un sistema generalizado N-periódico en tiempo discreto cuando se considera un retardo en el estado

Using the wavelet transform for T-wave alternans detection

[EN] This paper presents T-wave alternans (TWA) detection, applying the Wavelet Transform (WT) to electrocardiographic (ECG) synthetic signals. The TWA is generated with or without the sinusoidal addition of the wave with the required electrical level from 0.01 to 1 mV. The TWA is measured using the difference between the amplitudes of the augmented T-waves and the normal ones. (C) 2009 Elsevier Ltd. All rights reservedBoix García, M.; Cantó Colomina, B.; Cuesta Frau, D.; Micó Tormos, P. (2009). Using the wavelet transform for T-wave alternans detection. Mathematical and Computer Modelling. 50(5-6):738-742. https://doi.org/10.1016/j.mcm.2009.05.002S738742505-