Search

[EN] We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial condition is assumed to be a random variable too. We use the Dirac delta function to unify the treatment of hybrid (discrete-continuous) uncertainty. Under general hypotheses, we also compute the density of time until a certain value (usually representing the population) of the linear-quadratic logistic model is reached. The theoretical results are illustrated by means of several examples, including an application to modelling the number of users of Spotify using real data. We apply the Principle Maximum Entropy to assign plausible distributions to model parameters, This work has been supported by the Spanish Ministerio de Economa, Industria y Competitividad (MINECO) , the Agencia Estatal de Investigaci on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM201789664P. Computations have been carried thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF) /European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively

In this paper, we propose a modified Susceptible-Infected-Quarantine-Recovered (mSIQR) model, for the COVID-19 pandemic. We start by proving the well-posedness of the model and then compute its reproduction number and the corresponding sensitivity indices. We discuss the values of these indices for epidemiological relevant parameters, namely, the contact rate, the proportion of unknown infectious, and the recovering rate. The mSIQR model is simulated, and the outputs are fit to COVID-19 pandemic data from several countries, including France, US, UK, and Portugal. We discuss the epidemiological relevance of the results and provide insights on future patterns, subjected to health policies., The author CP was partially supported by CMUP (UID/-MAT/00144/2013), which is funded by Fundação para a Ciência e Tecnologia (FCT) (Portugal) with national (MEC) and European structural funds European Regional Development Fund (FEDER), under the partnership agreement PT2020. The revision of this paper has been overshadowed by Prof Tenreiro Machado's departure on October 2021. In sorrow, we dedicate this work to his memory

[EN] Bladder cancer is one of the most common malignant diseases in the urinary system and a highly aggressive neoplasm. The prognosis is not favorable usually, and its evolution for particular patients is very difficult to find out. In this paper, we propose a dynamic mathematical model that describes the bladder tumor growth and the immune response evolution. This model is customized for a single patient, determining appropriate model parameter values via model calibration. Due to the uncertainty of the tumor evolution, using the calibrated model parameters, we predict the tumor size and the immune response evolution over the next few months assuming three different scenarios: favorable, neutral, and unfavorable. In the former, it is not expected any trace of the cancer in the middle of September 2018 (after 16 mo). In the neutral scenario, at the same date, a 7- to 8-mm tumor is expected. In the worst case, a 40-mm tumor is expected. The patient was cited on 10 September 2018 to check the tumor size, and according to the doctors, there was no sign of recurrence. It seems that we are in the favorable scenario. The patient will be called again for follow-up in mid-2019., This work has been supported by the Ministerio de Economía, Industria y Competitividad grant MTM2017-89664-P.

Electroelastic materials, as for example, 3M VHB 4910, are attracting attention as actuators or generators in some developments and applications. This is due to their capacity of being deformed when submitted to an electric field. Some models of their actuation are available, but recently, viscoelastic models have been proposed to give an account of the dissipative behaviour of these materials. Their response to an external mechanical or electrical force field implies a relaxation process towards a new state of thermodynamic equilibrium, which can be described by a relaxation time. However, it is well known that viscoelastic and dielectric materials, as for example, polymers, exhibit a distribution of relaxation times instead of a single relaxation time. In the present approach, a continuous distribution of relaxation times is proposed via the introduction of fractional derivatives of the stress and strain, which gives a better account of the material behaviour. The application of fractional derivatives is described and a comparison with former results is made. Then, a double generalisation is carried out: the first one is referred to the viscoelastic or dielectric models and is addressed to obtain a nonsymmetric spectrum of relaxation times, and the second one is the adoption of the more realistic Mooney–Rivlin equation for the stress–strain relationship of the elastomeric material. A modified Mooney–Rivlin model for the free energy density of a hyperelastic material, VHB 4910 has been used based on experimental results of previous authors. This last proposal ensures the appearance of the bifurcation phenomena which is analysed for equibiaxial dead loads, time-dependent bifurcation phenomena are predicted by the extended Mooney–Rivlin equations.

[ES] La cuantificación de la incertidumbre está compuesta por una serie de métodos y técnicas computacionales cuyo objetivo principal es describir la aleatoriedad presente en problemas de diversa índole. Estos métodos son de utilidad en la modelización de procesos biológicos, físicos, naturales o sociales, ya que en ellos aparecen ciertos aspectos que no pueden ser determinados de manera exacta. Por ejemplo, la tasa de contagio de una enfermedad epidemiológica o el factor de crecimiento de un volumen tumoral dependen de factores genéticos, ambientales o conductuales. Estos no siempre pueden definirse en su totalidad y por tanto conllevan una aleatoriedad intrínseca que afecta en el desarrollo final. El objetivo principal de esta tesis es extender técnicas para cuantificar la incertidumbre en dos áreas de las matemáticas: el cálculo de ecuaciones diferenciales fraccionarias y la modelización matemática. Las derivadas de orden fraccionario permiten modelizar comportamientos que las derivadas clásicas no pueden, como por ejemplo los efectos de memoria o la viscoelasticidad en algunos materiales. En esta tesis, desde un punto de vista teórico, se extenderá el cálculo fraccionario a un ambiente de incertidumbre, concretamente en el sentido de la media cuadrática. Se presentarán problemas de valores iniciales fraccionarios aleatorios. El cálculo de la solución, la obtención de las aproximaciones de la media y varianza de la solución y la aproximación de la primera función de densidad de probabilidad de la solución son conceptos que se abordarán en los próximos capítulos. Sin embargo, no siempre es sencillo obtener la solución exacta de un problema de valores iniciales fraccionario aleatorio. Por ello en esta tesis también se dedicará un capítulo para describir un procedimiento numérico que aproxime su solución. Por otro lado, desde un punto de vista más aplicado, se desarrollan técnicas computacionales para cuantificar la incertidumbre en modelos matemáticos. Combinando estas técnicas junto con modelos matemáticos apropiados, se estudiarán problemas de dinámica biológica. En primer lugar, se determinará la cantidad de portadores de meningococo en España con un modelo de competencia de Lotka-Volterra fraccionario aleatorio. A continuación, el volumen de un tumor mamario se modelizará mediante un modelo logístico con incertidumbre. Finalmente ayudándonos de un modelo matemático que describe el nivel de glucosa en sangre de un paciente diabético, se pretende dar una recomendación de carbohidratos e insulina que se debe de ingerir para que el nivel de glucosa del paciente esté dentro de una banda de confianza saludable. Es importante subrayar que para poder realizar estos estudios se requieren datos reales, los cuales pueden estar alterados debido a los errores de medición o proceso que se han cometido para obtenerlos. Por este motivo, modelizar correctamente el problema junto con la incertidumbre en los datos es de vital importancia., [CA] La quantificació de la incertesa està composada per una sèrie de mètodes i tècniques computacionals, l'objectiu principal de les quals és descriure l'aleatorietat present en problemes de diversa índole. Aquests mètodes són d'utilitat en la modelització de processos biològics, físics, naturals o socials, ja que en ells apareixen certs aspectes que no poden ser determinats de manera exacta. Per exemple, la taxa de contagi d'una malaltia epidemiològica o el factor de creixement d'un volum tumoral depenen de factors genètics, ambientals o conductuals. Aquests no sempre poden definir-se íntegrament i per tant, comporten una aleatorietat intrínseca que afecta en el desenvolupament final. L'objectiu principal d'aquesta tesi doctoral és estendre tècniques per a quantificar la incertesa en dues àrees de les matemàtiques: el càlcul d'equacions diferencials fraccionàries i la modelització matemàtica. Les derivades d'ordre fraccionari permeten modelitzar comportaments que les derivades clàssiques no poden, com per exemple, els efectes de memòria o la viscoelasticitat en alguns materials. En aquesta tesi, des d'un punt de vista teòric, s'estendrà el càlcul fraccionari a un ambient d'incertesa, concretament en el sentit de la mitjana quadràtica. Es presentaran problemes de valors inicials fraccionaris aleatoris. El càlcul de la solució, l'obtenció de les aproximacions de la mitjana i, la variància de la solució i l'aproximació de la primera funció de densitat de probabilitat de la solució són conceptes que s'abordaran en els pròxims capítols. No obstant això, no sempre és senzill obtindre la solució exacta d'un problema de valors inicials fraccionari aleatori. Per això en aquesta tesi també es dedicarà un capítol per a descriure un procediment numèric que aproxime la seua solució. D'altra banda, des d'un punt de vista més aplicat, es desenvolupen tècniques computacionals per a quantificar la incertesa en models matemàtics. Combinant aquestes tècniques juntament amb models matemàtics apropiats, s'estudiaran problemes de dinàmica biològica. En primer lloc, es determinarà la quantitat de portadors de meningococ a Espanya amb un model de competència de Lotka-Volterra fraccionari aleatori. A continuació, el volum d'un tumor mamari es modelitzará mitjançant un model logístic amb incertesa. Finalment ajudant-nos d'un model matemàtic que descriu el nivell de glucosa en sang d'un pacient diabètic, es pretén donar una recomanació de carbohidrats i insulina que s'ha d'ingerir perquè el nivell de glucosa del pacient estiga dins d'una banda de confiança saludable. És important subratllar que per a poder realitzar aquests estudis es requereixen dades reals, els quals poden estar alterats a causa dels errors de mesurament o per la forma en que s'han obtés. Per aquest motiu, modelitzar correctament el problema juntament amb la incertesa en les dades és de vital importància., [EN] Uncertainty quantification collects different methods and computational techniques aimed at describing the randomness in real phenomena. These methods are useful in the modelling of different processes as biological, physical, natural or social, since they present some aspects that can not be determined exactly. For example, the contagious rate of a epidemiological disease or the growth factor of a tumour volume depend on genetic, environmental or behavioural factors. They may not always be fully described and therefore involve uncertainties that affects on the final result. The main objective of this PhD thesis is to extend techniques to quantify the uncertainty in two mathematical areas: fractional calculus and mathematical modelling. Fractional derivatives allow us to model some behaviours that classical derivatives cannot, such as memory effects or the viscoelasticity of some materials. In this PhD thesis, from a theoretical point of view, fractional calculus is extended into the random framework, concretely in the mean square sense. Initial value problems will be studied. The calculus of the analytic solution, approximations for the mean and for the variance and the computation of the first probability density function are concepts we deal with them thought the following chapters. Nevertheless, it is not always possible to obtain the analytic solution of an initial value problem. Therefore, in this dissertation a chapter is addressed to describe a numerical procedure to approximate the solution for an initial value problem. On the other hand, from a modelling point of view, computational techniques to quantify the uncertainty in mathematical models are developed. Merging these techniques with appropriate mathematical models, problems of biological dynamics are studied. Firstly, the carriers of meningococcus in Spain are determined using a competition Lotka-Volterra random fractional model. Then, the volume of breast tumours is modelled by a random logistic model. Finally, taking advantage of a mathematical model which describes the glucose level of a diabetic patient, a recommendation of insulin shots and carbohydrate intakes is proposed to a patient in order to maintain her/his glucose level in a healthy confidence range. An important observation is that to carry out these studies real data is required and they may include uncertainties contained in the measurements on the process to perform the corresponding study. This it is the reason why it is crucial to properly model the problem taking also into account the randomness of the data.

[EN] In this paper we propose a stochastic mathematical model with distributed delay in order to describe the transmission dynamics of cocaine consumption in Spain. We investigate conditions to guarantee the stability in probability of the equilibrium points under stochastic perturbations via the white noise processes. The results are applied to the model cocaine consumption using data retrieved from the Spanish Drug National Plan, http://www.pnsd.mscbs.gob.es/. The obtained results may be useful for policy health authorities in order to improve the strategies against the drug consumption in the long-run., 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. Computations have been carried thanks to the collaboration of Raul San Julian Garces and Elena Lopez Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014-2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively

[EN] The Baranyi-Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville-Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi-Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions., This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.

[EN] We consider a randomized discrete logistic equation to describe the dynamics of breast tumor volume. We propose a method, that takes advantage of the principle of maximum entropy, to assign reliable distributions to model inputs (initial condition and coefficients) and sample data, respectively. Since the distributions of coefficients depend on certain parameters, we design a computational procedure to determine the above mentioned parameters using the information of the probabilistic distributions. The proposed method is successfully applied to model the breast tumor volume using real data. The approach seems to be flexible enough to be adapted to other stochastic models in future contributions., 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) Grants MTM2017-89664-P and RTI2018-095180-B-I00.

Random fractional differential equations are useful mathematical tools to model problems involving memory effects and uncertainties. In this contribution, we present some results, which extent their deterministic counterpart, to fractional differential equations whose initial conditions and coefficients are random variables and/or stochastic process. The probabilistic analysis utilizes the random mean square calculus. For the sake of completeness, we study both autonomous and non-autonomous initial value problems. The analysis includes the computation of analytical and numerical solutions, as well as their main probabilistic information such as the mean, the variance and the first probability density function. Several examples illustrating the theoretical results are shown.

In this contribution, we revisit the Liouville-Gibbs theorem for dynamical systems. This theorem states that a partial differential equation that is satisfied by the probability density function of the solution stochastic process of an initial value problem with uncertainties in its initial condition, forcing term and coefficients. We show its key role in the setting of dynamical systems with uncertainties by means of a variety of illustrative models appearing in several scientific realms that include physics and biology. Specifically, we deal with the undamped and damped linear oscillator, and the logistic model. These models are formulated via random differential equations with a finite degree of randomness. Numerical simulations and computations are carried out to illustrate the capability of the Liouville-Gibbs theorem.

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

In the last years, industrial applications of chemometrics have largely increased due to their capacity to extract important information from complex records as chromatograms or spectra data. The use of chemometric methods also can avoid the use of detectors of elevated cost. In this work, a procedure to recognize the relevant chemical information contained in complex UV–vis chromatograms, after a trypsin digestion, to identify the three enzyme main classes (proteases, amylases and cellulases) commonly employed in the cleaning industry, has been developed. In order to recognize the chromatogram peaks, six indices of peak identity or identifiers were defined. A program written in MATLAB was elaborated to accomplish multiple comparisons between chromatograms to construct 3 rd order tensors, which contain the common peaks of two or more chromatograms. Using a training test and these tensors, the target and sample chromatograms were ordered according to the proximity to its respective class centroids. Further, the peaks with the best warranties of being correctly recognized as belonging to characteristic peptides, common to at least two chromatograms, were used to align the sample chromatograms. Afterwards, to construct an LDA model for enzyme classification, the relative peak area of the aligned and identified peaks were employed. The LDA model was validated showing a 100% of prediction capability by leave-one-out, by dividing the samples in training and evaluation sets and also by the successful prediction of some spiked real samples.

[EN] On advanced stages of the disease, diabetic patients have to inject insulin doses to maintain blood glucose levels inside of a healthy range. The decision of how much insulin is injected implies somehow to predict the level of glucose they will have after a certain time. Due to the sudden changes in the glucose levels, their estimation is a very difficult task. If we were able to give reliable estimations in advance, it would facilitate the process of taking therapeutic decisions to control the disease and improve the health of the patient. In this work, we present a technique to estimate the glucose level of a diabetic patient, capturing the measurement errors produced by continuous glucose monitoring systems (CGMSs), smart devices that measure glucose levels. To do that, we will use a model of glucose dynamics and we calibrate it with the aim to capture the glucose level data of the patient in a time interval of 30 minutes and the uncertainty given by the glucose measurement. Then, we use the calibrated parameters to predict the levels of glucose over the next 15 minutes. Repeating this procedure every 15 minutes, we are able to give short¿term accurate predictions., This work has been partially supported by the Spanish Ministerio de Economía y Competitividad under grant MTM2017-89664-P and RTI2018-095180-B-I00 and by Fundación Eugenio Rodriguez Pascual 2019 -GLENO Project

Bladder cancer is one of the most common malignant diseases in the urinary system and a highly aggressive neoplasm. The prognosis is not favourable usually and its evolution for particular patients is very difficult to find out. In this paper we propose a dynamic mathematical model that describes the bladder tumor growth and the immune response evolution. This model is customized for a single patient, determining appropriate model parameter values via model calibration. Due to the uncertainty of the tumor evolution, using the calibrated model parameters, we predict the tumor size and the immune response evolution over the next few months assuming three different scenarios: favourable, neutral and unfavourable. In the former, the cancer disappears; in the second a 5mm tumor is expected around the middle of August 2018; in the worst scenario, a 5mm tumor is expected around the end of May 2018. The patient has been cited around June 15th, 2018, to check the tumor size, if it exists.

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

[EN] Working in large networks applied to epidemiological-type models has led us to design a simple but e↵ective computed distributed environment to perform a large amount of model simulations in a reasonable time in order to study the behavior of these models and to calibrate them. Finding the model parameters that best fit the available data in the designed distributed computing environment becomes a challenge and it is necessary to implement reliable algorithms for model calibration. In this paper, we have adapted the random PSO algorithm to our distributed computing environment to be applied to the calibration of a Papillomavirus transmission dynamics model on a lifetime sexual partners network. And we have obtained a good fitting saving time and calculations compared with the exhaustive searching strategy we have been using so far., The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been partially supported by the Ministerio de Economa y Competitividad Grants MTM2013-41765-P and TIN 2014-54806-R.

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

[EN] In this paper, we retrieve data about the frequent users of electronic commerce during the period 2011-2016 from the Spanish National Institute of Statistics. These data,coming from surveys, have intrinsic uncertainty that we describe using appropriate random variables. Then, we propose a stochastic model to study the dynamics of frequent users of electronic commerce. The goal of this paper is to solve the inverse problem that consists of determining themodel parameters as suitable parametric random variables, in such a way the model output be capable of capturing the data uncertainty, at the time instantswhere sample data are available, via adequate probability density functions. To achieve the aforementioned goal, we propose a computational procedure that involves building a nonlinear objective function, based on statistical moment measures, to be minimized using a variation of the particle swarm optimization algorithm., This work has been supported by the Ministerio de Economía, Industria y Competitividad Grant MTM2017-89664-P.

[EN] In this paper a new version of the chain rule for calculat- ing the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the cor- responding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included., This work was completed with the support of our TEX-pert.

[EN] This paper extends both the deterministic fractional Riemann¿Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is ¿ ¿ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included., This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. The co-author Prof. L. Villafuerte acknowledges the support by Mexican Conacyt.

Competition among the different classes of genogroups of meningococcal and further bacteria takes place in the ecosystem they constitute within the human body environment. To the best of our knowledge, most of the epidemiological models neglect competition that could highly influence the dynamics of the meningococcal as well as its outbreak. In this paper, we propose a susceptible-carrier-susceptilbe epidemiological model in order to describe the evolution of the percentage of carriers in the population over the time. Moreover, we introduce a competition model to study the dynamics of the meningococcal genogroups in Spain among the carriers. As we are just interested in the meningococcus B, the latter model considers two subpopulations, meningococcal genogroup B and the rest. Both models depend on a number of parameters that are adjusted using real data from the distribution of the genogroups in Spain in years 2011 and 2012. Since these data survey contain sample errors, we further quantify the uncertainties associated to the model parameters taking advantage of a probabilistic fitting method that has been recently proposed by some of the authors. This allows us to give a probabilistic prediction for the evolution of the carriers of the two genogroup over the next few years by constructing a \(95\%\) confidence interval for the percentage of carriers of meningococcus B.

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

The aim of this paper is to study, in mean square sense, a class of random fractional linear differential equation where the initial condition and the forcing term are assumed to be second-order random variables. The solution stochastic process of its associated Cauchy problem is constructed combining the application of a mean square chain rule for differentiating second-order stochastic processes and the random Fröbenius method. To conduct our study, first the classical Caputo derivative is extended to the random framework, in mean square sense. Furthermore, a sufficient condition to guarantee the existence of this operator is provided. Afterwards, the solution of a random fractional initial value problem is built under mild conditions. The main statistical functions of the solution stochastic process are also computed. Finally, several examples illustrate our theoretical findings.

[EN] A probabilistic model is proposed to study the transmission dynamics of the cocaine consumption in Spain during the period of 1995–2011. Using the so-called probabilistic fitting technique, we study if the model is able to capture the data uncertainty coming from surveys. The proposed model is formulated through a nonlinear system of difference equations whose coefficients are treated as stochastic processes. A discussion regarding the usefulness and limitations of probabilistic fitting technique in order to capture the data uncertainty of the proposed model is presented., This work has been partially supported by the Ministerio de Econom´ıa y Competitividad (Grant MTM2013-41765-P).