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Memoria final del Proyecto de Innovación nº 450 (Proyectos de Innovación 2023-2024) BioStatSpot: Difusión y promoción del Máster en Bioestadística

Single server queues with repeated attempts arise naturally in computer and telecommunication systems. The increasing interest on this topic is mainly explained by the development of new technological facilities such as “repeat last number”, “repeat a fixed number of times”, “ring back when free”, etc. In this paper we analyze an M/G/1 queue operating under a quadratic retrial policy. The consideration of a quadratic policy extends the existing literature and provides more flexibility to the queueing design. When the service time distribution is exponential we show that the main performances measures can be expressed in terms of hypergeometric functions. The case of general service times is investigated with the help of a regenerative approach., Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

The paper deals with a stochastic SEIR model with nonlinear incidence rate and limited resources for a treatment. We focus on a long term study of two measures for the severity of an epidemic: The total number of cases of infection and the maximum of individuals simultaneously infected during an outbreak of the communicable disease. Theoretical and computational results are numerically illustrated., Ministerio de Ciencia, Innovación y Universidades de España, Comisión Europea, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

This paper uses the block-structured state-dependent event (BSDE) approach to generalize the scalar version of the single server retrial queue with finite population. The simple scalar version only involves exponential random variables, which make the underlying Markov chain tractable. However, this is a drawback in applications where the exponentiality is not a realistic assumption and the flows are correlated. The BSDE approach provides a versatile tool to deal with a non-exponential model with correlated flows, but keeping tractable the dimensionality of the block-structured Markov chain. We focus on the investigation of the limiting distribution of the system state and the waiting time. The theory is illustrated by numerical experiments, which demonstrate that the proposed BSDE approach can be applied efficiently, Ministerio de Cienca e Innovación de España, Comisión Europea, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary probabilities and moments of first-passage times to higher and lower levels. We provide a matrix-analytic scheme for numerically computing hitting probabilities, the number of upcrossings, sojourn time analysis, and the random area under the level trajectory. Our algorithmic solution is inspired from Gaussian elimination, which is applicable in all our descriptors since the underlying rate matrices have a block-structured form. Using the results obtained, numerical examples are given in the context of varicella-zoster virus infections., Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

We consider the classical deterministic susceptible-infective-susceptible epidemic model, where the infection and recovery rates depend on a background environmental process that is modeled by a continuous time Markov chain. This framework is able to capture several important characteristics that appear in the evolution of real epidemics in large populations, such as seasonality effects and environmental influences. We propose computational approaches for the determination of various distributions that quantify the evolution of the number of infectives in the population., Ministerio de Economía y Competitividad de España, Comisión Europea, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

We aim to quantify the spread of a direct contact infectious disease that confers permanent immunity after recovery, within a non-isolated finite and homogeneous population. Prior to the onset of the infection and to prevent the spread of this disease, a proportion of individuals was vaccinated. But the administered vaccine is imperfect and can fail, which implies that some vaccinated individuals get the infection when being in contact with infectious individuals. We study the evolution of the epidemic process over time in terms of a continuous-time Markov chain, which represents a general SIR model with an additional compartment for vaccinated individuals. In our stochastic framework, we study two bi-dimensional variables recording infection events, produced by a single infectious individual or by the whole infected group, taking into account if the newly infected individual was previously vaccinated or not. Theoretical schemes and recursive algorithms are derived in order to compute joint probability mass functions and factorial moments for these random variables. We illustrate the applicability of our techniques by means of a set of numerical experiments., Ministerio de Ciencia, Innovación y Universidades (España), British Spanish Society, Universidad Complutense de Madrid, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

We study the maximum number of infected individuals observed during an epidemic for a Susceptible–Infected–Susceptible (SIS) model which corresponds to a birth–death process with an absorbing state. We develop computational schemes for the corresponding distributions in a transient regime and till absorption. Moreover, we study the distribution of the current number of infected individuals given that the maximum number during the epidemic has not exceeded a given threshold. In this sense, some quasi-stationary distributions of a related process are also discussed., Government of Spain (Department of Science and Innovation), European Commission, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

In this paper, we give a survey of the use of information theoretic techniques for the estimation of the main performance characteristics of the M/G/1 retrial queue. We focus on the limiting distribution of the system state, the length of a busy period and the waiting time. Numerical examples are given to illustrate the accuracy of the maximum entropy estimations when they are compared versus the classical solutions., Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

The purpose of this paper is to propose new indicators of the dynamics of infectious disease spread in stochastic epidemic models, including both global system-oriented descriptors (e.g. the final size measured as the number of individuals infected on a least one occasion during an outbreak) and individual-oriented descriptors (e.g. the time to reach an individual run of infections). We focus on birth-and-death models and the basic SIR epidemic model but the methodology remains valid for other nonlinear stochastic epidemic models. The theory is illustrated by numerical experiments which demonstrate that the proposed behavioral indicators can be applied efficiently, Ministerio de Ciencia e Innovación de España, Comisión Europea, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

We present a recursive method of computation for the probability that at most k customers were served during the busy period of an M/G/1 retrial queue, DGES, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

This paper concerns the busy period of a single server queueing model with exponentially distributed repeated attempts. Several authors have analyzed the structure of the busy period in terms of the Laplace transform but, the information about the density function is limited to first and second order moments. We use the maximum entropy principle to find the least biased density function subject to several mean value constraints. We perform results for three different service time distributions: 3-stage Erlang, hyperexponential and exponential. Also a numerical comparative analysis between the exact Laplace transform and the corresponding maximum entropy density is presented., Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

We study a stochastic epidemic model of Susceptible-Exposed-Infective-Removed (SEIR) type and we quantify its behavior during an outbreak. More specifically, we model the epidemic by a continuous-time Markov chain and we develop efficient computational procedures for the distribution of the duration of an outbreak. We also study the evolution of the epidemic before its extinction using the ratio-of-expectations (RE) distribution for the number of individuals in the various classes of the model. The obtained results are illustrated by numerical examples including an application to an outbreak of Marburg hemorrhagic fever, Ministerio de Ciencia e Innovación de España, Comisión Europea, Universidad Complutense de Madrid, Banco Santander, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

In this paper we deal with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/c retrial queue., University of Athens, Greek Ministry of Education, European Union, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

An additional compartment of vaccinated individuals is considered in a SIS stochastic epidemic model with infection reintroduction. The quantification of the spread of the disease is modeled by a continuous time Markov chain. A well-known measure of the initial transmission potential is the basic reproduction number R, which determines the herd immunity threshold or the critical proportion of immune individuals required to stop the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the typical infective and previously infected individuals, R overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to R are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic., Ministerio de Ciencia, Innovación y Universidades (España), Comisión Europea, Banco Santander, Universidad Complutense de Madrid, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

Mathematical epidemic models are frequently used in biology for analyzing transmission dynamics of infectious diseases and assessing control measures to interrupt their expansion. In order to select and develop properly the above mathematical models, it is necessary to take into account the particularities of an epidemic process as type of disease, mode of transmission and population characteristics. In this thesis we focus on infectious diseases with stochastic transmission including vaccination as a control measure to stop the spread of the pathogen. To that end, we consider constant and moderate size populations where individuals are homogeneously mixed. We assume that characteristics related to the transmission/recovery of the infectious disease present a common probabilistic behavior for individuals in the population. To assure herd immunity protection, we consider that a percentage of the population is protected against the disease by a vaccine, prior to the start of the outbreak.The administered vaccine is imperfect in the sense that some individuals, who have been previously vaccinated, failed to increase antibody levels and, in consequence, they could be infected. Pathogenic transmission occurs by direct contact with infected individuals. As population is not isolated, disease spreads from direct contacts with infected individuals inside or outside the population..., Los modelos matemáticos epidemiológicos se usan frecuentemente en biología para analizar las dinámicas de transmisión de enfermedades infecciosas y para evaluar medidas de control con el objetivo de frenar su expansión. Para poder seleccionar y desarrollar adecuadamente estos modelos es necesario tener en cuenta las particularidades propias del proceso epidémico tales como el tipo de enfermedad, modo de transmisión y características de la población. En esta tesis nos centramos en el estudio de enfermedades de tipo infeccioso con transmisión por contacto directo, que disponen de una vacuna como medida de contención en la propagación del patógeno. Para ello, consideramos poblaciones de tamaño moderado, que permanece constante a lo largo de un brote y asumiremos que los individuos no tienen preferencia a la hora de relacionarse y que las características referentes a la transmisión de la enfermedad se representan en términos de variables aleatorias, comunes para todos los individuos. La población no está aislada y la transmisión del patógeno se produce mediante contacto directo con cualquier persona infectada, tanto de dentro de la población como fuera de ella. Asumimos que, antes del inicio del brote epidémico, se ha administrado la vacuna a un porcentaje suficiente de individuos de la población, de forma que se asegure la inmunidad de rebaño. Consideramos que la vacuna administrada es imperfecta en el sentido que algunos individuos vacunados no logran desarrollar anticuerpos frente a la enfermedad y por lo tanto, podrían resultar infectados al contactar con individuos enfermos...

The authors thank the support received from the research project MTM2005-01248. A. Economou was supported by the University of Athens grant ELKE/70/4/6415 and by the Greek Ministry of Education and European Union program PYTHAGORAS/2004., In this paper we deal with the main multiserver retrialqueue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busyperioddistribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busyperiod moments. The corresponding distributions for the total number of customers served during a busyperiod are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/cretrialqueue., University of Athens, Greek Ministry of Education, European Union program PYTHAGORAS/2004, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

Valoración de las presentaciones realizadas con funciones avanzadas de PowerPoint, tanto por los profesores para impartir sus clases en los estudios de Grado y Máster, como por los estudiantes con trabajos de asignaturas como exposiciones previas a sus Trabajos Fin de Máster (TFM) y Tesis Doctorales

This talk deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our purpose is to study the extinction time counterpart in discrete-time, that is the random variable that counts the total number of inspections that find an active epidemic process. As the underlying mathematical model involves a discrete-time Markov chain (DTMC) with a single absorbing state, the number of inspections in an outbreak is a first-passage time into this absorbing state. Cumulative probabilities are numerically determined from a recursive algorithm and expected values came from explicit expressions. Additionally, I provide several applications derived from the theoretical results. The talk is based on the paper: Gamboa M. and López-Herrero M.J. (2018). On the number of periodic inspections during outbreaks of discrete-time stochastic SIS epidemic models. Mathematics 6, article 128.DOI: 10.1007/s11538-013- 9836-3

Mathematical epidemic models are frequently used in biology for analyzing transmission dynamics of infectious diseases and assessing control measures to interrupt their expansion. In order to select and develop properly the above mathematical models, it is necessary to take into account the particularities of an epidemic process as type of disease, mode of transmission and population characteristics. In this thesis we focus on infectious diseases with stochastic transmission including vaccination as a control measure to stop the spread of the pathogen. To that end, we consider constant and moderate size populations where individuals are homogeneously mixed. We assume that characteristics related to the transmission/recovery of the infectious disease present a common probabilistic behavior for individuals in the population. To assure herd immunity protection, we consider that a percentage of the population is protected against the disease by a vaccine, prior to the start of the outbreak.The administered vaccine is imperfect in the sense that some individuals, who have been previously vaccinated, failed to increase antibody levels and, in consequence, they could be infected. Pathogenic transmission occurs by direct contact with infected individuals. As population is not isolated, disease spreads from direct contacts with infected individuals inside or outside the population...

En el ámbito biomédico, los modelos pronósticos se emplean habitualmente para predecir la probabilidad de que un paciente presente una determinada condición. Su validación interna es necesaria para estimar su rendimiento predictivo en nuevos individuos, y puede llevarse a cabo empleando la técnica de remuestreo bootstrap. Ante la presencia de valores perdidos, las técnicas estadísticas clásicas requieren su tratamiento previo, que puede abordarse mediante imputación múltiple: (I) los valores perdidos se imputan múltiples veces, (II) el análisis estadístico se realiza en cada una de las muestras completas resultantes y (III) las estimaciones obtenidas para el parámetro de interés se combinan. En el marco de la validación interna bootstrap, la forma en que la imputación múltiple debe integrarse en el proceso de remuestreo se encuentra actualmente en estudio. En el presente trabajo, se realiza un estudio de simulación para evaluar diferentes estrategias cuando se tienen valores perdidos tanto en los predictores como en la variable de interés de un modelo logístico. En la estrategia MI-BS, se aplica en primer lugar la imputación múltiple y el remuestreo se realiza sobre cada una de las muestras imputadas. En la estrategia BS-MI, se realiza en primer lugar el remuestreo y la imputación múltiple se aplica sobre cada una de las muestras bootstrap. La estrategia BS-MI proporciona estimadores de rendimiento de menor sesgo en la práctica mayoría de los escenarios estudiados. Las diferencias entre estrategias se encuentran cuando el número de eventos por variable (EPV) es reducido y se desdibujan conforme éste aumenta., In the biomedical field, prognostic models are commonly used to predict the probability of a patient having a certain condition. Internal validation is necessary to estimate their predictive performance in new individuals, and can be carried out using the bootstrap resampling technique. Missing data need to be handled prior to classical statistical analyses. Multiple imputation is a popular approach to addressing the presence of missing data: (I) missing values are imputed several times, (II) statistical analysis is performed on each of the resulting complete samples, and (III) the estimates obtained for the parameter of interest are pooled. In the framework of bootstrap internal validation, the way multiple imputation of missing values and resampling should be combined is currently under study. In this work, a simulation study is performed to evaluate different strategies when missing values are in both covariates and outcome of a logistic model. In the MI-BS strategy, multiple imputation is applied first and bootstrap resampling is performed on each of the complete samples. On the other hand, in the BS-MI strategy resampling is performed first and multiple imputation is applied on each of the bootstrap samples. BS-MI provides less biased estimators in most of the scenarios. Differences between both strategies are found when the number of events per variable (EPV) is low, and become smaller as it ncreases.

Actualmente el estudio de enfermedades contagiosas tiene un gran interés científico por las situaciones de emergencia que estas desencadenan, sobre todo, en el ámbito sanitario. Los modelos epidemiológicos y la simulación son herramientas que nos permiten mejorar nuestra comprensión de la dinámica de transmisión de estas enfermedades y así poder planificar situaciones de contingencia. En este trabajo nos proponemos conocer la evolución del Virus Varicela Zóster (VVZ) en una residencia. Nuestro interés se centra en poder cuantificar a largo plazo el número de infectados en una residencia, así como tener la información necesaria para controlar los brotes de la enfermedad. Para llevar a cabo el estudio se considera un modelo estocástico en tiempo continuo. Concretamente, se utilizan las cadenas de Markov, para describir la variación del estado de salud de los residentes durante los brotes de la enfermedad y el tiempo en el que la población en estudio se encuentra en un estado crítico., Currently the study of contagious diseases is of great scientific interest due to the emergency situations they trigger, especially in the health field. Epidemiological models and simulation are tools that allow us to improve our understanding of the transmission dynamics of these diseases and thus be able to plan contingency situations. In this work we propose to know the evolution of Varicella Zoster Virus (VZV) in a nursing home. Our interest arises in being able to quantify in the long term the number of infected people in a nursery home, as well as to have the necessary information to control the outbreaks of the disease. To carry out the study, a stochastic model in continuous time is considered. Specifically, Markov chains are used to describe the variation in the health status of residents during outbreaks of this disease and the time when the underlying population is in a critical state.

La combinación del envejecimiento de las sociedades y la tendencia a llevar un estilo de vida con alimentación irregular y horarios de sueño deficientes está aumentando la fragilidad de las personas mayores de 65 años. Este deterioro prolongado produce una cronodisrupción, un orden temporal interno de los ritmos circadianos bioquímicos, fisiológicos y de comportamiento. Esto puede tener graves consecuencias, no sólo a nivel individual sino también a nivel socioeconómico, afectando a los sistemas de atención sanitaria y de servicios sociales. Objetivo: Conocer si existe algún grado de concordancia entre dos tipos de mediciones de la calidad del sueño: métodos subjetivos y métodos objetivos. Método: Estudio descriptivo, en el que se reclutaron 480 sujetos de estudio de entre 65 y 80 años no ingresados y que acudieron a centros de Atención Primaria. Se midieron como variables principales la calidad de sueño y la somnolencia diurna. Se pretende sustituir un método por otro con la finalidad de eliminar el más costoso e incómodo. Resultados: Hemos obtenido resultados que no avalan ningún grado de concordancia, ya que no fue posible encontrarlo entre las mediciones. Para cuantificar dicha concordancia se han utilizado: gráficos de Bland y Altman, índice kappa y coeficiente de correlación intraclase. Sin embargo, al ver que no había concordancia, nos centramos en buscar relación numérica entre las mediciones, y a través de modelos mixtos con término independiente aleatorio se ha intentado encontrarla. Por cada unidad que aumente el índice PSQI, disminuirá en 0,0016 el valor del índice de salud cronobiológica.

This paper deals with a stochastic Susceptible-Infective-Vaccinated-Susceptible (SIVS) model with infection reintroduction. Health policies depend on vaccine coverage, v0, that guarantees herd immunity levels in the population. Vaccine failures occur when an organism develops a disease despite of being vaccinated against it. After vaccination, a proportion of healthy individuals unsuccessfully tries to increase antibody levels and, consequently these individuals are not immune to the vaccine preventable disease. When an infectious process is in progress, the initial vaccine coverage drops down and herd immunity will be lost. Our objective was to introduce a warning vaccination level and define random measures quantifying the time until the number of vaccinated descends to a warning vaccination level (i.e., the so-called sleeping period), and the epidemic size. A sensitivity analysis was performed to assess the influence of the model parameters on the variation and robustness of the sleeping period and the number of infections observed within it., Ministerio de Ciencia e Innovación (MICINN), Universidad Complutense de Madrid /Banco de Santander, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

Memoria final del proyecto de innovación docente titulado: "Gamificación en el aula con Kahoot! en la Facultad de Estudios Estadísticos"

A lo largo de la historia, las epidemias han sido grandes dificultades que los seres humanos hemos tenido que superar, ya que enfermedades como la viruela, el sarampión o la gripe española causaron un gran número de muertes en todo el mundo. El inter ́es por estudiarlas matemáticamente dio lugar a los modelos epidemiológicos, convertidos junto a la simulación en herramientas fundamentales para predecir el desarrollo de enfermedades contagiosas. La más reciente y que se ha extendido por gran parte de los países, es la causada por el coronavirus SARS-CoV 2 que fue detectado a finales del 2019 en la ciudad china de Wuhan, provincia de Hubei. Tomando como ejemplo el caso de la COVID-19 en Hubei, se ha implementado un modelo determinista de tipo SEIR para estudiar las características del brote epidémico y predecir su evolución futura bajo tres escenarios diferentes. En un primer caso, se ha simulado la propagación de la epidemia sin tomar ningún tipo de medidas de control y se ha comprobado, en un segundo escenario, que medidas como el confinamiento o la cancelación de vuelos, ayudan a controlar la propagación de la epidemia sin conseguir que la enfermedad desaparezca. Por último, se ha querido comprobar cómo influyen los cambios del entorno en la evolución de una epidemia y para describir la situación ambiental de cada día, se han utilizado un par de cadenas de Markov distintas que, a largo plazo, presentan el mismo comportamiento., Throughout history, humans have had to overcome great difficulties when dealing with an epidemic. Diseases such as smallpox, measles or the Spanish flu have caused a large number of deaths around the world. The interest in studying them mathematically has given a rise to epidemiological models which along with simulation has become fundamental tools for predicting the development of contagious diseases. The most recent disease that has spread to a large part of the world, is the one caused by the SARS-CoV 2 coronavirus. This was detected at the end of 2019 in the Chinese city of Wuhan, Hubei province. Using real information of COVID-19 outbreak, in Hubei, a deterministic SEIR-type model has been implemented to study characteristics of the epide- mic outbreak and predict its future evolution under three different scenarios. In the first scenario, the spread of the epidemic has been simulated without taking any type of control measures. In the second scenario, it has been found that measures such as confinement or flight cancellation help to control the spread of the epidemic without making the disease disappear. Finally in the third scenario, we have tried to verify how changes in the environment can influence the evolution of an epidemic. To describe the environmental situa- tion of each day, different Markov chains with the same long-term behaviour, have been considered.

This paper deals with an infective process of type SIS, taking place in a closed population of moderate size that is inspected periodically. Our aim is to study the number of inspections that find the epidemic process still in progress. As the underlying mathematical model involves a discrete time Markov chain (DTMC) with a single absorbing state, the number of inspections in an outbreak is a first-passage time into this absorbing state. Cumulative probabilities are numerically determined from a recursive algorithm and expected values came from explicit expressions., Ministerio de Ciencia e Innovación (MICINN)/FEDER, Depto. de Estadística y Ciencia de los Datos, Fac. de Estudios Estadísticos, TRUE, pub

Entre los grandes obstáculos que han tenido que superar los seres humanos se encuentran las epidemias, pues enfermedades como la peste negra, el cólera o el sida han sido las causantes de muchas muertes a lo largo de la historia. El estudio matemático de estas epidemias dio lugar a los modelos epidemiológicos. Existen dos enfoques mediante los que estudiar estos modelos, el determinista y el estocástico. Mientras el enfoque determinista se basa en realizar la modelización mediante sistemas de ecuaciones diferenciales, el enfoque estocástico realiza la modelización utilizando procesos estocásticos. En cualquier caso, disponemos de ciertas medidas de transmisión que nos permiten evaluar la expansión de la epidemia. Estas medidas son el factor de reproducción básico, el número reproductivo exacto y el número de transmisión poblacional. El objetivo principal de este trabajo es el estudio estocástico markoviano de una epidemia que se desarrolla en una población controlada y a la que se ha aplicado una política de vacunación parcial. Se persigue alcanzar los siguientes objetivos secundarios: analizar la transmisión de la enfermedad a partir de la caracterización de las variables que contabilizan la cantidad de individuos contagiosos por uno o por todos los infectados durante un intervalo de tiempo determinado, y estudiar la influencia de la política de vacunación, teniendo en cuenta que dicha vacuna es parcialmente eficaz.

A lo largo de toda la historia, la Humanidad se ha visto afectada por grandes epidemias como la peste, la viruela o el sida. Estas epidemias se han estudiado matemáticamente dando lugar a diversos modelos epidemiológicos. Estos modelos tienen dos posibles enfoques, el determinista que se modeliza mediante sistemas de ecuaciones diferenciales y el enfoque estocástico que nosotros lo modelizaremos mediante cadenas de Markov y cuyo modelo subyacente es el modelo SIS estocástico. En ambos casos disponemos de ciertos indicadores los cuales nos permitirán cuantificar la epidemia. Estos son el número reproductivo básico, el número reproductivo exacto y el número reproductivo poblacional. El objetivo principal de este trabajo es el estudio en tiempo discreto de una epidemia en poblaciones de tamaño moderado sin inmunidad y aunque la mayor parte de los estudios son en tiempo continuo estos no se ajustan bien a los datos disponibles en la vida real porque se suelen tener datos medidos sobre intervalos de tiempo como por ejemplo días o meses y no en cualquier instante (inspecciones). La metodología usada para el estudio del modelo epidemiológico, objeto de este trabajo, es la que concierne a las cadenas de Markov en tiempo discreto, en concreto a los procesos de nacimiento y muerte. Al modelizarse mediante cadenas de Markov se garantiza que todos los brotes epidémicos finalicen en un tiempo esperado finito y por ello se estudia el número de etapas hasta la finalización de la epidemia. Para su análisis hemos usado ecuaciones en diferencias finitas, método de la primera transición. Su posterior análisis numérico para evaluar la influencia de los parámetros del modelo se ha realizado con el software R.

El objetivo principal de este trabajo es la comparación de diversos métodos de tratamiento de datos faltantes en bases de datos de pruebas diagnósticas generadas a partir de una base de datos, sin pérdida de datos, mediante los diferentes mecanismos de pérdida de datos (MCAR, MAR y MNAR). La base de datos original recoge información sobre tumores mamarios malignos y benignos en una muestra de mujeres. La comparación se basa en los efectos que tienen los métodos de tratamiento de datos faltantes, tanto en el modelo diagnóstico que se obtenga mediante regresión logística con sus errores estándar, como en las curvas ROC y los valores asociados a dichas curvas. El objetivo secundario es obtener el mejor modelo de regresión logística para realizar el diagnóstico de los tumores. Se parte de la base de datos original sin datos faltantes, y el modelo fina obtenido es el que se aplica para comparar los efectos de los métodos de tratamiento de datos faltantes. Se explica la manera en la que se generan bases de datos con datos faltantes según cada mecanismo de pérdida de datos, tras detallar lo que representan las diferentes variables de la base de datos original. Se explican minuciosamente los métodos utilizados de tratamiento de datos faltantes con imputación múltiple, Missing Forest y MICE, y de manera más sencilla los demás métodos, el análisis de casos completos, el método de sustitución por la media y el uso de variables indicadoras de pérdida de datos.

We investigate stochastic and epidemic models, when there is a random environment that influences the spread of the infectious disease. The inclusion of an external environment into the epidemic model is done by replacing the constant transmission rates with dynamic rates governed by an environmental Markov chain. We put emphasis on the algorithmic evaluation of the influence of the environmental factors on the performance behavior of the epidemic model., Government of Spain (Department of Science and Innovation), European Commission, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

The basic reproduction number, R (0), is probably the most important quantity in epidemiology. It is used to measure the transmission potential during the initial phase of an epidemic. In this paper, we are specifically concerned with the quantification of the spread of a disease modeled by a Markov chain. Due to the occurrence of repeated contacts taking place between a typical infective individual and other individuals already infected before, R (0) overestimates the average number of secondary infections. We present two alternative measures, namely, the exact reproduction number, R (e0), and the population transmission number, R (p) , that overcome this difficulty and provide valuable insight. The applicability of R (e0) and R (p) to control of disease spread is also examined., Government of Spain (Department of Science and Innovation), European Commission, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

We deal with stochastic epidemic models having a set of absorbing states. The aim of the paper is to study some continuous characteristics of the epidemic. In this sense, we first extend the classical study of the length of an outbreak by investigating the whole probability distribution of the extinction time via Laplace transforms. Moreover, we also study two almost new epidemic descriptors, namely, the time until a non-infected individual becomes infected and the time until the individual is removed from the infective group. The obtained results are illustrated by numerical examples including an application to a stochastic SIS model for head lice infections., Government of Spain (Department of Science and Innovation), European Commission, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

We analyze the dynamics of infectious disease spread by formulating the maximum entropy (ME) solutions of the susceptible-infected-susceptible (SIS) and the susceptible-infected-removed (SIR) stochastic models. Several scenarios providing helpful insight into the use of the ME formalism for epidemic modeling are identified. The ME results are illustrated with respect to several descriptors, including the number of recovered individuals and the time to extinction. An application to infectious data from outbreaks of extended spectrum beta lactamase (ESBL) in a hospital is also considered., Government of Spain (Department of Science and Innovation), Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

The basic models of infectious disease dynamics (the SIS and SIR models) are considered. Particular attention is paid to the number of infected individuals that recovered and its relationship with the final epidemic size. We investigate this descriptor both until the extinction of the epidemic and in transient regime. Simple and efficient methods to obtain the distribution of the number of recovered individuals and its moments are proposed and discussed with respect to the previous work. The methodology could also be extended to other stochastic epidemic models. The theory is illustrated by numerical experiments, which demonstrate that the proposed computational methods can be applied efficiently. in particular, we use the distribution of the number of individuals removed in the SIR model in conjunction with data of outbreaks of ESBL observed in the intensive care unit of a Spanish hospital., Government of Spain (Department of Science and Innovation), European Commission, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

Many stochastic systems, including biological applications, use Markov chains in which there is a set of absorbing states. It is then needed to consider analogs of the stationary distribution of an irreducible chain. In this context, quasi-stationary distributions play a fundamental role to describe the long-term behavior of the system. The rationale for using quasi-stationary distribution is well established in the abundant existing literature. The aim of this study is to reformulate the ratio of means approach (Darroch and Seneta, 1965, 1967) which provides a simple alternative. We have a two-fold objective. The first objective is viewing quasi-stationarity and ratio of expectations as two different approaches for understanding the dynamics of the system before absorption. At this point, we remark that the quasi-stationary distribution and a ratio of means distribution may give or not give similar information. In this way, we arrive to the second objective; namely, to investigate the possibility of using the ratio of expectations distribution as an approximation to the quasi-stationary distribution. This second objective is explored by comparing both distributions in some selected scenarios, which are mainly inspired in stochastic epidemic models. Previously, the rate of convergence to the quasi-stationary regime is taking into account in order to make meaningful the comparison., Spanish Government (Department of Education and Innovation), European Commission, Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

We are concerned with the M/G/1 queue with repeated attempts where a customer who finds the server busy leaves the service area and repeats his request after a random amount of time. We concentrate on the study of the waiting time process. Its analysis in terms of Laplace transforms has been discussed in the literature. However, this solution has important limitations in practice. For instance, we cannot calculate the first moments of the waiting time, W, by direct differentiation. This paper supplements the existing work and provides a direct method of computation for the second moment of W. Then the maximum entropy approach is used to estimate the true waiting time distribution., Depto. de Estadística e Investigación Operativa, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we study the distribution of the maximum orbit size before emptiness in a stable M/G/1 retrial queue. By a recursive scheme, the computation of that distribution is reduced to solving systems of linear equations.