Person:
Gamboa Pérez, María

First Name

María

Last Name

Gamboa Pérez

URI

https://hdl.handle.net/20.500.14352/85701

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Estudios estadísticos

Department

Estadística y Ciencia de los Datos

Identifiers

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Now showing 1 - 10 of 18

Estudio en tiempo discreto de la expansión de una epidemia
(2017) Gamboa Pérez, María; Amador Pacheco, Julia; López Herrero, María Jesús
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.
Some exact measures to quantify the potential transmission of an epidemic in a stochastic SVIS model with infection reintroduction and imperfect vaccine
(2023) Gamboa Pérez, María
This communication is oriented to the study of contagious disease dynamics under a Markovian approach within the area of epidemic modelling. We consider infectious diseases that do not confer permanent immunity in constant size populations where individuals are homogeneous and uniformly mixed. Prior the start of the infectious process, a population proportion was vaccinated preventively to a communicable disease with an available vaccine that fails with a certain probability. This approach involves a stochastic SVIS model with infection reintroduction and imperfect vaccine. We represent the evolution of the epidemic, at each time point t, by the bi-dimensional CTMC, X = {(V (t), I(t)), t ! 0}, where the random variables V (t) and I(t) count the number of vaccinated and infected individuals at time t, respectively. The basic reproduction number, R0, is probably the most used descriptor of disease transmission. Due to repeated contacts between the marked infective and previously infected individuals, R0 overestimates the average number of secondary infections and leads to high immunization coverage. Our purpose is to describe alternative measures to R0 to estimate the potential transmission of an infectious disease. In more detail, we define the exact and population reproduction numbers, Re0 and Rp, under the effect of vaccination. For these measures, we provide theoretical schemes involving their mass probability and generating functions and factorial moments. We illustrate theoretical and algorithmic results with several numerical examples.
On the exact and population bi-dimensional reproduction numbers in a stochastic SVIR model with imperfect vaccine
(Applied Mathematics and Computation, 2024) Gamboa Pérez, María; López-García, M.; López Herrero, María Jesús; Simos, Theodore E.
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.
Measures to assess a warning vaccination level in a stochastic SIV model with imperfect vaccine
(Studies in Applied Mathematics, 2022) Gamboa Pérez, María; López Herrero, María Jesús
A stochastic Markovian Susceptible-Infectious-Susceptible (SIS) model, with infection reintroduction is considered to represent the evolution of an epidemic process within a finite population. Disease is assumed to be a contact disease whose effect can be prevented by a vaccine. Before the epidemic process emerges, individuals got vaccinated to assure that the population is protected by herd immunity. In consequence, we formulate the model by adding a new compartment for vaccine protected individuals. The administered vaccine is not a perfect one and consequently it fails in a proportion of vaccinated individuals that are not protected against the vaccine preventable communicable disease. Hence, while the infectious process is in progress, the initial vaccine coverage declines and herd immunity could be lost. A threshold on the size of the vaccinated group is included as a warning measure on the protection of the community. Our objective is to define and study random characteristics, depending on the vaccination eligible group, that could advise health authorities when to launch a new vaccination program to recover the initial immunity level.
The Effect of Setting a Warning Vaccination Level on a Stochastic SIVS Model with Imperfect Vaccine
(Mathematics, 2020) Gamboa Pérez, María; López Herrero, María Jesús
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.
The number of inspections until the extinction of an epidemic in a discrete-time stochastic SIS-type model with some applications
(2023) Gamboa Pérez, María; López Herrero, María Jesús
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
A stochastic SVIR model with imperfect vaccine and external source of infection
(Performance Engineering and Stochastic Modeling, 2021) Gamboa Pérez, María; López-García, Martín; López Herrero, María Jesús; Ballarini, Paolo
A stochastic SIR (Susceptible - Infected - Recovered) type model, with external source of infection, is considered for the spread of a disease in a finite population of constant size. Our interest is in studying this process in the situation where some individuals have been vaccinated prior to the start of the epidemic, but where the efficacy of the vaccine to prevent infection is not perfect. The evolution of the epidemic is represented by an absorbing three-dimensional continuous-time Markov chain. We focus on analysing the time for a threshold number of individuals to become infected, and carry out a global sensitivity analysis for the impact of varying model parameters on the summary statistic of interest.
Quantifying infection transmission in a stochastic SVIS model with infection reintroduction when vaccine is partially effective
(2023) Gamboa Pérez, María
This communication is framed within the area of epidemic modelling and studies infectious disease dynamics in a stochastic Markovian approach. We consider a constant size population where individuals are homogeneous and uniformly mixed. Prior the start of the epidemic, a percentage of the population was immunized preventively to an infectious disease with an available vaccine that fails with a certain probability. The underlying mathematical model is the stochastic SVIS model with infection reintroduction and imperfect vaccine. The evolution of the infectious disease, at each time point t, is represented in terms of the bidimensional CTMC, X = {(V (t), I(t)), t ! 0}, where the random variables V (t) and I(t) count the number of vaccinated and infected individuals at time t, respectively. The basic reproduction number, R0, is probably the most well-known descriptor of disease transmission and plays a privileged role in epidemiology. It is used to determine the herd immunity threshold or the vaccine coverage required to control the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the marked infective and previously infected individuals, R0 overestimates the average number of secondary infections and leads to high immunization coverage. In this sense, we propose alternatives exact measures to R0 to quantify the potential transmission of an infectious disease. Specifically, we describe the exact and population reproduction numbers, Re0 and Rp, in a post-vaccination context. For both random variables, we derive theoretical schemes involving their mass probability and generating functions, and moments distributions. We complement theoretical and algorithmic results with several numerical examples.
Modeling the propagation of an epidemic in a stochastic SVIS model when a re-vaccination of the susceptible population is considered
(2023) Gamboa Pérez, María; Pardo Fernández, Juan Carlos; Rodríguez Álvarez, María Xosé
This presentation is focused on the use of continuous-time Markov chains (CTMC) to model the transmission of contagious diseases that do not confer permanent immunity. Population is not isolated and in consequence, infections can arise from direct contact with infected individuals within the population or from an external source of infection. A proportion of the population receives an imperfect vaccine that fails with a certain probability in the sense that some individuals that have been previously vaccinated to prevent disease could be infected. We describe the evolution of the infectious process in terms of a bi-dimensional CTMC representing the number of vaccinated and infected individuals during the epidemic. The number of immunized individuals decreases over time due to the imperfect vaccine and external source of infection hypothesis, which can lead to the loss of herd immunity. To prevent this, it is established an alarm threshold for the number of protected individuals, which we refer to as the warning level. The viability of a re-vaccination program is evaluated in order to arise vaccine coverage to the initial situation under a Markovian approach. To achieve that objective it is analyzed the size of the susceptible population when the alarm threshold for vaccinated individuals is reached. We also quantify the time until a re-vaccination program can be launched. We provide theoretical and algorithmic results to obtain statistic characteristics for both random variables and also present some numerical results for several infectious disease outbreaks.
Vibraciones libres y forzadas: Simulación del acoplamiento de dos cuerdas vibrantes
(2023) Gamboa Pérez, María
La motivación de este libro ha partido de un hecho experimental: el acoplamiento de la vibración de una cuerda de guitarra en otra anudada a la anterior. Para este primer acercamiento se han considerado únicamente las vibraciones de una cuerda que parte del reposo mecánico y es forzada únicamente en un extremo. El trabajo se divide en dos partes en las que respectivamente se abordan los problemas continuos y discretos. Se ha comenzado con un estudio detallado del modelo sin forzamiento o de vibraciones libres, deteniéndonos en la obtención de soluciones clásicas hasta la frontera o en cualquier subconjunto compacto interior. En el siguiente capítulo se han estudiado las vibraciones producidas por un único forzamiento en el extremo derecho de la cuerda, partiendo de un estado inicial nulo. Mediante un cambio de variable se ha pasado el forzamiento a la ecuación. Al tratarse de un problema no homogéneo, se ha usado el Principio de Duhamel para abordarle mediante la acumulación de familia de soluciones de problemas homogéneos, vibraciones libres. En el último capítulo se ha ilustrado esta aproximación mediante la simulación del experimento con dos tipos de forzamientos: parabólicos y armónicos en las que la eventual coincidencia de su frecuencia con las naturales de vibración pueden producir el efecto de resonancia en frecuencia.