Optimal Social and Vaccination Control in the SVIR Epidemic Model

In this paper we introduce an approach to the management of infectious disease diffusion through the formulation of a controlled compartmental SVIR (Susceptible-Vaccinated-Infected-Recovered) model. We consider a cost functional encompassing three distinct yet interconnected dimensions: the social cost, the disease cost, and the vaccination cost. The proposed model addresses the pressing need for optimized strategies in disease containment, incorporating both social control measures and vaccination campaigns. Through the utilization of advanced control theory, we identify optimal control strategies that mitigate disease proliferation while considering the inherent trade-offs among social interventions and vaccination efforts. Finally, numerical implementation of the optimally controlled system through the Forward-Backward Sweep algorithm is presented.Comment: 22 pages, 8 figure

Estudio del efecto de la vacunación en modelos de epidemias con transmisión estocástica

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Estudios Estadísticos, leída el 15-12-2022Mathematical 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...Fac. de Estudios EstadísticosTRUEunpu

Effect of Population Density on the Model of the Spread of Measles

This study is expected to contribute to the health sector, specifically to describe the dynamics of the measles spread through the models that have been analyzed. One of the factors that became the focus of this study was reviewing the influence of population density on measles spread. The initial step formulated the model and then determined the primary reproduction number  and analyzed the stability of the model equilibrium point. The results of the analysis of this model show that there are two conditions for the value of  which is a requirement that the existence of two model equilibrium points as well as local stability is needed, namely  and .  When , there exists a unique equilibrium point, called the non-endemic equilibrium point denoted by . Conversely, when , there are two equilibrium points, namely  and the endemic equilibrium point characterized by . The results of local stability analysis show that when , the equilibrium point  is stable asymptotic locally. It means that if  hold, then in a long time there will not be a spread of disease in the susceptible and vaccinated sub-population, or in other words, the outbreak of the disease will stop. Conversely, when  equilibrium point is stable asymptotic locally. It means that if , then measles disease is still in the environment for an infinite time with the condition of the proportions of each sub-population approach to , ,  and

Simulating Influenza Epidemics with Waning Vaccine Immunity

In this study, we simulate an influenza epidemic that considers the effects of waning immunity by fitting epidemiological models to CDC secondary historical data aggregated on a weekly basis, and derive the transmission rates at which susceptible individuals become infected over the course of the influenza season. Using a system of differential equations, we define four groups of individuals in a population: susceptible, vaccinated, infected, and recovered. We show that a larger number of initially infected individuals might not only bring the influenza season to an end sooner but also reduce the epidemic size. Moreover, any influenza virus that entails a faster recovery rate does not necessarily lead to a smaller epidemic size

A cross-scale model for the evolution of influenza within a single season

In this thesis we develop a mathematical cross-scale model for the evolution of influenza within a single season. We model evolution as the emergence and spread of a mutant strain in a population that is already invaded by a parent resident strain. This allows us to investigate both the emergence dynamics of a mutant strain as well as the subsequent competition dynamics between the two strains. Our main research goal is to study the effects of a homologous vaccine against the resident strain on the epidemiological and evolutionary dynamics of the disease. Due to the complexity of cross-scale models, we first develop a simpler population-level SIR-type model for the evolution of influenza. Assuming an outbreak that is initiated by a single resident strain, we study the significance of the mutant’s emergence time by introducing it in the population at different times. We then also derive a probability density function for the emergence of the mutant in the population. Finally we incorporate vaccination to our model, and arrive at the conclusion that intermediate levels of vaccine- induced immuno-protection are the most beneficial for the emergence and spread of the mutant strain. We then start building towards a cross-scale model by developing a dynamical within-host model for the evolution of influenza. Our goal is for emergence to be a stochastic event, so we derive a probability density for the within-host emergence of a mutant strain. We also incorporate vaccination to our model and assess its impact on the viral loads of the two strains. Having analyzed our within-host model, we then couple it with a between-host SI model. The links between the two scales are the population-level transmission rates, which we assume are linear functions of the within-host viral load. We first investigate how varying the within-host parameters affects the population-level fitness of the two strains, and then we study our model’s results under different forms of the within-host emergence density. Finally we add vaccination to our cross-scale model, and arrive at the same conclusion that intermediate values of immuno-protection are the most inducive to the emergence and spread of a mutant strain in the population

A Mathematical Investigation of Vaccination Strategies to Prevent a Measles Epidemic

The purpose of this project is to quantitatively investigate vaccination strategies to prevent measles epidemics. A disease model which incorporates susceptible, vaccinated, infected, and recovered populations (SVIR) is used to investigate the process of how an epidemic of measles can spread within a closed population where a portion of the population has been vaccinated. The model is used to predict the number of infections and resulting reproductive number for the measles based on a variety of initial vaccination levels.  The model is further used to investigate the concept of herd immunity, which states that if a certain percentage of the population is vaccinated then it will provide protection for the entire population.  Results generated from these modeling efforts suggest that approximately 95\% of the population should be vaccinated against the measles in order to establish a herd immunity