Measuring infection transmission in a stochastic SIV model with infection reintroduction and imperfect vaccine

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 $R_0$, 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, $R_0$ overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to $R_0$ are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic

Measuring infection transmission in a stochastic SIV model with infection reintroduction and imperfect vaccine

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 $R_0$, 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, $R_0$ overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to $R_0$ are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic

The SEIQS stochastic epidemic model with external source of infection

This paper deals with a stochastic epidemic model for computer viruses with latent and quarantine periods, and two sources of infection: internal and external. All sojourn times are considered random variables which are assumed to be independent and exponentially distributed. For this model extinction and hazard times are analyzed, giving results for their Laplace transforms and moments. The transient behavior is considered by studying the number of times that computers are susceptible, exposed, infectious and quarantined during a period of time (0, t] and results for their joint and marginal distributions, moments and cross moments are presented. In order to give light this analysis, some numerical examples are showed

A new analytic numeric method solution for fractional modified epidemiological model for computer viruses

Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider the modified epidemiological model for computer viruses (SAIR) proposed by J. R. C. Piqueira and V. O. Araujo. This model includes an antidotal population compartment (A) representing nodes of the network equipped with fully effective anti-virus programs. The multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MSGDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that this method is very effective. Mathematica 9 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution

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

Markovian and stochastic differential equation based approaches to computer virus propagation dynamics and some models for survival distributions

This dissertation is divided in two Parts. The first Part explores probabilistic modeling of propagation of computer \u27malware\u27 (generally referred to as \u27virus\u27) across a network of computers, and investigates modeling improvements achieved by introducing a random latency period during which an infected computer in the network is unable to infect others. In the second Part, two approaches for modeling life distributions in univariate and bivariate setups are developed. In Part I, homogeneous and non-homogeneous stochastic susceptible-exposed-infectious- recovered (SEIR) models are specifically explored for the propagation of computer virus over the Internet by borrowing ideas from mathematical epidemiology. Large computer networks such as the Internet have become essential in today\u27s technological societies and even critical to the financial viability of the national and the global economy. However, the easy access and widespread use of the Internet makes it a prime target for malicious activities, such as introduction of computer viruses, which pose a major threat to large computer networks. Since an understanding of the underlying dynamics of their propagation is essential in efforts to control them, a fair amount of research attention has been devoted to model the propagation of computer viruses, starting from basic deterministic models with ordinary differential equations (ODEs) through stochastic models of increasing realism. In the spirit of exploring more realistic probability models that seek to explain the time dependent transient behavior of computer virus propagation by exploiting the essential stochastic nature of contacts and communications among computers, the present study introduces a new refinement in such efforts to consider the suitability and use of the stochastic SEIR model of mathematical epidemiology in the context of computer viruses propagation. We adapt the stochastic SEIR model to the study of computer viruses prevalence by incorporating the idea of a latent period during which computer is in an \u27exposed state\u27 in the sense that the computer is infected but cannot yet infect other computers until the latency is over. The transition parameters of the SEIR model are estimated using real computer viruses data. We develop the maximum likelihood (MLE) and Bayesian estimators for the SEIR model parameters, and apply them to the \u27Code Red worm\u27 data. Since network structure can be a possibly important factor in virus propagation, multi-group stochastic SEIR models for the spreading of computer virus in heterogeneous networks are explored next. For the multi-group stochastic SEIR model using Markovian approach, the method of maximum likelihood estimation for model parameters of interest are derived. The method of least squares is used to estimate the model parameters of interest in the multi-group stochastic SEIR-SDE model, based on stochastic differential equations. The models and methodologies are applied to Code Red worm data. Simulations based on different models proposed in this dissertation and deterministic/ stochastic models available in the literature are conducted and compared. Based on such comparisons, we conclude that (i) stochastic models using SEIR framework appear to be relatively much superior than previous models of computer virus propagation - even up to its saturation level, and (ii) there is no appreciable difference between homogeneous and heterogeneous (multi-group) models. The \u27no difference\u27 finding of course may possibly be influenced by the criterion used to assign computers in the overall network to different groups. In our study, the grouping of computers in the total network into subgroups or, clusters were based on their geographical location only, since no other grouping criterion were available in the Code Red worm data. Part II covers two approaches for modeling life distributions in univariate and bivariate setups. In the univariate case, a new partial order based on the idea of \u27star-shaped functions\u27 is introduced and explored. In the bivariate context; a class of models for joint lifetime distributions that extends the idea of univariate proportional hazards in a suitable way to the bivariate case is proposed. The expectation-maximization (EM) method is used to estimate the model parameters of interest. For the purpose of illustration, the bivariate proportional hazard model and the method of parameter estimation are applied to two real data sets

Dynamics of interacting diseases

Current modeling of infectious diseases allows for the study of complex and realistic scenarios that go from the population to the individual level of description. However, most epidemic models assume that the spreading process takes place on a single level (be it a single population, a meta-population system or a network of contacts). In particular, interdependent contagion phenomena can only be addressed if we go beyond the scheme one pathogen-one network. In this paper, we propose a framework that allows describing the spreading dynamics of two concurrent diseases. Specifically, we characterize analytically the epidemic thresholds of the two diseases for different scenarios and also compute the temporal evolution characterizing the unfolding dynamics. Results show that there are regions of the parameter space in which the onset of a disease's outbreak is conditioned to the prevalence levels of the other disease. Moreover, we show, for the SIS scheme, that under certain circumstances, finite and not vanishing epidemic thresholds are found even at the thermodynamic limit for scale-free networks. For the SIR scenario, the phenomenology is richer and additional interdependencies show up. We also find that the secondary thresholds for the SIS and SIR models are different, which results directly from the interaction between both diseases. Our work thus solve an important problem and pave the way towards a more comprehensive description of the dynamics of interacting diseases.Comment: 24 pages, 9 figures, 4 tables, 3 appendices. Final version accepted for publication in Physical Review

Optimal Control of a Delay-Varying Computer Virus Propagation Model

By incorporating the objective of keeping a low number of infected nodes and a high number of recovered nodes at a lower cost into a known computer virus model (the delay-varying SIRC model) extended by introducing quarantine, a novel model is described by means of the optimal control strategy and theoretically analyzed. Through the comparison of simulation results, it is shown that the propagation of computer virus with varying latency period can be suppressed effectively by the optimal control strategy