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Biological Examples Modeled by Discrete Markov Chains
Published in Lyle D. Broemeling, Bayesian Analysis of Infectious Diseases, 2021
This section deals with using Markov chains to model the evolution of an epidemic and then explores the use of Bayesian techniques to provide inferences for the unknown transition probabilities (which contain the basic parameters that describe the epidemic) of the process. First to be discussed is an explanation of the basic principles of an epidemic, which is the biological foundation of an epidemic. Next, a deterministic version of a simple epidemic is presented, which lays rudiments of the stochastic version of an epidemic model. Of primary importance in the study of epidemics is to determine the average duration of the epidemic. Various versions that generalize the simple epidemic are the chain binomial models which include the Greenwood model and the Reed–Frost model. The relationship between the epidemic model and the previously discussed birth and death process is elucidated.
Basic Stochastic Transmission Models and Their Inference
Published in Leonhard Held, Niel Hens, Philip O’Neill, Jacco Wallinga, Handbook of Infectious Disease Data Analysis, 2019
This result is true irrespective of the distribution of the infectious period I as long as . From branching process theory (conclusion 2 above), we see that the outbreak probability for the Reed-Frost model is the same as the final-size equation, so for this particular model the probability of a major outbreak (starting with one infective!) equals the final fraction getting infected in case of a major outbreak. As two numerical examples, if R0 = 1.5, then we have r(∞) = 0.583 so approximately 60 percent will get infected if an outbreak takes place in a community without any immunity, and r(∞) = 0.98 if R0 = 3.
Materialism and reductionism in science and medicine
Published in R. Paul Thompson, Ross E.G. Upshur, Philosophy of Medicine, 2017
R. Paul Thompson, Ross E.G. Upshur
Medicine encompasses both population-level research and interventions, as well as molecular-level research and intervention. As with the Bt resistance example, in many cases both levels are involved. For example, in dealing with an infectious disease (e.g. Ebola), there are numerous population-level activities that are undertaken, from quarantine to issuing protective masks or clothing. There are also numerous models employed, such as the Reed–Frost model of the spread of the disease and population genetic models. There are also molecular activities. Attempts, for example, are made to identify the molecular structure of the infectious agent, and where possible develop methods for killing it or interrupting its lifecycle. This often involves the use of pharmaceuticals but can also involve changes in lifestyle – hand washing, avoiding known sites of contamination, for instance. Again, as in the case of the Bt example, there are explanations and predictions that can be made at the population level that cannot be made at the molecular level and vice versa.
Modelling the reproductive power function
Published in Journal of Applied Statistics, 2021
To deal with the incompleteness of epidemic data, O'Neill and Roberts [15] propose a Bayesian approach to inference for both the Reed-Frost model and the general stochastic epidemic model. They assume that the observed data consist of a set of removal times so the unobserved infection times are treated as parameters in the model.This model can be used for prediction and needs the number of susceptibles at time zero or the assumption of a prior distribution on the initial number of susceptibles as well as assumptions about the other prior distributions. They also assume that the unobserved time of the first infection has an exponential distribution but other distributions might also be taken. They discuss estimating this model with Markov chain Monte-Carlo methods. Kypraios and O'Neill [11] extend this model to a non-parametric Bayesian model using an augmented likelihood (with a thinned homogeneous Poisson process) and a zero mean Gaussian process prior.