Statistical modeling of epidemic disease propagation via branching processes and Bayesian inference

Let us consider, a random model for the spread of a certain characteristic (disease) in a given population as follows. First, the characteristic of interest is transmitted to some members of a group from a source at an initial point in time. Then the individuals who have acquired the characteristic spread it, according to a probability distribution, to the members of other groups. The new “generation” spreads the same characteristic again and the process continues over and over until either it dies out or the entire population gets the characteristic. Under certain assumptions the number of individuals possessing the characteristic form a Bienaymé-Galton-Watson branching process. We study a new random variable concerning the population: maximum number of individuals infected by single host, which is an important measure of disease's spread.; Bayesian analysts are aware of the difficulties involved with respect to the choice of the loss function for Bayesian modeling. In a decision-theoretic framework, the problem of the specification of a loss function can be at least as important as that of choosing a prior. In the context of loss robustness one is faced with the task of evaluating the elements of a class of loss functions. We focus our attention to two of the most important such classes: LINEX and weighted squared-error loss functions. We study the sensitivity with respect to the loss function of the Bayes estimators for the offspring mean in branching processes. A smallpox disease data are used to illustrate the results.; The Borel-Tanner probability distribution was derived by Borel (1942) and Tanner (1953) to characterize the distribution behavior of the number of customers served in a queuing system with Poisson input and constant service time. Later this probability distribution was applied in some models for random trees and branching processes. In the latter case one of the parameters can be interpreted as the offspring mean in a Bienaymé-Galton-Watson process with Poisson reproduction law. We propose nonparametric empirical Bayes estimators for this parameter under LINEX and weighted squared-error loss functions. Asymptotic optimality of the estimators is proved.