Stochastic comparisons of mixtures of parametric families in stochastic epidemics.

The paper is first concerned with the stochastic comparisons for mixed Erlang random variables when the arbitrary mixing distributions are ordered by increasing directionally convex order or an univariate ordering. Similar results for mixtures of gamma, lognormal, geometric and Poisson families are given. The main results are applied for the analysis of the effect of the positive correlation and the variation of the parameters of some measures in stochastic epidemics, that are mixtures of parametric families as earlier with environmental parameters, arising from extensions that we provide of the SEIR model with vaccination and isolation for structured populations by [2] and the SIR model with term-time forcing, by [11]. Unlike the previous stochastic epidemic models, we consider parameter uncertainty with arbitrary mixing distributions, and stochastic dependencies among them. We rank the probabilities that the severity (active severity) of the epidemic in the household after the first removal exceeds a fixed level conditioning on a threshold parameter, we bound the expected value of increasing convex functions of the severity (active severity), we calculate and compare the basic reproduction numbers, for the SEIR model with vaccination and isolation; and in addition, we bound the number of type-i individuals infected from type-i infectives and the times until either a recovery or a state change happens, for the SIR model with term-time forcing. Using the positive quadrant dependence of the parameter vector, the mixture models are compared with models having the same marginal distributions for the mixing variables but independent components. They assess on the development of some public health policies (vaccination, household isolation, other structuring patterns).