The correlation structure of epidemic models.

For a general Markov SIS epidemic model, the fates of individuals at different times are shown to be positively correlated. When the population is subjected to two diseases, a certain condition, here called positive interference, results in positive correlations between individuals with respect to either disease, while another condition, called competition, gives negative correlation between diseases and positive correlation within each disease. The results generalize to two classes of disease, with positive interference within each class and competition between classes. A general (non-Markov) SIR model (which includes the general epidemic and generalized Reed-Frost models) exhibits positive correlation. The results for SIS models rely heavily on monotonicity properties and in some cases on a careful choice of partial order. For the SIR models a graphical construction of the models is used.