The Exact and Asymptotic Formulas for the State Probabilities in Simple Epidemics with m Kinds of Susceptibles

A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered. it is assumed that an individual's susceptibility at time t depends on the number of susceptible individuals at time t in his group, and on the total number of infective individuals at time t. The progress of this simple epidemic is modeled by an m-dimensional stochastic process. The components of this stochastic process represent the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained. It is shown that the approximate formulas are simple functions of time while, the derivations of the exact formulas involved tedious computations. (Author)