Threshold theorems for a stochastic model of an epidemic with natural immunization

A stochastic model of an epidemic is investigated, taking account of the removal of ill members of the population (by death, by recovery with immunization, by isolation) and natural immunization. Limiting distributions are found for the size $\nu$ of the epidemic, the number immunized $\nu_1$, and their sum, under the assumption that the original number of susceptible individuals $n\to\infty$ and the number of ill individuals $m\to\infty$, while $\lambda n\to1$, $\mu n\leqslant\alpha_0<\infty$,, where $\lambda$ and $\mu$ are the coefficients for the contraction of the disease and of immunization respectively.