The Asymptotic Evolution of the General Stochastic Epidemic

Abstract

Generalizing Sellke's construction, a general stochastic epidemic with non-Markovian transition behavior is considered. At time $t = 0$, the population of total size $K$ consists of $aK$ individuals that are infected by a certain disease (and infectious); the remaining $bK$ individuals are susceptible with respect to that disease. An initially susceptible individual $i$, when infected (call $A^K_i$ its time of infection), stays infectious for a period of length $r_i$, until it is removed. An initially infected individual $i$ stays infected for a period of length $\hat{r}_i$ until it is removed. Removed individuals can no longer be affected by the disease. A deterministic approximation as (as $K \rightarrow \infty$) to the empirical measure $\xi_K = \frac{1}{K} \sum^{aK}_{i=1} \delta_{(0,\hat{r}_i)} + \frac{1}{K} \sum^{bK}_{i=1} \delta_{(A^K_i, A^K_i + r_i)}$, describing the average path behavior, is established using Stein's method.