This paper deals with a stochastic SIR (Susceptible-Infected-Recovered) model with Erlang(k,μ) distributed infectious period commonly referred as SIkR model. We show that using the total number of remaining Erlang stages as the state variable, we do not need to keep track of the stages of individual infections, and can employ a first step analysis to efficiently obtain quantities of interest. We study the distribution of the total number of recovered individuals and the distribution of the maximum number of individuals who are simultaneously infected until the end of the disease. In the literature, final outbreak size is calculated only for a small population size exactly and derivations of approximate analytic solutions from asymptotic results are suggested for larger population sizes. We numerically demonstrate that our methods are implementable on large size problem instances.

中文翻译：

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在Erlang分布的传染期下，最终暴发规模分布的精确且可实现的计算。

本文研究了具有Erlang（k，μ）分布传染期的随机SIR（敏感感染-恢复）模型，通常称为SIkR模型。我们表明，使用剩余的Erlang阶段总数作为状态变量，我们不需要跟踪单个感染的阶段，并且可以采用第一步分析来有效地获得感兴趣的数量。我们研究了恢复的个体总数的分布以及同时感染直至疾病结束的最大个体数的分布。在文献中，仅精确地针对较小的人口规模计算了最终暴发规模，并建议针对较大的人口规模从渐近结果中得出近似解析解。