We discuss a continuous-time Markov branching model in which each individual can trigger an alarm according to a Poisson process. The model is stopped when a given number of alarms is triggered or when there are no more individuals present. Our goal is to determine the distribution of the state of the population at this stopping time. In addition, the state distribution at any fixed time is also obtained. The model is then modified to take into account the possible influence of death cases. All distributions are derived using probability-generating functions, and the approach followed is based on the construction of families of martingales.

中文翻译：

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关于具有警报触发的分支模型

我们讨论了一个连续时间马尔可夫分支模型，其中每个人都可以根据泊松过程触发警报。当触发给定数量的警报或没有更多人在场时，模型将停止。我们的目标是确定这个停止时间的人口状态分布。此外，还获得了任意固定时间的状态分布。然后修改模型以考虑死亡病例的可能影响。所有分布都是使用概率生成函数导出的，所遵循的方法是基于鞅族的构建。