Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$ - specifically the values of the process at crossing times, {\it{viz.}}, $\{(Z_{\tau_j}, Z_{\nu_j})\}$ - along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.

翻译：受COVID动力学的应用启发，我们描述了一个随机环境中的分支过程模型$\{Z_n\}$，其特性在跨越上下阈值时发生改变。这引入了涉及增长和衰减阶段的循环路径行为，导致超临界和亚临界状态。尽管该过程并非马尔可夫过程，我们识别了在随机时间点$\{(\tau_j, \nu_j)\}$处的子序列——具体而言，即过程在穿越时刻的值，即$\{(Z_{\tau_j}, Z_{\nu_j})\}$——在这些子序列上，过程保留了马尔可夫结构。在温和的矩条件和正则性条件下，我们证明了这些子序列具有再生结构，并证实了超临界和亚临界状态下过程增长率的极限正态分布是解耦的。基于此，我们建立了有关超临界和亚临界状态长度以及过程在这些状态下所花费时间比例的极限定理。作为分析的副产品，我们明确地用后代分布、阈值分布和环境序列的泛函表达了极限方差。