Much work has focused on the basic reproduction ratio [Formula: see text] for a variety of compartmental population models, but the theory of [Formula: see text] remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of [Formula: see text] for a class of such impulsive models. We first introduce [Formula: see text] and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of [Formula: see text]. Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.

中文翻译：

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具有脉冲的周期性和时滞间隔模型的基本复制率。

许多工作都集中在各种隔间人口模型的基本再生产比率上，但对于周期性和时滞的脉冲模型，[公式：见文本]的理论仍未解决。在本文中，我们针对此类脉冲模型开发了[公式：参见文本]理论。我们首先介绍[公式：参见文本]，并表明它是相关线性系统零解稳定性的阈值参数。然后，我们将此理论应用于经过脉冲处理的延时计算机病毒模型，并根据[公式：参见文本]获得其整体动力学的阈值结果。从数字上发现，时间平均延迟脉冲系统的基本复制率可能会高估病毒的传播风险。