Abstract Linear mapping is widely used in dynamic modeling and empirical data analysis, but it suffers from the serious shortcoming that it does not work in the common case of time intervals being large. The fundamental cause of the failure is that the linear mapping does not take into account the coupling effects of multiple events within a discrete time interval. Here, we develop a theoretical framework to provide a nonlinear mapping between continuous- and discrete-time dynamics by accounting for the coupling effect. We have verified the effectiveness of our mapping by exploring classical susceptible-infected-susceptible and susceptible-infected-recovered models. In particular, we give a quantitative criterion that the sum of two transition probabilities —from one state to the other and vice versa— must be strictly less than 1 for binary-state dynamics.

中文翻译：

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连续时间和离散时间动态之间的非线性映射

摘要线性映射广泛应用于动态建模和经验数据分析，但它有一个严重的缺点，即它不适用于时间间隔较大的常见情况。失败的根本原因是线性映射没有考虑离散时间间隔内多个事件的耦合效应。在这里，我们开发了一个理论框架，通过考虑耦合效应来提供连续时间和离散时间动力学之间的非线性映射。我们通过探索经典的易感者-感染者-易感者和易感者-感染者-恢复模型验证了我们的映射的有效性。特别是，我们给出了一个定量标准，即对于二元状态动力学，两个转移概率（从一种状态到另一种状态，反之亦然）的总和必须严格小于 1。