The majority of research on epidemics relies on models which are formulated in continuous-time. However, processing real-world epidemic data and simulating epidemics is done digitally and the continuous-time epidemic models are usually approximated by discrete-time models. In general, there is no guarantee that properties of continuous-time epidemic models, such as the stability of equilibria, also hold for the respective discrete-time approximation. We analyse the discrete-time NIMFA epidemic model on directed networks with heterogeneous spreading parameters. In particular, we show that the viral state is increasing and does not overshoot the steady-state, the steady-state is exponentially stable, and we provide linear systems that bound the viral state evolution. Thus, the discrete-time NIMFA model succeeds to capture the qualitative behaviour of a viral spread and provides a powerful means to study real-world epidemics.

中文翻译：

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离散时间 NIMFA 流行病模型的病毒状态动力学

大多数流行病研究依赖于连续时间制定的模型。然而，处理现实世界的流行病数据和模拟流行病是数字化的，连续时间流行病模型通常用离散时间模型近似。一般来说，不能保证连续时间流行模型的特性，例如均衡的稳定性，也适用于各自的离散时间近似。我们分析了具有异构传播参数的有向网络上的离散时间 NIMFA 流行病模型。特别是，我们表明病毒状态正在增加并且不会超过稳态，稳态是指数稳定的，并且我们提供了限制病毒状态演变的线性系统。因此，