We extend the $N$-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times $\underline{\mathrm{T}}\left(r\right)$ and $\overline{\mathrm{T}}\left(r\right)$ as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance $r$. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for $\overline{\mathrm{T}}\left(r\right)$. Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time $\overline{\mathrm{T}}\left(r\right)$ is almost entirely determined by the basic reproduction number $R_0$ of the network. The value of the upper-transition time $\overline{\mathrm{T}}\left(r\right)$ around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

中文翻译：

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从时变网络到静态网络的转变：易感-传染-易感流行病的N-交织平均场近似中的时间尺度分离

我们延长$氮$-时变网络的易感-传染-易感（SIS）流行病学过程的交织平均场近似（NIMFA）。时变网络上的过程通常是在过程和网络演化发生在不同时间尺度的假设下进行分析的。这种近似称为时间尺度分离。我们研究疾病传播和网络拓扑更新之间的时间尺度分离。我们介绍过渡时间$\underline{\mathrm{时间}}（r）$和$\stackrel{￣}{\mathrm{时间}}（r）$分别作为中间状态与退火（快速变化的网络）和淬火（静态网络）状态之间的边界，用于固定的精度公差$r$。通过分析静态 NIMFA 过程的收敛性，我们分析推导了$\stackrel{￣}{\mathrm{时间}}（r）$。我们的结果提供了静态 NIMFA SIS 过程收敛到稳定状态的时间的见解和界限。我们表明，在我们的假设下，上跃迁时间$\stackrel{￣}{\mathrm{时间}}（r）$几乎完全由基本再生数决定${右}_0$网络的。上跃迁时间的值$\stackrel{￣}{\mathrm{时间}}（r）$流行病阈值附近的误差很大，这与目前的理解一致，即一些现实世界的流行病无法用上述时间尺度分离来近似。