Cator E, van de Bovenkamp R, Van Mieghem P
Faculty of Science, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062816. doi: 10.1103/PhysRevE.87.062816. Epub 2013 Jun 24.
The classical, continuous-time susceptible-infected-susceptible (SIS) Markov epidemic model on an arbitrary network is extended to incorporate infection and curing or recovery times each characterized by a general distribution (rather than an exponential distribution as in Markov processes). This extension, called the generalized SIS (GSIS) model, is believed to have a much larger applicability to real-world epidemics (such as information spread in online social networks, real diseases, malware spread in computer networks, etc.) that likely do not feature exponential times. While the exact governing equations for the GSIS model are difficult to deduce due to their non-Markovian nature, accurate mean-field equations are derived that resemble our previous N-intertwined mean-field approximation (NIMFA) and so allow us to transfer the whole analytic machinery of the NIMFA to the GSIS model. In particular, we establish the criterion to compute the epidemic threshold in the GSIS model. Moreover, we show that the average number of infection attempts during a recovery time is the more natural key parameter, instead of the effective infection rate in the classical, continuous-time SIS Markov model. The relative simplicity of our mean-field results enables us to treat more general types of SIS epidemics, while offering an easier key parameter to measure the average activity of those general viral agents.
任意网络上的经典连续时间易感-感染-易感（SIS）马尔可夫流行病模型被扩展，以纳入感染时间和治愈或恢复时间，每个时间都由一般分布（而非马尔可夫过程中的指数分布）来表征。这种扩展后的模型称为广义SIS（GSIS）模型，据信它对现实世界中的流行病（如在线社交网络中的信息传播、实际疾病、计算机网络中的恶意软件传播等）具有更大的适用性，因为这些流行病可能不具有指数时间特征。虽然由于GSIS模型的非马尔可夫性质，其精确的控制方程难以推导，但我们推导出了精确的平均场方程，这些方程类似于我们之前的N交织平均场近似（NIMFA），因此使我们能够将NIMFA的整个分析机制应用于GSIS模型。特别是，我们建立了计算GSIS模型中流行病阈值的标准。此外，我们表明，恢复期间感染尝试的平均次数是更自然的关键参数，而不是经典连续时间SIS马尔可夫模型中的有效感染率。我们平均场结果的相对简单性使我们能够处理更一般类型的SIS流行病，同时提供一个更容易测量那些一般病毒媒介平均活动的关键参数。
