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The consideration of non-Poissonian infection or recovery processes does not lend itself easily to analytical approaches <ref>Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.</ref>. Some simple forms for the distribution of infectious periods, such as the Erlang distribution, which can be described as the convolution of identical Poisson processes <ref>Cox D R. Renewal Theory, 2-nd Edn[J]. 1967.</ref>, can be tackled analytically by postulating an extended epidemic model with different infective phases and Poissonian transitions among them <ref>Lloyd A L. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods[J]. Proceedings of the Royal Society of London. Series B: Biological Sciences, 2001, 268(1470): 985-993.</ref>. However, general non Poissonian forms lead to convoluted sets of integrodifferential equations <ref>Keeling M J, Grenfell B T. Disease extinction and community size: modeling the persistence of measles[J]. Science, 1997, 275(5296): 65-67.</ref>. As a consequence there are not many analytical results for non-Poissonian transitions in complex networks. We mention the results of Min, Goh, and Kim (2013)<ref name="Min2013">Min B, Goh K I, Kim I M. Suppression of epidemic outbreaks with heavy-tailed contact dynamics[J]. EPL (Europhysics Letters), 2013, 103(5): 50002.</ref> which consider the SIR process on a network in which infection events follow an interevent distribution P aðτaÞ. Assuming that infected nodes remain in that state for a fixed amount of time τi, it is possible to compute <ref name="Min2013"></ref>(Min, Goh, and Kim, 2013) the disease transmissibility as

The consideration of non-Poissonian infection or recovery processes does not lend itself easily to analytical approaches <ref>Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.</ref>. Some simple forms for the distribution of infectious periods, such as the Erlang distribution, which can be described as the convolution of identical Poisson processes <ref>Cox D R. Renewal Theory, 2-nd Edn[J]. 1967.</ref>, can be tackled analytically by postulating an extended epidemic model with different infective phases and Poissonian transitions among them <ref>Lloyd A L. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods[J]. Proceedings of the Royal Society of London. Series B: Biological Sciences, 2001, 268(1470): 985-993.</ref>. However, general non Poissonian forms lead to convoluted sets of integrodifferential equations <ref>Keeling M J, Grenfell B T. Disease extinction and community size: modeling the persistence of measles[J]. Science, 1997, 275(5296): 65-67.</ref>. As a consequence there are not many analytical results for non-Poissonian transitions in complex networks. We mention the results of Min, Goh, and Kim (2013)<ref name="Min2013">Min B, Goh K I, Kim I M. Suppression of epidemic outbreaks with heavy-tailed contact dynamics[J]. EPL (Europhysics Letters), 2013, 103(5): 50002.</ref> which consider the SIR process on a network in which infection events follow an interevent distribution P aðτaÞ. Assuming that infected nodes remain in that state for a fixed amount of time τi, it is possible to compute <ref name="Min2013"></ref> the disease transmissibility as

考虑了非泊松感染或恢复过程的疾病传播，其解析上的研究并不简单容易（Lambiotte等，2013）。一般的非泊松形式会导致卷积集的积分微分方程组（Keeling和Gren fall，1997）。因此，对于复杂网络中的非泊松过程，没有太多的解析结果。 可以提到的是，Min等人在2013年考虑了复杂网络上，感染事件遵循事件之间时间分布$P_a(\tau_a)$的SIR传播过程。假设感染态节点固定的时间$\tau_i$内状态不发生改变，则可以计算疾病传播率为（Min等人，2013年）

考虑了非泊松感染或恢复过程的疾病传播，其解析上的研究并不简单容易（Lambiotte等，2013）。一般的非泊松形式会导致卷积集的积分微分方程组（Keeling和Gren fall，1997）。因此，对于复杂网络中的非泊松过程，没有太多的解析结果。 可以提到的是，Min等人在2013年考虑了复杂网络上，感染事件遵循事件之间时间分布$P_a(\tau_a)$的SIR传播过程。假设感染态节点固定的时间$\tau_i$内状态不发生改变，则可以计算疾病传播率为（Min等人，2013年）

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For a power-law distribution $P_a(\tau_a)~\tau_a^{-\alpha}$, it is found that $\tau_{ic}$ diverges as $\alpha\to2$, implying that only diseases without recovery are able to spread through the network (Min, Goh, and Kim, 2013). An important step forward in the treatment of generic nonexponentially distributed recovery and transmission times in the SIR model is the application of a message-passing method, as reported by Karrer and Newman (2010). This approach leads to an exact description in terms of integrodifferential equations for trees and locally treelike networks, and to exact bounds for non-tree-like networks, in good agreement with simulations.

For a power-law distribution $P_a(\tau_a)~\tau_a^{-\alpha}$, it is found that $\tau_{ic}$ diverges as $\alpha\to2$, implying that only diseases without recovery are able to spread through the network <ref name="Min2013"></ref>. An important step forward in the treatment of generic nonexponentially distributed recovery and transmission times in the SIR model is the application of a message-passing method, as reported by Karrer and Newman (2010)<ref>Karrer B, Newman M E J. Message passing approach for general epidemic models[J]. Physical Review E, 2010, 82(1): 016101.</ref>. This approach leads to an exact description in terms of integrodifferential equations for trees and locally treelike networks, and to exact bounds for non-tree-like networks, in good agreement with simulations.

对于幂律分布$P_a(\tau_a)~\tau_a^{-\alpha}$，发现$\tau_{ic}$随$\alpha\to2$时发散，这意味着只有无法恢复的疾病才能在网络上将疾病传播开来（Min等人，2013）。2010年Karrer和Newman通过应用消息传递方法（message-passing method），对在SIR模型中处理一般的非指数分布的恢复和感染时间方面的研究迈出了重要的一步。 这种方法通过用积分微分方程来对树和局部树状网络的作精确描述，并且对非树状网络给出了精确边界，其结果与数值模拟结果符合得很好。

对于幂律分布$P_a(\tau_a)~\tau_a^{-\alpha}$，发现$\tau_{ic}$随$\alpha\to2$时发散，这意味着只有无法恢复的疾病才能在网络上将疾病传播开来（Min等人，2013）。2010年Karrer和Newman通过应用消息传递方法（message-passing method），对在SIR模型中处理一般的非指数分布的恢复和感染时间方面的研究迈出了重要的一步。 这种方法通过用积分微分方程来对树和局部树状网络的作精确描述，并且对非树状网络给出了精确边界，其结果与数值模拟结果符合得很好。

Finally, Cator, van de Bovenkamp, and Van Mieghem (2013) proposed an extension of the SIS IBMF theory for nonexponential distributions of infection or healing times.Using renewal theory, their main result is the observation that the functional form of the prevalence in the metastable state is the same as in the Poissonian SIS model, when the spreading rate $\lambda=\beta/\mu$ is replaced by the average number of infection attempts during a recovery time. The theory by Cator, van de Bovenkamp, and Van Mieghem (2013) also allows one to estimate the epidemic threshold in non-Markovian SIS epidemics.

Finally, Cator, van de Bovenkamp, and Van Mieghem (2013)<ref name="Cator2013">Cator E, Van de Bovenkamp R, Van Mieghem P. Susceptible-infected-susceptible epidemics on networks with general infection and cure times[J]. Physical Review E, 2013, 87(6): 062816.</ref> proposed an extension of the SIS IBMF theory for nonexponential distributions of infection or healing times.Using renewal theory, their main result is the observation that the functional form of the prevalence in the metastable state is the same as in the Poissonian SIS model, when the spreading rate $\lambda=\beta/\mu$ is replaced by the average number of infection attempts during a recovery time. The theory by Cator, van de Bovenkamp, and Van Mieghem (2013)<ref name="Cator2013"></ref> also allows one to estimate the epidemic threshold in non-Markovian SIS epidemics.

最后，Cator、van de Bovenkamp和Van Mieghem在2013年针对感染或恢复时间为非指数分布时的SIS模型提出了延伸性的基于个体的平均场理论。通过使用更新理论，他们主要观察到的结果是，当用一个恢复时间内的平均尝试感染次数代替传播率$\lambda=\beta/\mu$作为自变量时，亚稳态状态下的疾病传播范围的函数形式与泊松过程情况下的SIS模型形式相同。此外，他们在2013年的另一篇文献还通过理论预测了非马尔科夫SIS流行病的传播阈值。

最后，Cator、van de Bovenkamp和Van Mieghem在2013年针对感染或恢复时间为非指数分布时的SIS模型提出了延伸性的基于个体的平均场理论。通过使用更新理论，他们主要观察到的结果是，当用一个恢复时间内的平均尝试感染次数代替传播率$\lambda=\beta/\mu$作为自变量时，亚稳态状态下的疾病传播范围的函数形式与泊松过程情况下的SIS模型形式相同。此外，他们在2013年的另一篇文献还通过理论预测了非马尔科夫SIS流行病的传播阈值。