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Numerical results on non-Poissonian epidemics in networks are relatively scarce. Simple event-driven approaches rely on a time ordered sequence of events (tickets) that represent actions to be taken (recovery or infection) at given fixed times, which are computed from the interevent distributions <math>P_i(\tau_i)</math> and <math>P_a(\tau_a)</math>. These approaches are quite demanding, so only small system sizes can be considered. For example, Van Mieghem and van de Bovenkamp (2013)<ref name="Van2013">Van Mieghem P, Van de Bovenkamp R. Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks[J]. Physical review letters, 2013, 110(10): 108701.</ref> reported results for the SIS model with Poissonian recovery, with rate <math>\mu</math>, while infection happens with a nonexponential distribution following the Weibull form <math>P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}</math>. In this case, strong variations in the value of the prevalence and of the epidemic threshold are found when varying the parameter <math>\alpha</math>. A promising approach is provided by the general simulation framework proposed by <math>\mathrm{Bogu\tilde{n}\acute{a}}</math> et al. (2014)<ref name="Boguna2014"></ref>, based on the extension of the Gillespie algorithm for Poissonian processes <ref name="Gillespie1977"></ref>. This algorithm allows the simulation of much larger network sizes.

Numerical results on non-Poissonian epidemics in networks are relatively scarce. Simple event-driven approaches rely on a time ordered sequence of events (tickets) that represent actions to be taken (recovery or infection) at given fixed times, which are computed from the interevent distributions <math>P_i(\tau_i)</math> and <math>P_a(\tau_a)</math>. These approaches are quite demanding, so only small system sizes can be considered. For example, Van Mieghem and van de Bovenkamp (2013)<ref name="Van2013">Van Mieghem P, Van de Bovenkamp R. Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks[J]. Physical review letters, 2013, 110(10): 108701.</ref> reported results for the SIS model with Poissonian recovery, with rate <math>\mu</math>, while infection happens with a nonexponential distribution following the Weibull form <math>P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}</math>. In this case, strong variations in the value of the prevalence and of the epidemic threshold are found when varying the parameter <math>\alpha</math>. A promising approach is provided by the general simulation framework proposed by <math>\mathrm{Bogu\tilde{n}\acute{a}}</math> et al. (2014)<ref name="Boguna2014"></ref>, based on the extension of the Gillespie algorithm for Poissonian processes <ref name="Gillespie1977"></ref>. This algorithm allows the simulation of much larger network sizes.

复杂网络中关于非泊松流行病的数值结果相对较少。简单的事件驱动方法依赖于发生事件的时间顺序，这些事件表示的是在给定固定的时刻某些行为会发生，例如恢复或感染，而发生所需要的时间是由事件发生时间间隔分布<math>P_i(\tau_i)</math>和<math>P_a(\tau_a)</math>计算得出。这些方法的要求很高，因此只能考虑较小的系统来研究。例如，Van Mieghem和van de Bovenkamp（2013年）对SIS模型传播的研究结论是，恢复过程为泊松过程，速率为<math>\mu</math>，而感染过程是非泊松的，发生时间间隔服从韦布尔分布<math>P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}</math>。在这种传播过程情况下，当改变参数<math>\alpha</math>时，发现疾病传播力和传播阈值的有非常明显的差异性。在2014年，<math>\mathrm{Bogu\tilde{n}\acute{a}}</math>等人<ref name="Boguna2014">Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>提出的通用仿真框架提供了一种不错的方法，基于用于泊松过程的Gillespie算法的延伸算法<ref name="Gillespie1977">Gillespie D T. Exact stochastic simulation of coupled chemical reactions[J]. The journal of physical chemistry, 1977, 81(25): 2340-2361.</ref>，该算法可以适用于模拟更大的网络规模下的结果。

复杂网络中关于非泊松流行病的数值结果相对较少。简单的事件驱动方法依赖于发生事件的时间顺序，这些事件表示的是在给定固定的时刻某些行为会发生，例如恢复或感染，而发生所需要的时间是由事件发生时间间隔分布<math>P_i(\tau_i)</math>和<math>P_a(\tau_a)</math>计算得出。这些方法的要求很高，因此只能考虑较小的系统来研究。例如，Van Mieghem和van de Bovenkamp（2013年）对SIS模型传播的研究结论是，恢复过程为泊松过程，速率为<math>\mu</math>，而感染过程是非泊松的，发生时间间隔服从韦布尔分布<math>P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}</math>。在这种传播过程情况下，当改变参数<math>\alpha</math>时，发现疾病流行率和传播阈值的有非常明显的差异性。在2014年，<math>\mathrm{Bogu\tilde{n}\acute{a}}</math>等人<ref name="Boguna2014">Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>提出的通用仿真框架提供了一种不错的方法，基于用于泊松过程的Gillespie算法的延伸算法<ref name="Gillespie1977">Gillespie D T. Exact stochastic simulation of coupled chemical reactions[J]. The journal of physical chemistry, 1977, 81(25): 2340-2361.</ref>，该算法可以适用于模拟更大的网络规模下的结果。

For a power-law distribution <math>P_a(\tau_a)~\tau_a^{-\alpha}</math>, it is found that <math>\tau_{ic}</math> diverges as <math>\alpha\to2</math>, 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 name="Kar2010">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.

For a power-law distribution <math>P_a(\tau_a)~\tau_a^{-\alpha}</math>, it is found that <math>\tau_{ic}</math> diverges as <math>\alpha\to2</math>, 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 name="Kar2010">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<math>P_a(\tau_a)~\tau_a^{-\alpha}</math>，发现<math>\tau_{ic}</math>随<math>\alpha\to2</math>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<math>P_a(\tau_a)~\tau_a^{-\alpha}</math>，发现<math>\tau_{ic}</math>随<math>\alpha\to2</math>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）<ref name="Min2013"></ref>。2010年Karrer和Newman<ref name="Kar2010">Karrer B, Newman M E J. Message passing approach for general epidemic models[J]. Physical Review E, 2010, 82(1): 016101.</ref>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)<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 <math>\lambda=\beta/\mu</math> 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.

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 <math>\lambda=\beta/\mu</math> 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<math>\lambda=\beta/\mu</math>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（2013年）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> 对基于个体平均场理论的SIS模型，提出了一个扩展，运用于感染或恢复时间的非指数分布。使用新的理论，他们的主要结果是观察到：用恢复时间内的平均尝试感染次数代替传播率<math>\lambda=\beta/\mu</math>时，亚稳态流行率的函数形式与泊松SIS模型的形式相同。此外，他们的理论（2013年）<ref name="Cator2013"></ref> 还可以用来预测非马尔科夫SIS流行病的传播阈值。

==参考文献 References==   

==参考文献 References==