更改

在传播研究中，当平均传播速率已知时，则对传播过程作泊松近似，可能是合理的（Lambiotte等人，2013）<ref name=Lambiotte2013>Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.</ref>，但实际情况并非如此，实证的感染时间的分布特征，通常具有一个以平均值为中心的峰，同时显示出明显的非指数尾巴（Blythe和Anderson，1988）<ref name=Blythe1988>Blythe S P, Anderson R M. Variable infectiousness in HFV transmission models[J]. Mathematical Medicine and Biology: A Journal of the IMA, 1988, 5(3): 181-200.</ref>。此外，关于个体之间的社交和通信交流的模式，最近的证据，也观察到事件发生时间的间隔具有长尾特征，激发了人们对非指数传播过程研究的兴趣。

在传播研究中，当平均传播速率已知时，则对传播过程作泊松近似，可能是合理的（Lambiotte等人，2013）<ref name=Lambiotte2013>Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.</ref>，但实际情况并非如此，实证的感染时间的分布特征，通常具有一个以平均值为中心的峰，同时显示出明显的非指数尾巴（Blythe和Anderson，1988）<ref name=Blythe1988>Blythe S P, Anderson R M. Variable infectiousness in HFV transmission models[J]. Mathematical Medicine and Biology: A Journal of the IMA, 1988, 5(3): 181-200.</ref>。此外，关于个体之间的社交和通信交流的模式，最近的证据，也观察到事件发生时间的间隔具有长尾特征，激发了人们对非指数传播过程研究的兴趣。

 

The framework of non-Poissonian infection and recovery processes can be set up as follows, for either the SIS or SIR model <ref name=Bog2014>Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>: Infected individuals remain infective for a period of time <math>\tau_i</math>, after which they recover, that follows the (nonexponential)<math>P_i(\tau_i)</math> distribution. For simplicity, it is assumed that this distribution is the same for all nodes. Infection events take place along active links, connecting an infected to a susceptible node. Active links transmit the disease at times following the interevent distribution <math>P_a(\tau_a)</math>, i.e., a susceptible individual connected to an infected node becomes infected at a time <math>\tau_a</math>, measured from the instant the link became active. If a susceptible node is connected to more than one infected node, it becomes infected at the time of the first active link transmitting the disease. The complexity of this non-Markovian process is now evident: the infection of a node depends not only on the number of neighbors, but also on the time at which each connection became active.

The framework of non-Poissonian infection and recovery processes can be set up as follows, for either the SIS or SIR model <ref name="Bog2014">Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>: Infected individuals remain infective for a period of time <math>\tau_i</math>, after which they recover, that follows the (nonexponential)<math>P_i(\tau_i)</math> distribution. For simplicity, it is assumed that this distribution is the same for all nodes. Infection events take place along active links, connecting an infected to a susceptible node. Active links transmit the disease at times following the interevent distribution <math>P_a(\tau_a)</math>, i.e., a susceptible individual connected to an infected node becomes infected at a time <math>\tau_a</math>, measured from the instant the link became active. If a susceptible node is connected to more than one infected node, it becomes infected at the time of the first active link transmitting the disease. The complexity of this non-Markovian process is now evident: the infection of a node depends not only on the number of neighbors, but also on the time at which each connection became active.

对于SIS模型和SIR模型，非泊松感染和恢复过程的框架如下（boguna,2014）：被感染的个体在一段时间<math>\tau_i</math>内仍具有感染力，一旦过了<math>\tau_i</math>则就会恢复，<math>\tau_i</math>遵循非指数的分布<math>P_i(\tau_i)</math>。为了简单起见，通常可以假设所有节点服从同一分布。如果网络中一条连边的两端分别连接了易感个体和感染态个体，则称这样的连边为活跃边。在传播过程中，疾病是通过这样的活跃边，将疾病由感染态个体传播给易感个体。通过活跃边传播疾病的疾病感染时间服从分布<math>P_a(\tau_a)</math>，也就是说，例如在<math>t</math>时刻产生了一条活跃边，则该活跃边一端的易感态节点将在<math>t+\tau_a</math>时刻被感染为感染态节点。如果一个易感态节点连接了不止一个感染态节点，也就是说有多条活跃边，则它将被最早传播疾病过来的那条活跃边感染为感染态节点。因此，非马尔科夫过程的复杂性显而易见：一个节点的感染不止依赖于邻居节点数量，还依赖于其活跃边出现的时间等。

对于SIS模型和SIR模型，非泊松感染和恢复过程的框架如下（boguna,2014）<ref name="Bog2014">Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>：被感染的个体在一段时间<math>\tau_i</math>具有感染力，一旦过了<math>\tau_i</math>则就会恢复，<math>\tau_i</math>服从非指数的分布<math>P_i(\tau_i)</math>。简单起见，通常可以假设所有节点服从同一分布。传播事件发生在活跃连边上，活跃连边的两端分别连接了感染个体和易感个体。通过活跃连边传播疾病的时间间隔，服从分布<math>P_a(\tau_a)</math>，例如，从一条连边成为活跃连边开始计时，该活跃连边一端的易感节点将在<math>t+\tau_a</math>时间内成为感染节点。如果一个易感节点连接了不止一个感染节点，它将被最早的那条活跃连边感染为感染节点。因此，非马尔科夫过程的复杂性显而易见：一个节点的感染不仅依赖于邻居节点数量，还依赖于其活跃连边出现的时间等。

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>Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>, based on the extension of the Gillespie algorithm for Poissonian processes <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>. 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">Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.</ref>, based on the extension of the Gillespie algorithm for Poissonian processes <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>. 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在2013年研究了恢复过程为泊松过程，而感染过程是非泊松的且感染事件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>时，SIS模型传播的结果。在这种传播过程情况下，当改变参数<math>\alpha</math>时，发现疾病传播范围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，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在2013年研究了恢复过程为泊松过程，而感染过程是非泊松的且感染事件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>时，SIS模型传播的结果。在这种传播过程情况下，当改变参数<math>\alpha</math>时，发现疾病传播范围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，该算法可以适用于模拟更大的网络规模下的结果。

The consideration of non-Poissonian infection or recovery processes does not lend itself easily to analytical approaches <ref name=Lam2013>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 name=Cox1967>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 name=Lloyd2001>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 name=Kee1997>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 <math>P_a(\tau_a)</math>. 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

The consideration of non-Poissonian infection or recovery processes does not lend itself easily to analytical approaches <ref name="Lam2013">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 name="Cox1967">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 name="Lloyd2001">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 name="Kee1997">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 <math>P_a(\tau_a)</math>. 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<math>P_a(\tau_a)</math>的SIR传播过程。假设感染态节点固定的时间<math>\tau_i</math>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<math>P_a(\tau_a)</math>的SIR传播过程。假设感染态节点固定的时间<math>\tau_i</math>内状态不发生改变，则可以计算疾病传播率为（Min等人，2013年）

\end{equation}</math>

\end{equation}</math>

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）。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。

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。

== 参考文献 References ==   

==参考文献 References==   

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