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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）<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。

对于SIS模型和SIR模型，非泊松感染和恢复过程的框架可以按下面方式建立<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"></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.

网络中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一个按时间顺序排列的事件(票)序列，这些事件(票)表示在给定的固定时间内要采取的操作(恢复或感染)，这些操作是根据事件间分布<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。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。

网络中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一个按时间顺序排列的事件(票)序列，这些事件(票)表示在给定的固定时间内要采取的操作(恢复或感染)，这些操作是根据事件间分布<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。例如，Van Mieghem和van de Bovenkamp（2013年）<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>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。