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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）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。

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"></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 the disease transmissibility as:

<math>\begin{equation}

<math>\begin{equation}

T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta.

T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta.

\end{equation}</math>

\end{equation}</math>

其中<math>\Psi(\Delta)=\int_\Delta^\infty P_a(\tau_a)d\tau_a/\int_0^\infty P_a(\tau_a)d\tau_a</math>。等式（67）假设感染的动力学过程遵循平稳的更新过程（Cox，1967； Van Mieghem，2014b）。应用生成函数方法，从隐式方程中可以得到传播阈值作为<math>\tau_i</math>的函数表示为：

<math>\Psi(\Delta)=\int_\Delta^\infty P_a(\tau_a)d\tau_a/\int_0^\infty P_a(\tau_a)d\tau_a</math>

 

对非泊松感染或恢复过程的考虑，并不容易简单地得出它的解析方法<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>。一些简单的传染期分布形式，如Erlang分布，可描述为相同泊松过程的卷积<ref name="Cox1967">Cox D R. Renewal Theory, 2-nd Edn[J]. 1967.</ref>，可以通过假设一个具有不同传染期和泊松过渡的扩展传染病模型来解析地解决<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>。然而，一般的非泊松形式会导致复杂的积分微分方程组<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>。因此，对于复杂网络中的非泊松传播，并没有很多分析结果。可以提到的是，Min, Goh, 和Kim等人（2013年）<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> 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。假设感染节点处于该状态一段固定时间<math>\tau_i</math>，有可能计算出的疾病传播率为：

<math>\begin{equation}

<math>\begin{equation}

T(\tau_{ic})=\frac{\left<k\right>}{\left<k^2\right>-\left<k\right>}.

T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta.

\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.