非马尔科夫疾病传播（Non-Markovian Epidemic Spreading）

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马尔科夫疾病传播

The modeling framework presented in the previous sections is mostly based on the Poisson approximation[1] for both the transmission and recovery processes. The Poisson approximation assumes that the probabilities per unit time of transmitting the disease through a given edge, or recovering for a given infected node, are constant,[math]\displaystyle{ }[/math] and equal to [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \mu }[/math], respectively. Equivalently, the total time [math]\displaystyle{ \tau_i }[/math] that a given node [math]\displaystyle{ i }[/math] remains infected is a random variable with an exponential distribution [math]\displaystyle{ P_i(\tau_i)=\mu e^{-\tau_i \mu} }[/math], and that the time [math]\displaystyle{ \tau_a }[/math] for an infection to propagate from an infected node to a susceptible node along a given edge (the interevent time) is also exponentially distributed [math]\displaystyle{ P_a(\tau_a)=\mu e^{-\tau_a \mu} }[/math].

前面的章节中提出的建模框架，包括传染病的传播和恢复过程，大多是基于泊松近似[1]。这种泊松近似假设单位时间内通过给定连边传播疾病的概率或给定感染节点的恢复概率，是常数，分别等于[math]\displaystyle{ \beta }[/math]和[math]\displaystyle{ \mu }[/math]。对应等效地，网络中的节点[math]\displaystyle{ i }[/math]仍然为感染态的总时间[math]\displaystyle{ \tau_i }[/math]是服从指数分布[math]\displaystyle{ P_i(\tau_i)=\mu e^{-\tau_i \mu} }[/math]的随机变量，并且一个感染态节点沿着一条边传播疾病给一个易感染节点的所需要花费的时间(事件发生间隔)[math]\displaystyle{ \tau_a }[/math]也是服从指数分布[math]\displaystyle{ P_a(\tau_a)=\mu e^{-\tau_a \mu} }[/math]。


From a practical point of view, the Poisson assumption leads to an increased mathematical tractability. Indeed, since the rates of transmission and recovery are constant, they do not depend on the previous history of the individual, and thus lead to memoryless, Markovian processes [2][3][1][4].

从实用的角度看，泊松过程的假设，使得数学分析变得更易处理。实际上，由于传播和恢复的速率恒定，它们不依赖于个体的先前历史，因此这可以称为是无记忆的马尔科夫过程（Van Kampen，1981；Ross，1996； Tijms，2003； Van Mieghem，2014b）[2][3][1][4]。

非马尔科夫疾病传播

While the Poisson approximation may be justified when only the average rates are known [5], it is at odds with empirical evidence for the time duration of the infective period in most diseases [6], whose distribution usually features a peak centered on the average value but exhibits strongly nonexponential tails. Furthermore, the interest in nonexponential transmission processes has also been fueled by the recent evidence on the patterns of social and communication contacts between individuals, which have been observed to be ruled by heavy-tailed distributions of interevent times.

当知道平均传播速率时，泊松近似可能是合理的[5]，但它与大多数疾病感染期持续时间的经验证据不一致，其分布通常以平均值为中心的峰值，但表现出强烈的非指数尾部[6]。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 [7]: Infected individuals remain infective for a period of time [math]\displaystyle{ \tau_i }[/math], after which they recover, that follows the (nonexponential)[math]\displaystyle{ 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]\displaystyle{ P_a(\tau_a) }[/math], i.e., a susceptible individual connected to an infected node becomes infected at a time [math]\displaystyle{ \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可以按下面方式建立[7]：被感染的个体在一段时间[math]\displaystyle{ \tau_i }[/math]具有感染力，一旦过了[math]\displaystyle{ \tau_i }[/math]则就会恢复，[math]\displaystyle{ \tau_i }[/math]服从非指数的分布[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ P_i(\tau_i) }[/math] and [math]\displaystyle{ 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)[8] reported results for the SIS model with Poissonian recovery, with rate [math]\displaystyle{ \mu }[/math], while infection happens with a nonexponential distribution following the Weibull form [math]\displaystyle{ 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]\displaystyle{ \alpha }[/math]. A promising approach is provided by the general simulation framework proposed by [math]\displaystyle{ \mathrm{Bogu\tilde{n}\acute{a}} }[/math] et al. (2014)[9], based on the extension of the Gillespie algorithm for Poissonian processes [10]. 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]\displaystyle{ P_i(\tau_i) }[/math]和[math]\displaystyle{ 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）[8]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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \mathrm{Bogu\tilde{n}\acute{a}} }[/math]等人[9]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[10]，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 [11]. 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 [12], can be tackled analytically by postulating an extended epidemic model with different infective phases and Poissonian transitions among them [13]. However, general non Poissonian forms lead to convoluted sets of integrodifferential equations [14]. 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)[15]which consider the SIR process on a network in which infection events follow an interevent distribution [math]\displaystyle{ 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]\displaystyle{ \begin{equation} T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta. \end{equation} }[/math]

where, [math]\displaystyle{ \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[11]。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[12]，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[13]。然而，一般的非泊松形式会导致复杂的积分微分方程组[14]。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）[15] 考虑了复杂网络上，感染事件遵循事件之间时间分布[math]\displaystyle{ 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]\displaystyle{ \tau_i }[/math]，有可能计算出的疾病传播率为： [math]\displaystyle{ \begin{equation} T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta. \end{equation} }[/math]

其中[math]\displaystyle{ \Psi(\Delta)=\int_\Delta^\infty P_a(\tau_a)d\tau_a/\int_0^\infty P_a(\tau_a)d\tau_a }[/math]。

For a power-law distribution [math]\displaystyle{ P_a(\tau_a)~\tau_a^{-\alpha} }[/math], it is found that [math]\displaystyle{ \tau_{ic} }[/math] diverges as [math]\displaystyle{ \alpha\to2 }[/math], implying that only diseases without recovery are able to spread through the network [15]. 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)[16]. 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]\displaystyle{ P_a(\tau_a)~\tau_a^{-\alpha} }[/math]，发现[math]\displaystyle{ \tau_{ic} }[/math]随[math]\displaystyle{ \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）[15]。2010年Karrer和Newman[16]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)[17] 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]\displaystyle{ \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)[17] also allows one to estimate the epidemic threshold in non-Markovian SIS epidemics.

最后，Cator、Van de Bovenkamp和Van Mieghem（2013年）[17] 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]\displaystyle{ \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）[17] 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

  1. ↑ 1.0 1.1 1.2 1.3 Tijms H C. A first course in stochastic models[M]. John Wiley and sons, 2003.
  2. ↑ 2.0 2.1 Van Kampen N G. Stochastic processes in chemistry and physics[J]. Chaos, 1981.
  3. ↑ 3.0 3.1 Ross S M. Stochastic Processes. John Wiley & Sons[J]. New York, 1996.
  4. ↑ 4.0 4.1 Van Mieghem P. Performance analysis of complex networks and systems[M]. Cambridge University Press, 2014.
  5. ↑ 5.0 5.1 Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.
  6. ↑ 6.0 6.1 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.
  7. ↑ 7.0 7.1 Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.
  8. ↑ 8.0 8.1 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.
  9. ↑ 9.0 9.1 Boguná M, Lafuerza L F, Toral R, et al. Simulating non-Markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108.
  10. ↑ 10.0 10.1 Gillespie D T. Exact stochastic simulation of coupled chemical reactions[J]. The journal of physical chemistry, 1977, 81(25): 2340-2361.
  11. ↑ 11.0 11.1 Lambiotte R, Tabourier L, Delvenne J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(7): 320.
  12. ↑ 12.0 12.1 Cox D R. Renewal Theory, 2-nd Edn[J]. 1967.
  13. ↑ 13.0 13.1 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.
  14. ↑ 14.0 14.1 Keeling M J, Grenfell B T. Disease extinction and community size: modeling the persistence of measles[J]. Science, 1997, 275(5296): 65-67.
  15. ↑ 15.0 15.1 15.2 15.3 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.
  16. ↑ 16.0 16.1 Karrer B, Newman M E J. Message passing approach for general epidemic models[J]. Physical Review E, 2010, 82(1): 016101.
  17. ↑ 17.0 17.1 17.2 17.3 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.



此页参考来源： Pastor-Satorras R, Castellano C, Van Mieghem P, et al. Epidemic processes in complex networks[J]. Reviews of modern physics, 2015, 87(3): 925.

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