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

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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, and equal to $\beta$ and $\mu$, respectively. Equivalently, the total time $\tau_i$ that a given node $i$ remains infected is a random variable with an exponential distribution $P_i(\tau_i)=\mu e^{-\tau_i \mu}$, and that the time $\tau_a$ for an infection to propagate from an infected to a susceptible node along a given edge (the interevent time) is also exponentially distributed $P_a(\tau_a)=\mu e^{-\tau_a \mu}$.

在经典的流行病传播中,个体间疾病传播和恢复过程被近似假设为泊松过程,为马尔科夫的疾病传播。这种泊松近似假设了单位时间内通过给定连边传播疾病的概率或针对给定感染节点恢复的概率是常数的,分别等于$\beta$和$\mu$。对应等效地,网络中处于I态的节点$i$仍然为感染态的总时间$\tau_i$是服从指数分布$P_i(\tau_i)=\mu e^{-\tau_i \mu}$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$\tau_a$,即事件发生间隔(the interevent time)也是服从指数分布$P_a(\tau_a)=\mu e^{-\tau_a \mu}$的随机变量。

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][4][5].

这种泊松过程的假设实际上使得数学分析变得更易处理。由于传播和恢复的速率是恒定的,它们不依赖于个体的先前历史信息或经历,因此这可以称为是无记忆的马尔科夫过程(Ross,1996; Tijms,2003; van Kampen,1981; Van Mieghem,2014b)。


While the Poisson approximation may be justified when only the average rates are known [6], it is at odds with empirical evidence for the time duration of the infective period in most diseases [7], 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.


The framework of non-Poissonian infection and recovery processes can be set up as follows, for either the SIS or SIR model [8]: Infected individuals remain infective for a period of time $\tau_i$, after which they recover, that follows the (nonexponential)$P_i(\tau_i)$ 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 $P_a(\tau_a)$, i.e., a susceptible individual connected to an infected node becomes infected at a time $\tau_a$, 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.


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 $P_i(\tau_i)$ and $P_a(\tau_a)$. These approaches are quite demanding, so only small system sizes can be considered. For example, Van Mieghem and van de Bovenkamp (2013)[9] reported results for the SIS model with Poissonian recovery, with rate $\mu$, while infection happens with a nonexponential distribution following the Weibull form $P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}$. In this case, strong variations in the value of the prevalence and of the epidemic threshold are found when varying the parameter $\alpha$. A promising approach is provided by the general simulation framework proposed by $\mathrm{Bogu\tilde{n}\acute{a}}$ et al. (2014)[10], based on the extension of the Gillespie algorithm for Poissonian processes [11]. 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$P_i(\tau_i)$和$P_a(\tau_a)$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$P_a(\tau_a)~(x/b)^{\alpha-1}e^{-(x/b)^\alpha}$时,SIS模型传播的结果。在这种传播过程情况下,当改变参数$\alpha$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,$\mathrm{Bogu\tilde{n}\acute{a}}$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 [12]. 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 [13], can be tackled analytically by postulating an extended epidemic model with different infective phases and Poissonian transitions among them [14]. However, general non Poissonian forms lead to convoluted sets of integrodifferential equations [15]. 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)[16] which consider the SIR process on a network in which infection events follow an interevent distribution P aðτaÞ. Assuming that infected nodes remain in that state for a fixed amount of time τi, it is possible to compute [16] 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$P_a(\tau_a)$的SIR传播过程。假设感染态节点固定的时间$\tau_i$内状态不发生改变,则可以计算疾病传播率为(Min等人,2013年) \begin{equation} T(\tau_i)=1-\int^\infty_{\tau_i}\Psi(\Delta)d\Delta. \end{equation} 其中$\Psi(\Delta)=\int_\Delta^\infty P_a(\tau_a)d\tau_a/\int_0^\infty P_a(\tau_a)d\tau_a$。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$\tau_i$的函数表示为: \begin{equation} T(\tau_{ic})=\frac{\left<k\right>}{\left<k^2\right>-\left<k\right>}. \end{equation}

For a power-law distribution $P_a(\tau_a)~\tau_a^{-\alpha}$, it is found that $\tau_{ic}$ diverges as $\alpha\to2$, implying that only diseases without recovery are able to spread through the network [16]. 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)[17]. 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$P_a(\tau_a)~\tau_a^{-\alpha}$,发现$\tau_{ic}$随$\alpha\to2$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。

Finally, Cator, van de Bovenkamp, and Van Mieghem (2013)[18] 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 $\lambda=\beta/\mu$ is replaced by the average number of infection attempts during a recovery time. The theory by Cator, van de Bovenkamp, and Van Mieghem (2013)[18] 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$\lambda=\beta/\mu$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

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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.