免疫策略（Immunization Strategy）

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The fact that epidemic processes in heavy-tailed networks have a vanishing threshold in the thermodynamic limit, or a very small one in large but finite networks, prompted the study of immunization strategies leveraging on the network structure in order to protect the population from the spread of a disease. Immunization strategies are defined by specific rules for the identification of the individuals that shall be made immune, taking into account (local or nonlocal) information on the network connectivity pattern. Immunized nodes are in practice removed from the network, together with all the links incident to them, and each strategy is assessed by the effects of immunizing a variable fraction g of nodes in the network. The application of immunization does not only protect directly immunized individuals, but can also lead, for a sufficiently large fraction g, to an increase of the epidemic threshold up to an effective value [math]\displaystyle{ \lambda_c(g)\gt \lambda_c(g=0) }[/math], precluding the global propagation of the disease. This effect is called herd immunity. The main objective in this context is to determine the new epidemic threshold as a function of the fraction of immunized individuals. Indeed, for a sufficiently large value of g, any strategy for selecting immunized nodes will lead to an increased threshold. We define the immunization threshold [math]\displaystyle{ g_c(\lambda) }[/math], for a fixed value of [math]\displaystyle{ \lambda }[/math] such that, for values of [math]\displaystyle{ g\gt g_c(\lambda) }[/math] the average prevalence is zero, while for [math]\displaystyle{ g\lt =g_c(\lambda) }[/math] the average prevalence is finite.


重尾网络(heavy-tailed network)的流行病过程，在热力学极限上，传播阀值(threshold)趋近于零，又或者，在大而有限的网络中，传播阈值(threshold)非常小。这一事实，促进了一种免疫策略的研究：结合网络结构来保护人们免受疾病传播的影响。免疫策略(Immunization strategy)是由特定规则定义的，它考虑了网络链接模式（局域或全局）的信息，用来识别应被免疫的个体（网络中的节点）。在实践中，免疫节点及其所有连边，都被从网络中移除。每一种策略，都会评估网络中的一定比例g的免疫节点（被移除的节点）的效果。这种免疫方法，不仅可以保护被直接免疫的个体，而且，对于足够大的比例值g，还可以增大流行病传播阈值(epidemic threshold)，大到一个有效值[math]\displaystyle{ \lambda_c(g)\gt \lambda_c(g=0) }[/math]，能防止全局范围的疾病传播。这种效果，称为群体免疫。这种情况下，主要目标是确定新的传播阈值与免疫个体比例g之间的函数关系。实际上，对于足够大的g值，选择移除任何免疫节点的策略，都会使传播阈值增大。我们定义免疫的传播阈值[math]\displaystyle{ g_c(\lambda) }[/math]，那么，对于固定的[math]\displaystyle{ \lambda }[/math]值，当[math]\displaystyle{ g\gt g_c(\lambda) }[/math]，平均传播率为零，当[math]\displaystyle{ g≤g_c(\lambda) }[/math]，平均传播率不为零。

随机免疫策略(Random Immunization Strategy)

The simplest immunization protocol, using essentially no information at all, is the random immunization, in which a number gN of nodes is randomly chosen and made immune. While random immunization in the SIS model (under the DBMF approximation) can depress the prevalence of the infection, it does so too slowly to increase the epidemic threshold substantially. Indeed, from Eq. (20), an epidemics in a randomly immunized network is equivalent to a standard SIS process in which the spreading rate is rescaled as [math]\displaystyle{ \lambda\to\lambda(1-g) }[/math], i.e., multiplied by the probability that a given node is not immunized, so that the immunization threshold becomes [math]\displaystyle{ \begin{equation} g_c(\lambda)=1-\frac{\left\lt k\right\gt }{\lambda\left\lt k^2\right\gt } \end{equation} }[/math]

（Pastor-Satorras和Vespignani，2002b）[1]


最简单的免疫方案是随机免疫，即不依赖任何信息，在网络中随机选择gN个节点，使其免疫。然而在SIS模型（基于度的平均场(DBMF)的近似）中，这种方案，可以降低传播率，但是效果太慢，无法大幅度提高传播阈值。【译注：①基于度的平均场(DBMF)也称为异质平均场(HMF)。②SIS模型，将特定人群的人口结构，划分为了易感染者（S）、感染者（I）。但感染者治愈成为康复者后，康复者会以一定概率，重新加入易感染者(S)。通过对三类人的动态变化的研究，来模拟流行病的发展规律。】实际上，从该等式，可以得出，在随机免疫网络中，疾病的传播，等效于将标准SIS模型的传播速率重新调整[math]\displaystyle{ \lambda\to\lambda(1-g) }[/math]，也就是，乘以网络中的给定节点未被免疫的概率。因此，免疫阈值就变为 [math]\displaystyle{ \begin{equation} g_c(\lambda)=1-\frac{\left\lt k\right\gt }{\lambda\left\lt k^2\right\gt } \end{equation} }[/math] （Pastor-Satorras和Vespignani，2002b）[1]


For heterogeneous networks, for which [math]\displaystyle{ \left\lt k^2\right\gt }[/math] diverges and any value of [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ g_c(\lambda) }[/math]tends to 1 in the limit [math]\displaystyle{ N\to\infty }[/math], indicating that almost the whole network must be immunized to suppress the disease.


对于异质网络(heterogeneous network)，当[math]\displaystyle{ \left\lt k^2\right\gt }[/math]发散，对于任意[math]\displaystyle{ \lambda }[/math]，在[math]\displaystyle{ N\to\infty }[/math]时，[math]\displaystyle{ g_c(\lambda) }[/math]趋近于1。这表明，必须对整个网络进行免疫，才能抑制该疾病的传播。

目标免疫策略（Targeted Immunization Strategy）

This example shows that an effective level of protection in heavy-tailed networks must be achieved by means of optimized immunization strategies [2], taking into account the network heterogeneity. Large degree nodes (the hubs leading to the large degree distribution variance) are potentially the largest spreaders. Intuitively, an optimized strategy should be targeting those hubs rather than small degree vertices. Inspired by this observation, the targeted immunization protocol proposed by Pastor-Satorras and Vespignani (2002b)[1] considers the immunization of the gN nodes with largest degree. A simple DBMF analysis leads to an immunization threshold given, for the SIS model, by the implicit equation [1]


这个例子表明，在重尾网络(heavy-tailed network)中，考虑到网络异质性，必须采用优化的免疫策略，才能实现有效的保护水平（Anderson和May，1992）[2]。大度节点(连线多的节点，导致大影响的分布差异的枢纽节点)，可能是影响最大的传播者。因此，直观地，优化策略应针对这种大度节点，而不是小度节点（连线少的节点）。受此启发，Pastor-Satorras和Vespignani（2002b）提出的目标免疫方案，考虑对网络中的gN数量的大度节点，进行免疫[1]。对于SIS模型，一个简单的基于度的平均场（DBMF）的分析，得到对应的免疫阈值，由如下的隐方程给出：

[math]\displaystyle{ \begin{equation} \frac{\left\lt k^2\right\gt _{g_c}}{\left\lt k\right\gt _{g_c}}=\frac{1}{\lambda} \end{equation} }[/math]


where [math]\displaystyle{ \left\lt k^n\right\gt _{g} }[/math]is the [math]\displaystyle{ n }[/math]the moment of the degree distribution [math]\displaystyle{ P_g(k) }[/math] of the network resulting after the deletion of the gN nodes of highest degree, which takes the form [3]


其中[math]\displaystyle{ \left\lt k^n\right\gt _{g} }[/math]是网络的度分布(degree distribution)【译注：在图论及网络中，度(degree)指入射到一个节点的连线（边）的多少。一个节点的度值，就是入射到该节点的连线数量】[math]\displaystyle{ P_g(k) }[/math]的第n阶矩，它是在删除度值最高的gN个节点后得到的。[math]\displaystyle{ P_g(k) }[/math]的形式为[3]：

[math]\displaystyle{ \begin{equation} P_g(k)=\sum_{k'\ge k}^{k_c}P(k')\dbinom{k'}{k}(1-g)^kg^{k'-k} \end{equation} }[/math]


Equation (64) can be readily solved in the case of scale-free networks. For a degree exponent [math]\displaystyle{ \gamma=3 }[/math], the immunization threshold reads [math]\displaystyle{ g_c(\lambda)\simeq exp[-2/(m\lambda)] }[/math], where [math]\displaystyle{ m }[/math] is the minimum degree in the network. This result highlights the convenience of targeted immunization, with an immunization threshold that is exponentially small over a large range of the spreading rate [math]\displaystyle{ \lambda }[/math]. A similar effect can be obtained with a proportional immunization strategy [1] [see also Dezsö and Barabási (2002)[4] for a similar approach involving the cure of infected individuals with a rate proportional to their degree], in which nodes of degree k are immunized with probability [math]\displaystyle{ g_k }[/math], which is some increasing function of k. In this case, the infection is eradicated when [math]\displaystyle{ g_k\ge1-1/(\lambda k) }[/math], leading to an immunization threshold [1]


在无标度网络(scale-free network)的情况下，该方程很容易求解。当一个无标度网络的度分布指数(degree exponent)[math]\displaystyle{ \gamma=3 }[/math]时，免疫阈值[math]\displaystyle{ g_c(\lambda)\simeq exp[-2/(m\lambda)] }[/math]，其中m是网络中最小的度值。该结果突显出了目标免疫(targeted immunization)的便利性，其免疫阈值(immunization threshold)在传播速率[math]\displaystyle{ \lambda }[/math]的较大范围内呈指数减小。此外，采用比例免疫(proportional immunization)的策略，也可以获得类似的效果（Pastor-Satorras和Vespignani，2002b）[1] 。[另见Dezsö和Barabási（2002年）[5]的相似方案，以感染者的度值为比例，来治愈感染者。]其中，度值为k的节点，以概率[math]\displaystyle{ g_k }[/math]被免疫，[math]\displaystyle{ g_k }[/math]是k的某种递增函数。在这种情况下，当[math]\displaystyle{ g_k\ge1-1/(\lambda k) }[/math]时，感染被完全消除，从而，传播阀值达到了免疫阈值(immunization threshold)（Pastor-Satorras and Vespignani，2002b）[1]。


[math]\displaystyle{ \begin{equation} g_c(\lambda)=\sum_{k\gt \lambda^{-1}}(1-\frac{1}{k\lambda})P(k) \end{equation} }[/math]


which takes the form [math]\displaystyle{ g_c(\lambda)\simeq(m\lambda)^2/3 }[/math] for scale-free networks with [math]\displaystyle{ \gamma=3 }[/math].


对于[math]\displaystyle{ \gamma=3 }[/math]的无标度网络来说，其形式为[math]\displaystyle{ g_c(\lambda)\simeq(m\lambda)^2/3 }[/math]。


Other approaches to immunization stress that not only the behavior close to the critical point should be taken into account, but also the entire prevalence curve (the so-called viral conductance) [6][7][8]. Additionally, strategies involving possible different interventions on different nodes have been analyzed within a game-theoretic formalism [9][10][11].


其他免疫方法强调，不仅要考虑接近临界点的行为，还要考虑整个流行曲线(所谓的病毒传导)（Kooij R E.等，2009；Youssef，Kooij和Scoglio，2011; Van Mieghem，2012b）[12][13][14]。此外，对不同节点采取不同干预措施的策略，也有研究者通过博弈论体系(game-theoretic formalism)进行分析（Van Mieghem和Omic，2008； Omic，Orda和Van Mieghem，2009； Gourdin，Omic和Mieghem，2011）[15][16][17]。


The previously discussed immunization protocols are based on a global knowledge of the network properties (the whole degree sequence must be known to selectively target the nodes to be immunized). Actually, the more a global knowledge of the network is available, the more effective is the immunization strategy. For instance, one of the most effective targeted immunization strategies is based on the betweenness centrality (see Sec. III.B.5), which combines the bias toward high degree nodes and the inhibition of the most probable paths for infection transmission [18]. This approach can even be improved by taking into account the order in which nodes are immunized in a sequential scheme in which the betweenness centrality is recomputed after the removal of every single node, and swapping the order of immunization in different immunization sequences, seeking to minimize a properly defined size for the connected component of susceptible individuals. This approach has been proven to be highly efficient in the case of the SIR model [19]. Improved immunization performance in the SIR model has been found with an “equal graph partitioning” strategy [20] which seeks to fragment the network into connected components of approximately the same size, a task that can be achieved by a much smaller number of immunized nodes, compared with a targeted immunization scheme.


前面讨论的免疫策略是基于网络特征的全局信息（必须知道全部节点的度值的大小顺序，从而选择性地选中要免疫的目标节点）。实际上，对一个网络的全局信息掌握得越多，目标免疫策略就越有效。例如，最有效的目标免疫策略之一是基于介数中心性（参见第二部分第III.B.5），它结合考虑了对大度节点的偏好和对最可能的传播路径的抑制（Holme等 等人，2002）[21]。该方法甚至可以考虑方案的实施顺序而改进：首先，节点是按顺序免疫的，删除每个节点后，就要重新计算剩下节点的介数中心性。其次，在不同的免疫顺序中，可交换免疫次序，以尽量减少子网connected component的易感个体的规模。在SIR模型中，这种方法已被证明是高效的（Schneider等，2011）[22]。研究已经发现，通过“等图分割”策略，可提高SIR模型中的免疫性能（Chen等，2008）[23] ，它将整个网络分割成大小大致相等的子网connected component。与目标免疫方案相比，它免疫少得多的节点，也能达到相同的效果。【译注：子网(connected component)指的是，整个网络被免疫节点分割为几个更少节点的“部分”，每个“部分”的内部，节点之间仍相互连接，但一个“部分”的节点与其它“部分”的节点之间，没有连接。一个“部分”就是一个子网】

局部免疫策略（local immunization strategies）

The information that makes targeted strategies very effective also makes them hardly feasible in real-world situations, where the network structure is only partially known. In order to overcome this drawback, several local immunization strategies have been considered. A most ingenious one is the acquaintance strategy proposed by Cohen, Havlin, and ben-Avraham (2003)[24], and applied to the SIR model. In this protocol, a number gN of individuals is chosen at random and each one is asked to point to one of his or her nearest neighbors. Those nearest neighbors, instead of the nodes, are selected for immunization. Given that a randomly chosen edge points with high probability to a large degree node, this protocol realizes in practice a preferential immunization of the hubs that results in being effective in hampering epidemics. An analogous result can be obtained by means of a random walk immunization strategy [25][26], in which a random walker diffuses in the network and immunizes every node that it visits, until a given degree of immunization is reached. Given that a random walk visits a node of degree ki with probability proportional to ki [27], this protocol leads to the same effectiveness as the acquaintance immunization.


现实中网络结构仅部分已知，使得对目标策略非常有效的全局信息，在现实世界中几乎行不通。为了克服该缺陷，研究者考虑了几种局部免疫策略。其中，最巧妙的一个方法是科恩等人（2003）[28]提出的熟人免疫策略(acquaintance strategy)，它已应用于SIR模型。该方案，在网络种随机选择[math]\displaystyle{ gN }[/math]个个体（节点），并要求每个个体都指出他或她的一个最近的邻居。然后，对那些最近的邻居，而不是那[math]\displaystyle{ gN }[/math]个个体（节点），进行免疫。考虑到随机选定的连边，有很大的概率会指向大度节点，该策略在实际操作中，实现了对大度节点的优先免疫，从而可以有效地抑制流行病的传播。类似的结果，也可以通过一个随机游走免疫策略(random walk immunization strategy)实现（Holme，2004； Ke和Yi，2006）[29][30]，其中随机游走者在网络中扩散并对其访问的每个节点进行免疫，直到达到给定的免疫比例。一个节点的连边数量为[math]\displaystyle{ k_i }[/math]，随机游走访问到它的概率，与[math]\displaystyle{ k_i }[/math]成正比（Noh and Rieger，2004）[31]，该方案会产生与熟人免疫策略相同的效果。

The acquaintance immunization protocol can be improved by allowing for the consideration of additional information, always at the local level. For example, allowing for each node to have knowledge of the number of connections of its nearest neighbors, a large efficiency is attained by immunizing the neighboring nodes with the largest degree [32]. As more information is available, one can consider the immunization of the nodes with highest degree found within short paths of length l starting from a randomly selected node [33]. The random walk immunization strategy, on the other hand, can be improved by allowing a bias favoring the exploration of high degree nodes during the random walk process [34]. Variations of the acquaintance immunization scheme have also been used for weighted networks.The acquaintance immunization for weighted networks is outperformed by a strategy in which the immunized neighbors are selected among those with large edge weights [35].


考虑网络结构的更多信息，通常是局域信息，可以进一步改进熟人免疫策略。例如，允许每个节点提供它的各邻居节点的连边数量，这样可以对它相邻的大度节点进行免疫，获得最好的免疫效果（Holme，2004）[36]。当能够获得更多有用信息时，可以考虑，从随机选择的节点开始，在长度为[math]\displaystyle{ l }[/math]的路径范围之内，找到大度节点，并对它进行免疫（Gomez-Gardenes，Echenique和Moreno，2006）[37]。另一方面，随机游走免疫策略，在随机游走过程中，允许偏向于搜索大度节点，从而获得优化（Stauffer and Barbosa，2006）[38]。熟人免疫方案的变体，可用于加权网络(weighted network)。选择连边权重更大的邻居节点进行免疫，可以使熟人免疫策略在加权网络(weighted network)上表现出色（Deijfen，2011）[39]。

A different approach to immunization, the high-risk immunization strategy, applied by Nian and Wang (2010)[40] to the SIRS model, considers a dynamical formulation, in which nodes in contact with one or more infected individuals are immunized with a given probability. Again, by immunizing only a small fraction of the network, a notable reduction of prevalence and increase of the epidemic threshold can be achieved.


Nian F.和Wang X.（2010）[41] 将高风险免疫策略(high-risk immunization strategy)应用于SIRS模型，它一种不同的免疫方法。它考虑了一种动态方案，当一个节点接触了一个或多个感染者时，要对它进行特定概率的免疫。同样，它只需要对网络中的一小部分节点进行免疫接种，就可以显著降低流行率并提高传染病的传播阈值。【译注：SIRS模型，将特定人群的人口结构，划分为了易感染者（S）、感染者（I）、康复者(R)。康复者(R)也许不具备终身免疫力，会在一段时间后，或者会以一定的概率，重新加入易感染者(S)。通过对四类人的动态变化的研究，来模拟流行病的发展规律。】

Finally, for the SIR model, the mapping to percolation suggests which nodes to target in a vaccination campaign, depending on whether the probability of an outbreak or its size are to be minimized [42]. A targeted vaccination of nodes in the GSCC (giant strongly connected component) implies a reduction of both the probability of a major epidemics and its size.


最后，对于SIR模型，可以被转变为渗流问题【译注：即，把易感者(S)，一个个渐渐地转变为感染者(I)，再转变为康复者(R)，或通过免疫接种而移出该人群，就像水的渗流一样，渐渐“消除”所有的易感者(S)】，因此，针对哪些节点进行免疫接种，应由(流行病)可能爆发的概率或规模来决定（Kenah和Miller，2011）[43]。对超级链接子网(giant strongly connected component)进行目标免疫，就意味着，能使流行病爆发的概率和规模都减小。

参考文献 References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Pastor-Satorras R, Vespignani A. Immunization of complex networks[J]. Physical review E, 2002, 65(3): 036104.
  2. ↑ 2.0 2.1 Anderson R M, Anderson B, May R M. Infectious diseases of humans: dynamics and control[M]. Oxford university press, 1992.
  3. ↑ 3.0 3.1 Cohen R, Erez K, Ben-Avraham D, et al. Breakdown of the internet under intentional attack[J]. Physical review letters, 2001, 86(16): 3682.
  4. ↑ Dezső Z, Barabási A L. Halting viruses in scale-free networks[J]. Physical Review E, 2002, 65(5): 055103.
  5. ↑ Dezső Z, Barabási A L. Halting viruses in scale-free networks[J]. Physical Review E, 2002, 65(5): 055103.
  6. ↑ Kooij R E, Schumm P, Scoglio C, et al. A new metric for robustness with respect to virus spread[C]//International Conference on Research in Networking. Springer, Berlin, Heidelberg, 2009: 562-572.
  7. ↑ Youssef M, Kooij R, Scoglio C. Viral conductance: Quantifying the robustness of networks with respect to spread of epidemics[J]. Journal of Computational Science, 2011, 2(3): 286-298.
  8. ↑ Van Mieghem P. The viral conductance of a network[J]. Computer Communications, 2012, 35(12): 1494-1506.
  9. ↑ Van Mieghem, P., and J. Omic, 2008, arXiv:1306.2588.
  10. ↑ Omic J, Orda A, Van Mieghem P. Protecting against network infections: A game theoretic perspective[C]//IEEE INFOCOM 2009. IEEE, 2009: 1485-1493.
  11. ↑ Moradi A, Shah-Mansouri V. Opportunistic content dissemination in mobile social networks via adjustment of user selfishness[J]. IET Networks, 2018, 8(2): 126-137.
  12. ↑ Kooij R E, Schumm P, Scoglio C, et al. A new metric for robustness with respect to virus spread[C]//International Conference on Research in Networking. Springer, Berlin, Heidelberg, 2009: 562-572.
  13. ↑ Youssef M, Kooij R, Scoglio C. Viral conductance: Quantifying the robustness of networks with respect to spread of epidemics[J]. Journal of Computational Science, 2011, 2(3): 286-298.
  14. ↑ Van Mieghem P. The viral conductance of a network[J]. Computer Communications, 2012, 35(12): 1494-1506.
  15. ↑ Van Mieghem, P., and J. Omic, 2008, arXiv:1306.2588.
  16. ↑ Omic J, Orda A, Van Mieghem P. Protecting against network infections: A game theoretic perspective[C]//IEEE INFOCOM 2009. IEEE, 2009: 1485-1493.
  17. ↑ Moradi A, Shah-Mansouri V. Opportunistic content dissemination in mobile social networks via adjustment of user selfishness[J]. IET Networks, 2018, 8(2): 126-137.
  18. ↑ Holme, P., B. J. Kim, C. N. Yoon, and S. K. Han, 2002, Phys. Rev. E 65, 056109.
  19. ↑ Schneider, C. M., T. Mihaljev, S. Havlin, and H. J. Herrmann, 2011, Phys. Rev. E 84, 061911.
  20. ↑ Chen, Y., G. Paul, S. Havlin, F. Liljeros, and H. E. Stanley, 2008, Phys. Rev. Lett. 101, 058701.
  21. ↑ Holme, P., B. J. Kim, C. N. Yoon, and S. K. Han, 2002, Phys. Rev. E 65, 056109.
  22. ↑ Schneider, C. M., T. Mihaljev, S. Havlin, and H. J. Herrmann, 2011, Phys. Rev. E 84, 061911.
  23. ↑ Chen, Y., G. Paul, S. Havlin, F. Liljeros, and H. E. Stanley, 2008, Phys. Rev. Lett. 101, 058701.
  24. ↑ Cohen R, Havlin S, Ben-Avraham D. Efficient immunization strategies for computer networks and populations[J]. Physical review letters, 2003, 91(24): 247901.
  25. ↑ Holme P. Efficient local strategies for vaccination and network attack[J]. EPL (Europhysics Letters), 2004, 68(6): 908.
  26. ↑ Ke H, Yi T. Immunization for scale-free networks by random walker[J]. Chinese Physics, 2006, 15(12): 2782.
  27. ↑ Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.
  28. ↑ Cohen R, Havlin S, Ben-Avraham D. Efficient immunization strategies for computer networks and populations[J]. Physical review letters, 2003, 91(24): 247901.
  29. ↑ Holme P. Efficient local strategies for vaccination and network attack[J]. EPL (Europhysics Letters), 2004, 68(6): 908.
  30. ↑ Ke H, Yi T. Immunization for scale-free networks by random walker[J]. Chinese Physics, 2006, 15(12): 2782.
  31. ↑ Noh J D, Rieger H. Random walks on complex networks[J]. Physical review letters, 2004, 92(11): 118701.
  32. ↑ Holme P. Efficient local strategies for vaccination and network attack[J]. EPL (Europhysics Letters), 2004, 68(6): 908.
  33. ↑ Gómez-Gardenes J, Echenique P, Moreno Y. Immunization of real complex communication networks[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2006, 49(2): 259-264.
  34. ↑ Stauffer A O, Barbosa V C. Dissemination strategy for immunizing scale-free networks[J]. Physical Review E, 2006, 74(5): 056105.
  35. ↑ Deijfen M. Epidemics and vaccination on weighted graphs[J]. Mathematical biosciences, 2011, 232(1): 57-65.
  36. ↑ Holme P. Efficient local strategies for vaccination and network attack[J]. EPL (Europhysics Letters), 2004, 68(6): 908.
  37. ↑ Gómez-Gardenes J, Echenique P, Moreno Y. Immunization of real complex communication networks[J]. The European Physical Journal B-Condensed Matter and Complex Systems, 2006, 49(2): 259-264.
  38. ↑ Stauffer A O, Barbosa V C. Dissemination strategy for immunizing scale-free networks[J]. Physical Review E, 2006, 74(5): 056105.
  39. ↑ Deijfen M. Epidemics and vaccination on weighted graphs[J]. Mathematical biosciences, 2011, 232(1): 57-65.
  40. ↑ Nian F, Wang X. Efficient immunization strategies on complex networks[J]. Journal of theoretical biology, 2010, 264(1): 77-83.
  41. ↑ Nian F, Wang X. Efficient immunization strategies on complex networks[J]. Journal of theoretical biology, 2010, 264(1): 77-83.
  42. ↑ Kenah E, Miller J C. Interdiscip. Perspect[J]. Infect. Dis, 2011, 543520.
  43. ↑ Kenah E, Miller J C. Interdiscip. Perspect[J]. Infect. Dis, 2011, 543520.




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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