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近年来,随着对复杂网络科学认识的深入,科学家们发现很多的真实系统都可以用多层网络结构进行更精确的描述。例如,人际关系网中的线下朋友关系以及线上朋友关系等;不同运输工具(航空网、铁路网和公路网)构成的交通网络等。因此关于多层网络的结构及动力学的研究已经成为网络科学领域的一个研究热点和重点[1-11]。
传染性疾病的爆发会引发相关的信息通过人群内部、新闻媒介以及在线社交网络等多种渠道传播,而人们获得该信息后会警觉且采取一定的预防措施自我保护,从而对疾病传播产生深远的影响。已有学者在研究传染病动力学的时候开始从不同角度考虑信息因素的影响,如:个体根据对疾病信息的了解程度来调整与外界的接触情况[12],个体行为方式的改变[13],人群内部、新闻媒介以及在线社交网络传播信息对疾病传播的影响[14];文献[15]基于传统的“S(susceptible)-I(infected)-S(susceptible)”疾病传播模型,考虑了节点会通过邻居的感染状况而出现警惕状态,而警觉态易感者被感染的概率显然和易感态不同,提出了S(susceptible)-A(aware)-I(infected)-S(susceptible)”模型,但其模型的研究局限于单个网络,没有考虑警惕信息的传播问题;文献[16]又研究了信息传播对疾病传播的影响,并找出在不同网络拓扑下最理想的信息传播机制;文献[17]在双层网络中研究疾病和信息两类传播动力学的相互作用,揭示信息传播对疾病传播及传播阈值的影响;文献[18]进一步研究了一种局部警觉控制传染模型(LACS),其中当节点的警惕邻居数目与信息层节点度的比值达到局部警惕率或已被疾病感染,才能变成警惕态传播警惕信息;文献[19]研究了双层网络上具有自激发和扩散机制的警觉行为对传播动力学的影响;文献[20]考虑了双层网络上个体的异质性对警惕信息传播动力学的影响等。
之前基于双层网络对疾病传播与警觉意识扩散的研究多存在一个普遍性假设:在同一时刻警觉意识先传播,疾病后传播。虽然信息传播途径广一些,传播也更加便利一些,但是并非所有情况都如此。比如有些疾病虽然已经在人群中传播,但由于疾病的外部表现不容易被发现或者危险性没有引起充分注意,可能导致警觉不能快速传播。再比如,对于有些疾病而言,感染疾病的人不太愿意告诉他人自己的得病情况(比如性病等),因此警觉也不能快速传播。故与之前的研究都不同,本文摒弃之前的假设,而是认为警觉意识与疾病的传播是不分先后次序的,进而比较两种机制对传播阈值和传播范围的影响。
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根据UAU-SIS模型,节点处于3种不同状态:无警觉且易感态(US),警觉且易感态(AS),警觉且感染态(AI);未警觉的感染态,即UI态是不存在的,因为假设节点一旦感染就立即变为警觉态,所以UI对应于AI态。在时刻t,每个节点
$i$ 都以一个确定的概率处于这3种状态之一,分别表示为$p_i^{\rm {US}}({\rm{t}})$ 、$p_i^{\rm {AS}}({\rm{t}})$ 和$p_i^{\rm {AI}}(t)$ 。假设没有动力的相关性[16],同时记处于U态的节点$i$ 未被警觉的概率为${\gamma _i}(t)$ ,处于U态的节点$i$ 未被感染的概率为$q_i^{\rm U}(t)$ ,处于A态的节点$i$ 未被感染的概率为$q_i^{\rm A}(t)$ ,则:$${\gamma _i}(t) = \prod\limits_j {(1 - {a_{ji}}p_j^{\rm A}(t)\lambda )}$$ (1) $$q_i^{\rm U}(t) = \prod\limits_j {(1 - {b_{ji}}p_j^{\rm I}(t){\beta ^{\rm U}})}$$ (2) $$q_i^{\rm A}(t) = \prod\limits_j {(1 - {b_{ji}}p_j^{\rm I}(t){\beta ^{\rm A}})}$$ (3) 下面将分析双层网络中信息与疾病的传播次序性对传播动力学的影响。
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文献[17]在进行理论分析时,假设警觉意识的传播优先于疾病的传播(简称为ordered model),即在同一时刻警觉意识先传播,疾病后传播,从而有每个节点可能处的状态以及概率转化图如图2所示,其中
${\gamma _i}(t)$ 为处于U态的节点$i$ 未警觉的概率,$q_i^{\rm U}(t)$ 为处于U态的节点$i$ 未被感染的概率,$q_i^{\rm A}(t)$ 为处于A态的节点$i$ 未被感染的概率,$\delta $ 为警觉信息恢复率,$\mu $ 为感染恢复率。列出如下马尔可夫方程:$$ \begin{split} & p_i^{\rm {US}}(t + 1) = p_i^{\rm {US}}(t){\gamma _i}(t)q_i^{\rm U}(t) + p_i^{\rm {AS}}(t)\delta q_i^{\rm U}(t) + p_i^{\rm {AI}}(t)\delta \mu \\ &\qquad\qquad p_i^{\rm {AS}}(t + 1) = p_i^{\rm {US}}(t)\left[ {1 - {\gamma _i}(t)} \right]q_i^{\rm A}(t) +\\ &\qquad\qquad p_i^{\rm {AS}}(t)(1 - \delta )q_i^{\rm A}(t) + p_i^{\rm {AI}}(t)(1 - \delta )\mu \end{split} $$ $$\begin{split} &\;\; p_i^{\rm {AI}}(t + 1) = p_i^{\rm {US}}(t)\left\{ {[ {1 - {\gamma _i}(t)} ][ {1 - q_i^{\rm A}(t)} ] +}\right. \\ &\qquad\;\; \left.{{\gamma _i}(t)[ {1 - q_i^{\rm U}(t)} ]} \right\} + p_i^{\rm {AS}}(t)\\ & {\left\{ {\delta [ {1 - q_i^{\rm U}(t)} ] + (1 - \delta )[ {1 - q_i^{\rm A}(t)} ]} \right\} + p_i^{\rm {AI}}(t)(1 - \mu )} \end{split} $$ 通过理论分析,得出传播阈值
$\,\beta _c^U = \dfrac{\mu }{{{ \wedge _{\max }}({ H})}}$ ,其中矩阵H的元素为${h_{ji}} = \left( {1 - (1 - \gamma )p_i^{\rm A}} \right){b_{ji}}$ 。故传播阈值受到双层网络结构、警觉意识的传播率等因素的影响。 -
文献[17]的研究以及基于此工作的推广都存在一个假设:在同一时刻警觉意识先传播,疾病后传播。虽然信息传播途径更便利一些,但在有些情况下,这种人为规定警觉传播的优先性是欠妥的,因为只有疾病发生了,警觉意识才会传播,另外,当人们没有充分意识到疾病的风险时,警觉意识的传播也未必快于疾病传播。故与之前的研究都不同,本文摒弃之前的假设,而是认为警觉意识与疾病的传播是不分先后次序的,提出一种不考虑传播次序的模型(简称为concurrent model)。
对于concurrent model模型,图3刻画了每个节点可能处的状态以及概率转化图,其中
${\gamma _i}(t)$ 为处于U态的节点$i$ 未警觉的概率,$q_i^{\rm U}(t)$ 为处于U态的节点$i$ 未被感染的概率,$q_i^{\rm A}(t)$ 为处于A态的节点$i$ 未被感染的概率,$\delta $ 为警觉信息恢复率,$\mu $ 为感染恢复率。根据图3的概率转移图并结合马尔可夫链方法[17-19],可以列出如下马尔可夫方程:
$$p_i^{\rm {US}}(t + 1) = p_i^{\rm {US}}(t){\gamma _i}(t)q_i^{\rm U}(t) + + p_i^{\rm {AS}}(t)\delta q_i^{\rm A}(t) + p_i^{\rm {AI}}(t)\delta \mu $$ (4) $$ \begin{split} & p_i^{\rm {AS}}(t + 1) = p_i^{\rm {US}}(t)\left[ {1 - {\gamma _i}(t)} \right]q_i^{\rm U}(t) +\\ & p_i^{\rm {AS}}(t)(1 - \delta )q_i^{\rm A}(t) + p_i^{\rm {AI}}(t)(1 - \delta )\mu \end{split} $$ (5) $$ \begin{split} & p_i^{\rm {AI}}(t + 1) = p_i^{\rm {US}}(t)\left[ {1 - q_i^{\rm U}(t)} \right] + + p_i^{\rm {AS}}(t)\times\\ &\qquad\; \left[ {1 - q_i^{\rm A}(t)} \right] + p_i^{\rm {AI}}(t)(1 - \mu ) \end{split} $$ (6) 这里
$p_i^{\rm {US}}(t) + p_i^{\rm {AS}}(t) + p_i^{\rm {AI}}(t) \equiv 1$ 。$p_j^{\rm A} = p_j^{\rm {AS}} + p_j^{\rm {AI}},$ $p_j^{\rm I} = p_j^{\rm {AI}}$ 。当系统稳定时,$p_i^{\rm {US}}(t + 1) = p_i^{\rm {US}}(t) = p_i^{\rm {US}}$ ,$p_i^{\rm {AS}}(t + 1) = p_i^{\rm {AS}}(t){\rm{ = }}p_i^{\rm {AS}}$ ,$p_i^{\rm {AI}}(t + 1) = p_i^{\rm {AI}}(t){\rm{ = }}p_i^{\rm {AI}}$ 。传播阈值决定疾病的爆发或灭亡,因此分析各参数对阈值的影响十分重要。在阈值附近,节点被感染的概率很低,使得
$p_i^{\rm I} = {\varepsilon _i} \ll {\rm{1}}$ ,因此有$q_i^{\rm A} \approx 1 - {\beta ^{\rm A}}$ $\displaystyle\sum {\left( {{b_{ji}}{\varepsilon _j}} \right)} $ 和$q_i^{\rm U} \approx 1 - {\beta ^U}\displaystyle\sum {\left( {{{\rm{b}}_{ji}}{\varepsilon _j}} \right)} $ 。假定$p_i^{\rm I} = {\varepsilon _i} \to {\rm{0}}$ ,进一步近似$q_i^{\rm A} \approx 1$ 和$q_i^{\rm U} \approx 1$ ,代入并整理式(4)和式(5)得到:$$ p_i^{\rm {US}} = p_i^{\rm {US}}{\gamma _i} + p_i^{\rm {AS}}\delta $$ (7) $$ p_i^{\rm {AS}} = p_i^{\rm {US}}(1 - {\gamma _i}) + p_i^{\rm {AS}}(1 - \delta ) $$ (8) 联合式(6)~式(8),则得一个更简单的式子:
$$\mu {\varepsilon _i} = (p_i^{\rm {US}}{\beta ^{\rm U}} + p_i^{\rm {AS}}{\beta ^{\rm A}})\sum {{b_{ji}}{\varepsilon _j}} $$ (9) 又因为
$\,{\beta ^{\rm A}} = \gamma {\beta ^{\rm U}}$ ,$p_i^{\rm U} = p_i^{\rm {US}} $ ,$p_i^{\rm A} = p_i^{\rm {AS}} \!+\! p_i^{\rm {AI}} \!\approx p_i^{\rm {AS}} $ ,$p_i^{\rm I} = {\varepsilon _i} \ll {\rm{1}}$ ,则式(9)整理成:$$\sum\limits_{} {\left[ {\left( {1 - (1 - \gamma )p_i^{\rm A}} \right){b_{ji}} - \frac{\mu }{{{\beta ^U}}}{\delta _{ji}}} \right]} {\varepsilon _j} = 0$$ (10) 式中,当
$i = j$ 时,${\delta _{ij}} = 1$ ;否则为0。定义矩阵H,其元素为
${h_{ji}} = \left( {1 - (1 - \gamma )p_i^{\rm A}} \right){b_{ji}}$ 。式(10)即为:$$H\varepsilon = \frac{\mu }{{{\beta ^{\rm U}}}}\varepsilon $$ (11) 式中,
$\varepsilon = {({\varepsilon _1},{\varepsilon _2},\cdots,{\varepsilon _N})^{\rm T}}$ 。式(11)的非平凡解就是矩阵H对应于特征值为$\dfrac{\mu }{{{\beta ^{\rm U}}}}$ 的特征向量,记${ \wedge _{\max }}({ H})$ 为矩阵H的最大特征值,则感染阈值就可以写成:$$\beta _c^{\rm U} = \frac{\mu }{{{ \wedge _{\max }}({ H})}}$$ (12) 从式(10)~式(12)可以得出,传播阈值依赖于实际接触层的网络结构
${({{{b}}_{ij}})_{N \times N}}$ 、参数$\gamma $ 以及警觉密度${\rho ^{\rm A}}$ 。其中${\rho ^{\rm A}}$ 是由信息层网络结构、传播率$\lambda $ 以及恢复率$\delta $ 等进一步确定的。故传播阈值受到双层网络结构、警觉意识的传播率等因素的影响。同时研究结果表明,两种模型得到的传播阈值是相同的。
Effects of the Order of Awareness Diffusion and Disease Propagation on the Spreading Dynamics
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摘要: 近年来,基于双层网络研究疾病传播与警觉传播的相互作用已引起广泛关注。在该模型框架下,疾病通过物理接触网络传播,而有关疾病的警觉信息则通过虚拟接触网络传播,两个网络具有相同的节点,但对应的连边不同。已有的模型在进行理论分析时,多假设警觉意识的传播先于疾病的传播(ordered model),考虑到在真实情况下,疾病传播和警觉意识的传播难以区分先后顺序,因此该文提出了一种不考虑传播次序的模型(concurrent model)。通过研究发现,两种模型给出相同的疾病爆发阈值,但却给出不同的传播范围,当警觉意识传播率较小的时候,无序模型对应的感染范围会小于有序模型对应的感染范围,但是随着警觉意识传播率的增加,结果会发生反转,即无序性模型会导致疾病的感染范围大于有序性模型。Abstract: Recently, studies on the interaction between disease transmission and awareness transmission based on two-layer networks have attracted much attention. Within such a framework, infectious disease is propagated through physical contact networks, while the diffusion of awareness is transmitted through virtual contact network. Moreover, the nodes on two layers are the same, but the edges connecting nodes in the two layers are different. Most of the existing models assume that the diffusion of awareness precedes the spread of disease (hereinafter referred to as ordered model). In real cases, it is difficult to distinguish the order of disease transmission from that of awareness diffusion, thus, this paper proposes a model without considering the order of two spreading processions (concurrent model for short). We find that the two models yield the same epidemic threshold, but have different influence on the spreading sizes. When the transmission rate of awareness is low, the epidemic size of concurrent model is smaller than that of ordered model. However, with the increase of the transmission rate of awareness, the result is reversed, that is, the epidemic size of the concurrent model is larger than that of ordered model.
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Key words:
- awareness diffusion /
- epidemic threshold /
- infectious diseases /
- two-layer networks
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