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Epidemic outbreaks in the real world are often accompanied by rapid information diffusion, which will change individual behavior patterns and affect the spread of epidemics. The community phenomenon in human society will also have an important influence on the spread of epidemics. The above factors to construct a new bilayer network are considered in this work. The activity-driven model is used to generate time-varying online information contact layer network and offline physical contact layer network. The information diffusion of individual online contact layer is used to affect the epidemic spreading dynamics of offline physical contact layer, and the individual mobility factor is used to control the community structure characteristics. In order to obtain the spreading dynamic equation of the whole network and analyze the spreading threshold of the network effectively, the microscopic Markov chain (MMC) approach is improved and extended to time-varying networks. Experimental verification of Monte Carlo simulations shows that the proposed method is highly accurate in predicting epidemic outbreak thresholds. The results show that individual mobility has no effect on the epidemic outbreak threshold, but it will affect the final number of infections in each community. The greater the individual contact capability of the online contact layer, the smaller the individual contact capability of the offline contact layer that can effectively suppress the epidemic spread. The above findings can present an important reference for effectively preventing and controlling the epidemic transmission in the real world.
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Keywords:
- microscopic Markov chain approach /
- bilayer network /
- epidemic spreading /
- community structure
[1] 李翔, 李聪, 王建波 2020 复杂网络传播理论-流行的隐秩序(上卷) (北京: 高等教育出版社) 第10页
Li X, Li C, Wang J B 2020 Theory of Spreading on Complex Networks: Hidden Rules of Epidemics (Vol. 1) (Beijing: Higher Education Press) p10 (in Chinese)
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[12] Wang B, Gou M, Han Y 2021 Nonlinear Dyn. 105 3835
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[13] Zhang Q P, Zhong L, Gao S Y, Li X M 2018 IEEE Trans. Cybern. 48 3411
Google Scholar
[14] Salehi M, Sharma R, Marzolla M, Magnani M, Siyari P, Montesi D 2015 IEEE Trans. Netw. Sci. Eng. 2 65
Google Scholar
[15] Wang B H, Chen W S, Wang J C, Zhang B, Zhang Z Q, Qiu X G 2019 IEEE Trans. Cybern. 49 4308
Google Scholar
[16] 孙皓宸, 刘肖凡, 许小可, 吴晔 2020 69 240201
Google Scholar
Sun H C, Liu X F, Xu X K, Wu Y 2020 Acta Phys. Sin. 69 240201
Google Scholar
[17] Wang H, Zhang H F, Zhu P C, Ma C 2022 Chaos 32 083110
Google Scholar
[18] Wang W, Liu Q H, Cai S M, Tang M, Braunstein L A, Stanley H E 2016 Sci. Rep. 6 29259
Google Scholar
[19] Pan Y H, Yan Z J 2018 Physica A 491 45
Google Scholar
[20] Funk S, Gilad E, Watkins C, Jansen V A 2009 Proc. Natl. Acad. Sci. USA 106 6872
Google Scholar
[21] Guo Q T, Lei Y J, Jiang X, Ma Y F, Huo G Y, Zheng Z M 2016 Chaos 26 043110
[22] Yang B, Shang K K, Small M, Chao N P 2022 National Science Open 62 254491591
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[23] Fortunato S 2010 Phys. Rep. 486 75
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Chang Z C, Chen H C, Liu Y, Yu H T, Huang R Y 2015 Acta Phys. Sin. 64 218901
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Google Scholar
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图 1 具有社团结构的信息-流行病共演化模型
Figure 1. Information-epidemic co-evolutionary model with community structure.
图 2 状态为US, AI和AS的概率转移树
Figure 2. Transition probability trees for the states US, AI, and AS, respectively.
图 3 不同的信息扩散率下MC模拟和理论阈值的对比
Figure 3. Comparison of Monte Carlo simulation and theoretical thresholds for different information diffusion rates.
图 4 最终感染规模热力图 (a) MC模拟结果; (b) MMC方法结果
Figure 4. Final infection scale heat map: (a) MC simulation results; (b) MMC approach results.
图 5 (a)—(c) 个体接触能力在不同的
$\lambda $ 下对流行病爆发阈值的影响; (d)—(f)线上信息接触层接触能力在不同的$\lambda $ 下对流行病爆发阈值影响的波动率; (g)—(i)线下物理接触层接触能力在不同的$\lambda $ 下对流行病爆发阈值影响的波动率Figure 5. (a)–(c) Influence of individual contact ability on epidemic outbreak threshold under different conditions; (d)–(f) volatility of the impact on epidemic outbreak thresholds under different virtual network layer contact capabilities; (g)–(i) volatility of impact on epidemic outbreak thresholds for different physical contact layer contact capacities.
图 6 (a)自我保护率对流行病爆发阈值的影响; (b)自我保护率对稳态感染个体比例的影响
Figure 6. (a) Impact of self-protection rate on epidemic outbreak threshold; (b) effect of self-protection rate on the proportion of individuals with stable infection.
图 7 稳态感染个体比例
${\rho ^{\text{I}}}$ 作为流行病传染率$\beta $ 和参数h的函数 (a)社团1的MC结果; (b)社团2的MC结果Figure 7. Steady-state proportion of infected individuals as a function of the epidemic transmission rate and the parameter h: (a) Monte Carlo results for community 1; (b) Monte Carlo results for community 2.
表 1 符号的含义
Table 1. Description of all symbols
符号 描述 $N$ 网络中的个体数 $t$ 时间步 ${k_{\text{v}}}$ 线上信息接触层中的活跃个体在单位时间步长中产生的连边数 ${k_{\text{p}}}$ 线下物理接触层中的活跃个体在单位时间步长中产生的连边数 $a_{l, i}^{\text{V}}$ 线上信息接触层中个体i 在$l$社团的活跃性类别 $a_{l, i}^{\text{P}}$ 线下物理接触层中个体i 在$l$社团的活跃性类别 $ \left\langle {{k_{\text{v}}}} \right\rangle $ 线上信息接触层的平均度 $\left\langle {{k_{\text{p}}}} \right\rangle $ 线下物理接触层的平均度 $\lambda $ 信息扩散率 $\delta $ 信息遗忘率 $\beta $ 流行病传染率 $\mu $ 流行病恢复率 ${\beta ^{\text{U}}}$ 状态为U的个体的流行病传染率 ${\beta ^{\text{A}}}$ 状态为A的个体的流行病传染率 $\sigma $ 捕捉${\beta ^{\text{U}}}$和${\beta ^{\text{A}}}$之间差异的参数, 又称为自我保护率: ${\beta ^{\text{A}}} = \sigma {\beta ^{\text{U}}}$ $h$ 个体流动率 $P_{l, i}^X$ 个体i 在$l$社团处于状态X的概率 ${r_{l, i}}\left( t \right)$ 处于$l$社团的个体i 在时间$t$内没有被任何邻居告知的概率 $q_{l, i}^{\text{A}}\left( t \right)$ 处于$l$社团且状态为A的个体i 在时间$t$内没有被任何邻居感染的概率 $q_{l, i}^{\text{U}}\left( t \right)$ 处于$l$社团且状态为U的个体i在时间$t$内没有被任何邻居感染的概率 
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[1] 李翔, 李聪, 王建波 2020 复杂网络传播理论-流行的隐秩序(上卷) (北京: 高等教育出版社) 第10页
Li X, Li C, Wang J B 2020 Theory of Spreading on Complex Networks: Hidden Rules of Epidemics (Vol. 1) (Beijing: Higher Education Press) p10 (in Chinese)
[2] Garten R J, Davis C T, Russell C A, et al. 2009 Science 325 197
Google Scholar
[3] Machens A, Gesualdo F, Rizzo C, Tozzi A E, Barrat A, Cattuto C 2013 BMC Infect. Dis. 13 185
Google Scholar
[4] Park B J, Wannemuehler K A, Marston B J, Govender N, Pappas P G, Chiller T M 2009 AIDS 23 525
Google Scholar
[5] Schwarzkopf Y, Rákos A, Mukamel D 2010 Phys. Rev. E 82 036112
Google Scholar
[6] Anderson R M, Anderson B, May R M 1992 Infectious Diseases of Humans: Dynamics and Control (Oxford: Oxford University Press) p127
[7] Hethcote H W 2000 SIAM Rev. 42 599
Google Scholar
[8] Grabowski A, Kosiński R A 2004 Phys. Rev. E 70 031908
Google Scholar
[9] Yang H, Gu C G, Tang M, Cai S M, Lai Y C 2019 Appl. Math. Model. 75 806
Google Scholar
[10] Granell C, Gómez S, Arenas A 2013 Phys. Rev. Lett. 111 128701
Google Scholar
[11] Davis J T, Perra N, Zhang Q, Moreno Y, Vespignani A 2020 Nat. Phys. 16 590
Google Scholar
[12] Wang B, Gou M, Han Y 2021 Nonlinear Dyn. 105 3835
Google Scholar
[13] Zhang Q P, Zhong L, Gao S Y, Li X M 2018 IEEE Trans. Cybern. 48 3411
Google Scholar
[14] Salehi M, Sharma R, Marzolla M, Magnani M, Siyari P, Montesi D 2015 IEEE Trans. Netw. Sci. Eng. 2 65
Google Scholar
[15] Wang B H, Chen W S, Wang J C, Zhang B, Zhang Z Q, Qiu X G 2019 IEEE Trans. Cybern. 49 4308
Google Scholar
[16] 孙皓宸, 刘肖凡, 许小可, 吴晔 2020 69 240201
Google Scholar
Sun H C, Liu X F, Xu X K, Wu Y 2020 Acta Phys. Sin. 69 240201
Google Scholar
[17] Wang H, Zhang H F, Zhu P C, Ma C 2022 Chaos 32 083110
Google Scholar
[18] Wang W, Liu Q H, Cai S M, Tang M, Braunstein L A, Stanley H E 2016 Sci. Rep. 6 29259
Google Scholar
[19] Pan Y H, Yan Z J 2018 Physica A 491 45
Google Scholar
[20] Funk S, Gilad E, Watkins C, Jansen V A 2009 Proc. Natl. Acad. Sci. USA 106 6872
Google Scholar
[21] Guo Q T, Lei Y J, Jiang X, Ma Y F, Huo G Y, Zheng Z M 2016 Chaos 26 043110
[22] Yang B, Shang K K, Small M, Chao N P 2022 National Science Open 62 254491591
Google Scholar
[23] Fortunato S 2010 Phys. Rep. 486 75
Google Scholar
[24] 常振超, 陈鸿昶, 刘阳, 于洪涛, 黄瑞阳 2015 64 218901
Google Scholar
Chang Z C, Chen H C, Liu Y, Yu H T, Huang R Y 2015 Acta Phys. Sin. 64 218901
Google Scholar
[25] Huang H, Chen Y, Ma Y 2021 Appl. Math. Comput. 388 125536
[26] Digital 2022: Another Year of Bumper Growth < a href="https://wearesocial.com/cn/blog/2022/01/digital-2022-another-year-of-bumper-growth/">https://wearesocial.com/cn/blog/2022/01/digital-2022-another-year-of-bumper-growth/ [2022-11-17]
[27] Aleta A, Martín-Corral D, Pastore Y Piontti A, et al. 2020 Nat. Hum. Behav. 4 964
Google Scholar
[28] Metcalf C J E, Morris D H, Park S W 2020 Science 369 368
Google Scholar
[29] Perra N, Gonçalves B 2012 Sci. Rep. 2 469
Google Scholar
[30] Mossong J, Hens N, Jit M, et al. 2008 PLOS Med. 5 e74
Google Scholar
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