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疾病的广泛传播给人类带来了巨大的损失, 因此抑制疾病的传播非常重要. 本文考虑了个体接种疫苗意愿的差异性, 并结合博弈理论建立了一个基于节点度信息的自愿免疫模型. 理论解析结果证明当感染率超过某个阈值时, 该模型与忽略个体接种意愿差异性的经典模型(Zhang et al 2010 New J. Phys. 12 023015) 传播效果(感染节点数)一样. 继而考虑疫苗永久有效和有效期有限两种情况, 在Barabási-Albert网络中利用SIS传播模型对疾病的传播进程进行了数值模拟, 发现数值模拟结果与理论解析结果非常符合. 实验证明, 当感染耗费和接种疫苗耗费相同时, 该模型比忽略个体接种意愿差异性的经典模型能够更好的抑制疾病的传播, 且感染人数下降比例超过65%, 更重要的是,疫苗有效期越长本文的模型 (与忽略个体接种意愿差异性的经典模型相比)抑制疾病传播效果越好.The widespread of epidemics bring tremendous losses to the mankind, thus it is very important to prevent the spread of epidemics. In this paper, the differences between individual tendency of vaccination is taken into account to propose a voluntary vaccination model based on the node degree information. Further, the theoretical analysis result shows that if propagation rate exceed a threshold value, the effectiveness of epidemic spreading (the number of infectious nodes) of the model above and the classical model ignoring the difference between the individual vaccination willingness [Zhang et al 2010 New J. Phys. 12 023015] will be the same. Both the permanent vaccination and the temporary vaccination are considered to analyze the process of epidemic spreading for the Barabási-Albert network by using the SIS model. The numerical simulation results are consistent with the empirical ones very well. Experiments prove that when the infection cost and vaccine cost is the same, the model can prevent the spread of the epidemic more effective as compared with the classical one, and the proportion of the infections decreases over 65% than the classical one. In addition, the longer the live of vaccine, the more effective the prevention of the spread of the epidemic using this model (compared with the classical model ignoring the difference between the individual vaccination willingness).
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Keywords:
- epidemic spreading /
- voluntary vaccination /
- vaccination inclination /
- node degree
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[2] Meyers L A, Pourbohloul B, Newman M E J, Skowronski D M, Brunham R C 2005 J. Theor. Biol. 232 71
[3] Li X, Wang X F 2006 IEEE Trans. Automat. Control 51 534
[4] Liu J G, Wu Z X, Wang F 2007 Int. J. Mod. Phys. C 18 1087
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[6] Ren Z M, Shao F, Liu J G, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 128901 (in Chinese) [任卓明, 邵凤, 刘建国, 郭强, 汪秉宏 2013 62 128901]
[7] Liu J G, Ren Z M, Guo Q 2013 Physica A 392 4154
[8] Hu Q C, Yin Y S, Ma P F, Zhang Y, Xing C X 2013 Acta Phys. Sin. 62 140101 (in Chinese) [庆成, 尹龑燊, 马鹏, 斐高旸, 张勇, 邢春晓 2013 62 140101]
[9] Ren Z M, Liu J G, Shao F, Hu Z L, Guo Q 2013 Acta Phys. Sin. 62 108902 (in Chinese) [任卓明, 刘建国, 邵凤, 胡兆龙, 郭强 2013 62 108902]
[10] Liu J G, Ren Z M, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 178901 (in Chinese) [刘建国, 任卓明, 郭强, 汪秉宏 2013 62 178901]
[11] Mller J, Schönfisch B, Kirkilionis M 2000 J. Math. Biol. 41 143
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[14] Salathé M, Jones J H 2010 PLoS Comput. Biol. 6(4) 1000736
[15] Jiang Z H, Wang H, Gao C 2011 Acta Phys. Sin. 60 058903 (in Chinese) [姜志宏, 王晖, 高超 2011 60 58903]
[16] Bauch C T 2005 Proc. R. Soc. B 272 1669
[17] Wang Y Q, J G P 2010 Acta Phys. Sin. 59 6734 (in Chinese) [王亚奇, 蒋国平 2010 59 6734]
[18] Perisic A, Bauch C T 2009 BMC Infect. Dis. 9 77
[19] Dushoff J, Plotkin J B, Levin S A, Earn D J D 2004 Proc. Natl Acad. Sci. USA 101 16915
[20] Fu F, Rosenbloom D I, Wang L, Nawak M A 2011 Proc. R. Soc. B 278 42
[21] Bauch C T, Galvani A P, Earn D J D 2003 Proc. Natl Acad. Sci. USA 100 10564
[22] Zhang H, Zhang J, Zhou C, Small M, Wang B 2010 New J. Phys. 12 023015
[23] Anderson R M, May R M, Anderson B 1992 Infectious Diseases of Humans: Dynamics and Control (Oxford : Oxford Science Publications) p66
[24] Zhou T, Liu J G, Bai W J, Chen G R, Wang B H 2006 Phys. Rev. E 74 056109
[25] Albert R, Jeong H, Barabási A L 2000 Nature 406 378
[26] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
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[1] Ma Z N, Zhou Y C, Wang W D 2004 The mathematical modeling and research on dynamics of infectious diseases (Beijing: science press) pp1–5 (in Chinese) [马知恩, 周义仓, 王稳地 2004 传染病动力学的数学建模与研究(北京: 科学出版社) 第1–5页]
[2] Meyers L A, Pourbohloul B, Newman M E J, Skowronski D M, Brunham R C 2005 J. Theor. Biol. 232 71
[3] Li X, Wang X F 2006 IEEE Trans. Automat. Control 51 534
[4] Liu J G, Wu Z X, Wang F 2007 Int. J. Mod. Phys. C 18 1087
[5] Yu H, Liu Z, Li Y J 2013 Acta Phys. Sin. 62 020204 (in Chinese) [于会, 刘尊, 李勇军 2013 62 020204]
[6] Ren Z M, Shao F, Liu J G, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 128901 (in Chinese) [任卓明, 邵凤, 刘建国, 郭强, 汪秉宏 2013 62 128901]
[7] Liu J G, Ren Z M, Guo Q 2013 Physica A 392 4154
[8] Hu Q C, Yin Y S, Ma P F, Zhang Y, Xing C X 2013 Acta Phys. Sin. 62 140101 (in Chinese) [庆成, 尹龑燊, 马鹏, 斐高旸, 张勇, 邢春晓 2013 62 140101]
[9] Ren Z M, Liu J G, Shao F, Hu Z L, Guo Q 2013 Acta Phys. Sin. 62 108902 (in Chinese) [任卓明, 刘建国, 邵凤, 胡兆龙, 郭强 2013 62 108902]
[10] Liu J G, Ren Z M, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 178901 (in Chinese) [刘建国, 任卓明, 郭强, 汪秉宏 2013 62 178901]
[11] Mller J, Schönfisch B, Kirkilionis M 2000 J. Math. Biol. 41 143
[12] Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104
[13] Cohen R, Havlin S, Ben-Avraham D 2003 Phys. Rev. Lett. 91 247901
[14] Salathé M, Jones J H 2010 PLoS Comput. Biol. 6(4) 1000736
[15] Jiang Z H, Wang H, Gao C 2011 Acta Phys. Sin. 60 058903 (in Chinese) [姜志宏, 王晖, 高超 2011 60 58903]
[16] Bauch C T 2005 Proc. R. Soc. B 272 1669
[17] Wang Y Q, J G P 2010 Acta Phys. Sin. 59 6734 (in Chinese) [王亚奇, 蒋国平 2010 59 6734]
[18] Perisic A, Bauch C T 2009 BMC Infect. Dis. 9 77
[19] Dushoff J, Plotkin J B, Levin S A, Earn D J D 2004 Proc. Natl Acad. Sci. USA 101 16915
[20] Fu F, Rosenbloom D I, Wang L, Nawak M A 2011 Proc. R. Soc. B 278 42
[21] Bauch C T, Galvani A P, Earn D J D 2003 Proc. Natl Acad. Sci. USA 100 10564
[22] Zhang H, Zhang J, Zhou C, Small M, Wang B 2010 New J. Phys. 12 023015
[23] Anderson R M, May R M, Anderson B 1992 Infectious Diseases of Humans: Dynamics and Control (Oxford : Oxford Science Publications) p66
[24] Zhou T, Liu J G, Bai W J, Chen G R, Wang B H 2006 Phys. Rev. E 74 056109
[25] Albert R, Jeong H, Barabási A L 2000 Nature 406 378
[26] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
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