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The imperfect immunization includes immune failure and immune invalidity on complex networks. In this paper, based on mean-field theory, we propose a new susceptible-infected-removed model to study epidemic spreading in networks with both immune failure and immune invalidity. Theoretical analysis shows that the immune failure and immune invalidity can significantly reduce the epidemic threshold and enhance the epidemic prevalence in the networks. According to the relationships between epidemic threshold, immune density of nodes, immune success rate and immune invalid rate, some suggestions on how to effectively control the propagation of epidemics are presented. Numerical simulations have confirmed the theoretical results.
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Keywords:
- complex network /
- epidemic spreading /
- imperfect immunization /
- epidemic threshold
[1] Li X, Wang X F 2006 IEEE Trans. Automatic Control 51 534
[2] Li X, Wang X F 2007 Int. J. Syst. Sci. 38 400
[3] Song Y R, Jiang G P 2009 Acta Phys. Sin. 58 5911 (in Chinese) [宋玉蓉、蒋国平 2009 58 5911]
[4] Yang R, Wang B H, Ren J, Bai W J, Shi Z W, Xu W, Zhou T 2007 Phys. Lett. A 386 189
[5] Pei W D, Chen Z Q, Yang Z Z 2008 Chin. Phys. B 17 373
[6] Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A 2005 J. Theo. Bio. 235 275
[7] Zhou T, Liu J G, Bai W J, Chen G R, Wang B H 2006 Phys. Rev. E 74 056109
[8] Hayashi Y, Minoura M, Matsukubo J 2004 Phys. Rev. E 69 016112
[9] Huang W, Jiang R, Hu M B, Wu Q S 2009 Chin. Phys. B 18 1306
[10] Zhang H F, Michael S, Fu X C 2009 Commun. Theor. Phys. 52 180
[11] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200
[12] Wang Y, Zheng Z G 2009 Acta Phys. Sin. 58 4421 (in Chinese) [王 延、郑志刚 2009 58 4421]
[13] Watts D J, Strogatz S H 1998 Nature 393 440
[14] Barabsi A L, Albert R 1999 Science 286 509
[15] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[16] Lloyd A L, May R M 2001 Science 292 1316
[17] Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104
[18] Guo W P, Li X, Wang X F 2004 Physica A 380 684
[19] Cohen R , Havlin S, ben-Avraham D 2003 Phys. Rev. Lett. 91 247901
[20] Bose T, Trimper S 2009 Phys. Rev. E 79 051903
[21] Gumel A B, McCluskey C C, van den Driessche P 2006 Bull. Math. Biol. 68 2105
[22] Yip P, Watson R, Chen Q Z 2007 Statist. Med. 26 4475
[23] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
[24] Gandon S, Day T 2008 Vaccine 26 54
[25] Ball F G, Knock E S, O’Neill P D 2008 Math. Biosci. 216 100
[26] Wang Y B, Xiao G X, Hu J, Cheng T H, Wang L S 2009 Physica A 388 2535
[27] Xiao S, Xiao G, Cheng T H 2008 IEEE Commun. Mag. 46 146
[28] Gandon S, Mackinnon M, Nee S, Read A 2003 Proc. Roy. Soc. Lond. B 270 1129
[29] Ganusov V V, Antia R 2006 Evolution 60 957
[30] Newman M E J 2002 Phys. Rev. Lett. 89 208701
[31] Barabsi A L, Albert R, Jeong H 1999 Physica A 272 173
[32] Takeuchi Y, Ma W B, Beretta E 2000 Nonl. Anal. 42 931
[33] Barabsi A L, Albert R 2002 Rev. Mod. Phys. 74 47
[34] Abramowitz M, Stegun I 1972 Handbook of Mathematical Functions (New York: Dove Publisher)
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[1] Li X, Wang X F 2006 IEEE Trans. Automatic Control 51 534
[2] Li X, Wang X F 2007 Int. J. Syst. Sci. 38 400
[3] Song Y R, Jiang G P 2009 Acta Phys. Sin. 58 5911 (in Chinese) [宋玉蓉、蒋国平 2009 58 5911]
[4] Yang R, Wang B H, Ren J, Bai W J, Shi Z W, Xu W, Zhou T 2007 Phys. Lett. A 386 189
[5] Pei W D, Chen Z Q, Yang Z Z 2008 Chin. Phys. B 17 373
[6] Barthélemy M, Barrat A, Pastor-Satorras R, Vespignani A 2005 J. Theo. Bio. 235 275
[7] Zhou T, Liu J G, Bai W J, Chen G R, Wang B H 2006 Phys. Rev. E 74 056109
[8] Hayashi Y, Minoura M, Matsukubo J 2004 Phys. Rev. E 69 016112
[9] Huang W, Jiang R, Hu M B, Wu Q S 2009 Chin. Phys. B 18 1306
[10] Zhang H F, Michael S, Fu X C 2009 Commun. Theor. Phys. 52 180
[11] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200
[12] Wang Y, Zheng Z G 2009 Acta Phys. Sin. 58 4421 (in Chinese) [王 延、郑志刚 2009 58 4421]
[13] Watts D J, Strogatz S H 1998 Nature 393 440
[14] Barabsi A L, Albert R 1999 Science 286 509
[15] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[16] Lloyd A L, May R M 2001 Science 292 1316
[17] Pastor-Satorras R, Vespignani A 2002 Phys. Rev. E 65 036104
[18] Guo W P, Li X, Wang X F 2004 Physica A 380 684
[19] Cohen R , Havlin S, ben-Avraham D 2003 Phys. Rev. Lett. 91 247901
[20] Bose T, Trimper S 2009 Phys. Rev. E 79 051903
[21] Gumel A B, McCluskey C C, van den Driessche P 2006 Bull. Math. Biol. 68 2105
[22] Yip P, Watson R, Chen Q Z 2007 Statist. Med. 26 4475
[23] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
[24] Gandon S, Day T 2008 Vaccine 26 54
[25] Ball F G, Knock E S, O’Neill P D 2008 Math. Biosci. 216 100
[26] Wang Y B, Xiao G X, Hu J, Cheng T H, Wang L S 2009 Physica A 388 2535
[27] Xiao S, Xiao G, Cheng T H 2008 IEEE Commun. Mag. 46 146
[28] Gandon S, Mackinnon M, Nee S, Read A 2003 Proc. Roy. Soc. Lond. B 270 1129
[29] Ganusov V V, Antia R 2006 Evolution 60 957
[30] Newman M E J 2002 Phys. Rev. Lett. 89 208701
[31] Barabsi A L, Albert R, Jeong H 1999 Physica A 272 173
[32] Takeuchi Y, Ma W B, Beretta E 2000 Nonl. Anal. 42 931
[33] Barabsi A L, Albert R 2002 Rev. Mod. Phys. 74 47
[34] Abramowitz M, Stegun I 1972 Handbook of Mathematical Functions (New York: Dove Publisher)
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