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自适应复杂网络是以节点状态与拓扑结构之间存在反馈回路为特征的网络. 针对自适应网络病毒传播模型, 利用非线性微分动力学系统研究病毒传播行为; 通过分析非线性系统对应雅可比矩阵的特征方程, 研究其平衡点的局部稳定性和分岔行为, 并推导出各种分岔点的计算公式. 研究表明, 当病毒传播阈值小于病毒存在阈值, 即R0R0c时, 网络中病毒逐渐消除, 系统的无病毒平衡点是局部渐近稳定的; R0cR01时, 网络出现滞后分岔, 产生双稳态现象, 系统存在稳定的无病毒平衡点、较大稳定的地方病平衡点和较小不稳定的地方病平衡点; R01时, 网络中病毒持续存在, 系统唯一的地方病平衡点是局部渐近稳定的. 研究发现, 系统先后出现了鞍结分岔、跨临界分岔、霍普夫分岔等分岔行为. 最后通过数值仿真验证所得结论的正确性.Adaptive network is characterized by feedback loop between states of nodes and topology of the network. In this paper, for adaptive epidemic spreading model, epidemic spreading dynamics is studied by using a nonlinear differential dynamic system. The local stability and bifurcation behavior of the equilibrium in this network model are investigated and all kinds of bifurcation point formula are obtained by analyzing its corresponding characteristic equation of Jacobian matrix of the nonlinear system. It is shown that, when the epidemic threshold is less than epidemic persistence threshold R00c, the disease always dies out and the disease-free equilibrium is asymptotically locally stable. If R0c01, a backward bifurcation leading to bistability possibly occurs, and there are possibly three equilibria: a stable disease-free equilibrium, a larger stable endemic equilibrium, and a smaller unstable endemic equilibrium. If R01, the disease is uniformly persistent and only one endemic equilibrium is asymptotically locally stable. It is also found that the system has saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Numerical simulations are given to verify the results of theoretical analysis.
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Keywords:
- adaptive network /
- stability /
- bifurcation /
- basic reproduction number
[1] Albert R, Baraba'si A L 2002 Rev. Mod. Phys. 74 47
[2] Newman M E J 2003 SIAM Rev. 45 167
[3] Barthelemy M, Barrat A, Pastor-Satorras R, Vespihnani A 2004 Phys. Rev. Lett. 92 178701
[4] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[5] Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521
[6] Bruno B, Salvatore R 2010 Appl. Math and Computation 217 4010
[7] Zhang H F, Fu X C 2009 Nonl. Anal TMA 70 3273
[8] Wang Y Q, Jiang G P 2010 Acta Phys．Sin. 59 6734 (in Chinese) [王亚奇, 蒋国平 2010 59 6734]
[9] Wang Y Q, Jiang G P 2010 Acta Phys．Sin. 59 6725 (in Chinese) [王亚奇, 蒋国平 2010 59 6725]
[10] Li M Y, Smith H L, Wang L 2001 SIMA J. Appl. Math. 62 58
[11] Li M Y, Wang L 2002 IMA 126 295
[12] Li C G, Chen G, Chao S 2004 Solitons and Fractals 20 353
[13] Li X, Chen G, Li C G 2004 Int. J. of Systems Science 35 527
[14] Van den Driessche P, Watmough J 2000 J. Math. Biol. 40 525
[15] Hadeler K P, van den Driessche P 1997 Math. Biosei. 146 15
[16] Van den Driessche P, Watmough J 2002 Math. Biosei. 180 29
[17] Tu S, Reiss E, SIAM J 1986 Appl. Math. 46 189
[18] Gross T, D' Lima C J D, Blasius B 2006 Phys. Rev. Lett. 96 208701
[19] Shaw L B, Schwartz I B 2008 Phys. Rev. E 77 066101
[20] Gross T, Blasius B, Soc J R 2008 Interface 5 259
[21] Gross Sayama T H, eds 2009 Adaptive Networks: Theory, Models and Applications (Springer)
[22] Gross T, Kevrekidis I G 2008 Eur. Lett. 82 38004
[23] Risau-Gusman S, Zanette D H 2009 J. Theor. Biol. 257 52
[24] Zanette D, Risau-Gusm_an S 2008 J. Biol. Phys. 34 135
[25] Schwartz I B, Shaw L B 2010 Physics 3
[26] Song Y R, Jiang G P, Xu J G 2011 Acta Phys. Sin. 60 120509 (in Chinese) [宋玉蓉, 蒋国平, 徐加刚 2010 60 120509]
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[1] Albert R, Baraba'si A L 2002 Rev. Mod. Phys. 74 47
[2] Newman M E J 2003 SIAM Rev. 45 167
[3] Barthelemy M, Barrat A, Pastor-Satorras R, Vespihnani A 2004 Phys. Rev. Lett. 92 178701
[4] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[5] Moreno Y, Pastor-Satorras R, Vespignani A 2002 Eur. Phys. J. B 26 521
[6] Bruno B, Salvatore R 2010 Appl. Math and Computation 217 4010
[7] Zhang H F, Fu X C 2009 Nonl. Anal TMA 70 3273
[8] Wang Y Q, Jiang G P 2010 Acta Phys．Sin. 59 6734 (in Chinese) [王亚奇, 蒋国平 2010 59 6734]
[9] Wang Y Q, Jiang G P 2010 Acta Phys．Sin. 59 6725 (in Chinese) [王亚奇, 蒋国平 2010 59 6725]
[10] Li M Y, Smith H L, Wang L 2001 SIMA J. Appl. Math. 62 58
[11] Li M Y, Wang L 2002 IMA 126 295
[12] Li C G, Chen G, Chao S 2004 Solitons and Fractals 20 353
[13] Li X, Chen G, Li C G 2004 Int. J. of Systems Science 35 527
[14] Van den Driessche P, Watmough J 2000 J. Math. Biol. 40 525
[15] Hadeler K P, van den Driessche P 1997 Math. Biosei. 146 15
[16] Van den Driessche P, Watmough J 2002 Math. Biosei. 180 29
[17] Tu S, Reiss E, SIAM J 1986 Appl. Math. 46 189
[18] Gross T, D' Lima C J D, Blasius B 2006 Phys. Rev. Lett. 96 208701
[19] Shaw L B, Schwartz I B 2008 Phys. Rev. E 77 066101
[20] Gross T, Blasius B, Soc J R 2008 Interface 5 259
[21] Gross Sayama T H, eds 2009 Adaptive Networks: Theory, Models and Applications (Springer)
[22] Gross T, Kevrekidis I G 2008 Eur. Lett. 82 38004
[23] Risau-Gusman S, Zanette D H 2009 J. Theor. Biol. 257 52
[24] Zanette D, Risau-Gusm_an S 2008 J. Biol. Phys. 34 135
[25] Schwartz I B, Shaw L B 2010 Physics 3
[26] Song Y R, Jiang G P, Xu J G 2011 Acta Phys. Sin. 60 120509 (in Chinese) [宋玉蓉, 蒋国平, 徐加刚 2010 60 120509]
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