Abstract
In this paper, we propose a susceptible-infected-susceptible (SIS) model on complex networks, small-world (WS) networks and scale-free (SF) networks, to study the epidemic spreading behavior with time delay which is added into the infected phase. Considering the uniform delay, the basic reproduction number R 0 on WS networks and \(\bar R_0\) on SF networks are obtained respectively. On WS networks, if R 0 ≤ 1, there is a disease-free equilibrium and it is locally asymptotically stable; if R 0 > 1, there is an epidemic equilibrium and it is locally asymptotically stable. On SF networks, if \(\bar R_0 \leqslant 1\), there is a disease-free equilibrium; if \(\bar R_0 > 1\), there is an epidemic equilibrium. Finally, we carry out simulations to verify the conclusions and analyze the effect of the time delay τ, the effective rate λ, average connectivity 〈k〉 and the minimum connectivity m on the epidemic spreading.
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References
Albert, R., Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys., 74: 47–97 (2002)
Barabási, A.L. Albert, R. Emergence of scaling in random networks. Science, 286: 509–512 (1999)
Dorogovtsev, S.N., Mendes, J.F.F. Evolution of networks. Adv. Phys., 51: 1079–1084 (2002)
Kephart, J.O., Sorkin, G.B., Chess, D.M., et al. Fighting computer viruses. Scientific American, 277(5): 56–61 (1997)
Ma, Z.E., Zhou, Y.C., Wang, W.D., et al. Epidemic dynamic mathematical model and study. Science Press, Beijing, 2004
Newman, M.E.J. Spread of epidemic disease on networks. Phys. Rev. E, 66: 116–128 (2002)
Pastor-Satorras, R., Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett., 86(14): 3200–3203 (2001)
Watts, D.J., Strogatz, S.H. Collective Dynamics of Small-World Networks. Nature (London), 393: 440–442 (1998)
Xia, C.Y., Liu, Z.X., Chen, Z.Q., Yuan, Z.Z. Dynamic spreading behaviors of homogeneous and heterogeneous networks. Prog. Nat. Sci., 17(3): 358–365 (2007)
Xia, C.Y., Liu, Z.X., Chen, Z.Q., Sun, S.W., Yuan, Z.Z. Epidemic spreading behavior in local-world evolving networks. Prog. Nat. Sci., 18(6): 763–768 (2008)
Xia, C.Y., Wang, L., Sun, S.W., Wang, J. An SIR model with infection delay and propagation vector in complex networks. Nonlinear Dyn., 69: 927–934 (2012)
Xu, X.J., Peng, H.O., Wang, X.M., Wang, Y.H. Epidemic spreading with time delay in complex networks. Phys. A, 367: 525–530 (2006)
Xia, C.Y., Liu, Z.X., Chen, Z.Q., Sun, S.W., Yuan, Z.Z. Epidemic spreading behavior with time delay on local-world evolving networks. Front. Electr. Electron. Eng., 3(2): 129–135 (2008)
Zanette, D.H. Dynamics of rumor propagation on small-world networks. Phy. Rev. E, 65: 1–9 (2002)
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This work is supported by the National Natural Sciences Foundation of China (Nos. 11331009, 61272095, 61175067, 61573231, 61432011, U1435212, 11471197, 11401541), Shanxi Scholarship Council of China (2013-014) and Shanxi Science and Technology Infrastructure (2015091001-0102).
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Zhang, Rx., Jin, Z. & Li, Sp. Epidemic spreading with time delay on complex networks. Acta Math. Appl. Sin. Engl. Ser. 32, 319–326 (2016). https://doi.org/10.1007/s10255-016-0554-1
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DOI: https://doi.org/10.1007/s10255-016-0554-1