臺灣博碩士論文加值系統

在這篇論文中，我們考慮一個有分布時滯的傳染病模型在均勻
網路裡，這個模型由兩個均勻的網路所組成，且假設時滯只在疾病
傳染時才發生，在資訊傳播時不發生時滯。 在這個模型裡有三個正
的固定點E1、E2、E3，分別代表疾病與資訊不爆發、疾病不爆發資
訊爆發和疾病與資訊爆發。 以下是我們主要的結果。 第一、證明
了這個模型裡E3 是一致持續生存的。 第二、證明了E1 的全局穩定
性。 第三、證明了E2 的全局穩定性。 第四、證明了對於夠小的時
滯，E3 的全局穩定性。

In this paper, we consider an epidemic model in well-mixed multiplex networks with distributed time delay. Specifically, the model consists of two layers of well-mixed networks, where two diffusive processes on the
same individual interacting and affecting each other. We assume that there is a distributed time delay for an individual getting infected and no delay for an individual changing one’s status from unawareness to
awareness. There are three possible equilibria E1, E2 and E3, called disease and information free equilibrium, disease free and information saturated equilibrium and endemic and information saturated equilibrium, in this model. Our main result contain the following. First, we prove the uniform persistence of the model for parameter region yielding E3. Second, it is shown that E1 is globally stable for any time delay. Third, we prove that E2 is globally stable for any time delay. Finally, With help of the uniform persistence, we shoe that there exists a H*>0 such that E3 is globally stable for time delay less than H*.

Chinese Abstract ……………………… i
English Abstract ……………………… ii
Acknowledgement ………………………… iii
Table of Contents …………………… iv
1. Introduction…………………………… 1
2. Uniform persistence………… 3
3. Global stability………………… 11
References ……………………………………… 18

[1] R. M. Anderson and R. M. May, Infections Disease of Humans: Dynamics and control, Oxford University Press, Oxford, 1992.

[2] S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, and M. Zanin, The structure and dynamics of
multilayer networks, Phys. Rep. 544 (2014), pp. 1–122, doi:10.1016/j.physrep.2014.07.001.

[3] E. Cozzo, A. Arenas, and Y. Moreno, Stability of boolean multilevel networks, Phys. Rev. E, 86 (2012), p. 036115, doi:10.1103/PhysRevE.86.036115.

[4] S. Funk, E. Gilad, C. Watkins, and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proc. Natl. Acad. Sci. U.S.A., 106 (2009), pp. 6872–6877, doi: 10.1073/pnas.0810762106.

[5] C. Granell, S. Gómez, and A. Arenas, Dynamical interplay between awareness and epidemic spreading in multiplex networks, Phys. Rev. Lett., 111 (2013), p. 128701, doi:10.1103/ PhysRevLett.111.128701.

[6] C. Granell, S. Gómez, and A. Arenas, Competing spreading processes on multiplex networks: Awareness and epidemics, Phys. Rev. E, 90 (2014), p. 012808, doi:10.1103/PhysRevE. 90.012808.

[7] S. Gómez, A. Díaz-Guilera, J. Gómez-Gardeñes, C. J. Pérez-Vicente, Y. Moreno, and A. Arenas, Diffusion dynamics on multiplex networks, Phys. Rev. Lett., 110 (2013), p. 028701, doi:10.1103/PhysRevLett.110.028701.

[8] Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, and Z. Zhang, Two-stage effects of awareness cascade on epidemic spreading in multiplex networks, Phys. Rev. E, 91 (2015), p. 012822, doi: 10.1103/PhysRevE.91.012822.

[9] H. W. Hethcote, The mathematics of infectious disease, SIAM Rev., 42 (2002), pp. 955–960,
doi:10.1137/S0036144500371907.

[10] Y.-J. Huang, J. Juang, Y.-H. Liang, and H.-Y.Wang, Stability for epidemic models on multiplex networks, submitted.

[11] J. Juang and Y.-H. Liang, The impact of vaccine success and awareness on epidemic dynamics,
Chaos, 26 (2016), p. 113105, doi:10.1063/1.4966945.

[12] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.

[13] Y.-H. Liang and J. Juang, The impact of vaccine failure rate on epidemic dynamics in responsive networks, Chaos, 25 (2015), p. 043116, doi:10.1063/1.4919245.

[14] P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, and J.P. Onnela, Community structure in time-dependent, multiscale, and multiplex networks, Science, 328 (2010), pp. 876–878, doi:10.1126/science.1184819.

[15] H. L. Smith and X.-Q. Zhao, Robust persistence for semi-dynamical systems, Nonlinear Anal., 47 (2001), pp. 6169–6179, doi:10.1016/S0362-546X(01)00678-2.

[16] M. Szell, R. Lambiotte, S. Thurner, Multirelational organization of large-scale social networks in a online world, Proc. Natl. Acad. Sci. U.S.A., 107 (2010), pp. 13636–13641, doi: 10.1073/pnas.1004008107.

[17] Z. Wang, L. Wang, and M. Perc, Degree mixing in multilayer networks impedes the evolution of cooperation, Phys. Rev. E, 89 (2014), p. 052813, doi:10.1103/PhysRevE.89.052813.

[18] Z. Wang, H. Zhang, and Z. Wang, Multiple effects of self-protection on the spreading of epidemics, Chaos Solitons Fract., 61 (2014), pp. 1–7, doi:10.1016/j.chaos.2014.01.004.

[19] Q. Wu, X. Fu, M. Small, and X. J. Xu, The impact of awareness on epidemic spreading in networks, Chaos, 22 (2012), p. 013101, doi:10.1063/1.3673573.

[20] X.-Q. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.