Reverse-feeding effect of epidemic by propagators in two-layered networks
Dayu Wu(吴大宇), Yanping Zhao(赵艳萍), Muhua Zheng(郑木华), Jie Zhou(周杰), Zonghua Liu(刘宗华)
Department of Physics, East China Normal University, Shanghai 200062, China

Abstract Epidemic spreading has been studied for a long time and is currently focused on the spreading of multiple pathogens, especially in multiplex networks. However, little attention has been paid to the case where the mutual influence between different pathogens comes from a fraction of epidemic propagators, such as bisexual people in two separated groups of heterosexual and homosexual people. We here study this topic by presenting a network model of two layers connected by impulsive links, in contrast to the persistent links in each layer. We let each layer have a distinct pathogen and their interactive infection is implemented by a fraction of propagators jumping between the corresponding pairs of nodes in the two layers. By this model we show that (i) the propagators take the key role to transmit pathogens from one layer to the other, which significantly influences the stabilized epidemics; (ii) the epidemic thresholds will be changed by the propagators; and (iii) a reverse-feeding effect can be expected when the infective rate is smaller than its threshold of isolated spreading. A theoretical analysis is presented to explain the numerical results.
Key words ：
propagators
complex network
two pathogens
epidemic spreading
Received: 04 September 2015
PACS:
87.18.Yt
(Circadian rhythms)
05.45.Xt
(Synchronization; coupled oscillators)
87.18.Sn
(Neural networks and synaptic communication)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11135001, 11375066, and 11405059) and the National Basic Key Program of China (Grant No. 2013CB834100).
Corresponding Authors: Zonghua Liu
E-mail: zhliu@phy.ecnu.edu.cn

[1]
Anderson R M and May R M 1991 Infectious diseases of humans: dynamics and control (Oxford, New York: Oxford University Press)
[2]
Albert R and Barabási A L 2002 Rev. Mod. Phys. 74 47
[3]
Dorogovtsev S N, Goltsev A V and Mendes J F F 2008 Rev. Mod. Phys. 80 1275
[4]
Pastor-Satorras R and Vespignani A 2001 Phys. Rev. Lett. 86 3200
[5]
Eguiluz V M and Klemm K 2002 Phys. Rev. Lett. 89 108701
[6]
Newman M E J 2002 Phys. Rev. E 66 016128
[7]
Boguna M, Pastor-Satorras R and Vespignani A 2003 Phys. Rev. Lett. 90 028701
[8]
Gross T, D'Lima C J D and Blasius B 2006 Phys. Rev. Lett. 96 208701
[9]
Parshani R, Carmi S and Havlin S 2010 Phys. Rev. Lett. 104 258701
[10]
Castellano C and Pastor-Satorras R 2010 Phys. Rev. Lett. 105 218701
[11]
Schwarzkopf Y, Rakos A and Mukamel D 2010 Phys. Rev. E 82 036112
[12]
Pastor-Satorras R and Vespignani A 2002 Phys. Rev. E 65 036104
[13]
Dezso Z and Barabasi A L 2002 Phys. Rev. E 65 055103
[14]
Liu Z, Lai Y C and Ye N 2003 Phys. Rev. E 67 031911
[15]
Cohen R, Havlin S and ben-Avraham D 2003 Phys. Rev. Lett. 91 247901
[16]
Gallos L K et al. 2007 Phys. Rev. E 75 045104(R)
[17]
Ferguson N 2007 Nature 446 12
[18]
Meloni S et al. 2011 Sci. Rep. 1 62
[19]
Schwartz I B and Shaw L B 2010 Physics 3 17
[20]
Ruan Z, Tang M and Liu Z 2012 Phys. Rev. E 86 036117
[21]
Li K, Xu Z, Zhu G and Ding Y 2014 Chin. Phys. B 23 118904
[22]
Wang W, Tang M, Yang H, Do Y, Lai Y and Lee G W 2014 Sci. Rep. 4 5097
[23]
Liu Q, Wang W, Tang M and Zhang H 2015 arXiv: 1509.08183
[24]
Tang M, Liu Z and Li B 2009 Europhys. Lett. 87 18005
[25]
Liu Z 2010 Phys. Rev. E 81 016110
[26]
Ruan Z, Hui P M, Lin H Q and Liu Z 2013 Eur. Phys. J. B 86 13
[27]
Ruan Z, Tang M and Liu Z 2013 Eur. Phys. J. B 86 149
[28]
Grassberger P 2013 J. Stat. Mech. 2013 04004
[29]
Colizza V, Pastor-Satorras R and Vespignani A 2007 Nat. Phys. 3 276
[30]
Colizza V and Vespignani A 2007 Phys. Rev. Lett. 99 148701
[31]
Colizza V and Vespignani A 2008 J. Theor. Biol. 251 450
[32]
Baronchelli A, Catanzaro M and Pastor-Satorras R 2008 Phys. Rev. E 78 016111
[33]
Tang M, Liu L and Liu Z 2009 Phys. Rev. E 79 016108
[34]
Wang Z, Wang L and Perc M 2014 Phys. Rev. E 89 052813
[35]
Chen Z, Du W, Cao X and Zhou X 2015 Chaos, Solitons and Fractals 80 7
[36]
Du W, Zhou X, Chen Z, Cai K and Cao X 2014 Chaos, Solitons and Fractals 68 72
[37]
Chen L, Ghanbarnejad F, Cai W and Grassberger P 2013 Europhys. Lett. 104 50001
[38]
Mills H L, Ganesh A and Colijn C 2013 J. Theor. Biol. 320 47
[39]
Funk S and Jansen V A A 2010 Phys. Rev. E 81 036118
[40]
Dickison M, Havlin S and Stanley H E 2012 Phys. Rev. E 85 066109
[41]
Saumell-Mendiola A, Serrano M A and Boguna M 2012 Phys. Rev. E 86 026106
[42]
Newman M E J 2005 Phys. Rev. Lett. 95 108701
[43]
Sulkowski M S 2008 J. Hepatol. 48 353
[44]
Abu-Raddad L J, Patnaik P and Kublin J G 2006 Science 314 1603
[45]
Brundage J F and Shanks G D 2008 Emerg. Infect. Dis. 14 1193
[46]
Oei W and Nishiura H 2012 Comput. Math. Methods Med. 2012 124861
[47]
Martcheva M and Pilyugin S S 2006 SIAM J. Appl. Math. 66 843
[48]
Marceau V, Noel P A, Hebert-Dufresne L, Allard A and Dube L J 2011 Phys. Rev. E 84 026105
[49]
Miller J C 2013 Phys. Rev. E 87 060801
[50]
Newman M E J and Ferrario C R 2013 PLoS ONE 8 e71321
[51]
Britton T, Nordvik M K and Liljeros F 2007 Theor. Popul. Biol. 72 389
[52]
Liljeros F, Edling C R, Amaral L A N, Stanley H E and Aberg Y 2001 Nature 411 907
[53]
Sanz J, Xia C, Meloni S and Moreno Y 2014 Phys. Rev. X 4 041005
[54]
Zhao Y, Zheng M and Liu Z 2014 Chaos 24 043129
[55]
Goodenow C, Netherland J and Szalacha L 2002 Amer. J. Public Health 92 203
[56]
Hightow L B, Leone P A, Macdonald P D, McCoy S I, Sampson L A and Kaplan A H 2006 Sexu. Trans. Diseases 33 585
[57]
Jeffries W L 2011 Perspect. Sex. Reprod Health 43 151
[58]
Jeffries W L and Dodge B 2007 J. Sex Research 44 278
[59]
Bauch C T and Galvani A P 2013 Science 342 47
[60]
Olinky R and Stone L 2004 Phys. Rev. E 70 030902
[61]
Liu Z and Hu B 2005 Europhys. Lett. 72 315
[62]
Wu Q, Fu X, Small M and Xu X J 2012 Chaos 22 013101
[63]
Cai C, Wu Z X and Guan J 2014 Chaos, Solitons and Fractals 62-63 36
[64]
Lacroix R, Mukabana W R, Gouagna L C and Koella J C 2005 PLoS Biol. 3 e298
[65]
Zhou J, Chung N, Chew L and Lai C 2012 Phys. Rev. E 86 026115
[66]
Pastor-Satorras R and Vespignani A 2001 Phys. Rev. E 63 066117

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