In order to better study the interaction between epidemic propagation and information diffusion, a new coupling model on multiplex networks with time delay is put forward in this paper. One layer represents the information diffusion about epidemics. There is not only information about the positive prevention of infectious diseases but also negative preventive information. Meanwhile, the dissemination of information at this layer will be influenced by the mass media, which can convey positive and reliable preventive measures to help the public avoid exposure to contagion. The other layer represents the transmission of infectious diseases, and the public in this layer no longer only exchange information related to infectious diseases in the virtual social network like the information layer but spread infectious diseases through contact among people. The classical SIR model is used to model for epidemic propagation. Since each infected individual needs to spend enough time to recover, the infected one at one time does not necessarily change to the recovered one at the next time, so time delay is an essential factor to be considered in the model. Based on the microscopic Markov chain approach, this paper obtains an explicit expression for epidemic threshold in the two-layered multiplex networks with time delay, which reveals some main factors affecting epidemic threshold. In particular, the time delay has a noticeable effect on the epidemic threshold to some extent. Finally, the influence of these main factors on the epidemic threshold and their interaction are proved through numerical simulations.

1.
D. A.
Henderson
,
T. V.
Inglesby
,
J. G.
Bartlett
,
M. S.
Ascher
, and
K.
Tonat
, “
Smallpox as a biological weapon: Medical and public health management
,”
J. Am. Med. Assoc.
281
,
2127
2137
(
1999
).
2.
D. J.
Alexander
, “
A review of avian influenza in different bird species
,”
Vet. Microbiol.
74
,
3
13
(
2000
).
3.
D. M.
Morens
,
G. K.
Folkers
, and
A. S.
Fauci
, “
The challenge of emerging and re-emerging infectious diseases
,”
Nature
430
,
242
(
2004
).
4.
B.
Bramanti
,
N. C.
Stenseth
,
L.
Walløe
, and
X.
Lei
, “
Plague: A disease which changed the path of human civilization
,” in
Yersinia Pestis: Retrospective and Perspective
(Springer, 2016), Vol. 918, pp. 1–26.
5.
H. W.
Hethcote
, “
The mathematics of infectious diseases
,”
SIAM Rev.
42
,
599
653
(
2000
).
6.
L.
Hall-Stoodley
,
J. W.
Costerton
, and
P.
Stoodley
, “
Bacterial biofilms: From the natural environment to infectious diseases
,”
Nat. Rev. Microbiol.
2
,
95
108
(
2004
).
7.
J.
Mossong
,
N.
Hens
,
M.
Jit
,
P.
Beutels
,
K.
Auranen
,
R.
Mikolajczyk
,
M.
Massari
,
S.
Salmaso
,
G. S.
Tomba
, and
J.
Wallinga
, “
Social contacts and mixing patterns relevant to the spread of infectious diseases
,”
PLoS Med.
5
,
381
391
(
2008
).
8.
L. J.
Allen
and
A. M.
Burgin
, “
Comparison of deterministic and stochastic SIS and SIR model in discrete time
,”
Math. Biosci.
163
,
1
33
(
2000
).
9.
M. Y.
Li
,
H. L.
Smith
, and
L.
Wang
, “
Global dynamics of an SEIR epidemic model with vertical transmission
,”
SIAM J. Appl. Math.
62
,
58
69
(
2002
).
10.
Y.
Zhou
and
H.
Liu
, “
Stability of periodic solutions for an SIS model with pulse vaccination
,”
Math. Comput. Model.
38
,
299
308
(
2003
).
11.
S.
Gómez
,
A.
Arenas
,
J.
Borge-Holthoefer
,
S.
Meloni
, and
Y.
Moreno
, “
Discrete-time Markov chain approach to contact-based disease spreading in complex networks
,”
Europhys. Lett.
89
,
38009
(
2010
).
12.
H.
Kang
and
X.
Fu
, “
Epidemic spreading and global stability of an SIS model with an infective vector on complex networks
,”
Commun. Nonlinear Sci. Numer. Simul.
27
,
30
39
(
2015
).
13.
C.
Wesley
,
A. S.
Mata
, and
S. C.
Ferreira
, “
Robustness and fragility of the susceptible-infected-susceptible epidemic models on complex networks
,”
Phys. Rev. E
98
,
012310
(
2018
).
14.
Y.
Wang
,
Z.
Wei
, and
J.
Cao
, “
Epidemic dynamics of influenza-like diseases spreading in complex networks
,”
Nonlinear Dyn.
101
,
1801
1820
(
2020
).
15.
X.
Luo
and
Z.
Jin
, “
A new insight into isolating the high-degree nodes in network to control infectious diseases
,”
Commun. Nonlinear Sci. Numer. Simul.
91
,
105363
(
2020
).
16.
M.
Salehi
,
R.
Sharma
,
M.
Marzolla
,
M.
Magnani
,
P.
Siyari
, and
D.
Montesi
, “
Spreading processes in multilayer networks
,”
IEEE Trans. Netw. Sci. Eng.
2
,
65
83
(
2015
).
17.
P.
Hu
,
L.
Ding
, and
X.
An
, “
Epidemic spreading with awareness diffusion on activity-driven networks
,”
Phys. Rev. E
98
,
062322
(
2018
).
18.
M.
De Domenico
,
C.
Granell
,
M. A.
Porter
, and
A.
Arenas
, “
The physics of spreading processes in multilayer networks
,”
Nat. Phys.
12
,
901
906
(
2016
).
19.
S.
Funk
,
M.
Salathé
, and
V. A.
Jansen
, “
Modelling the influence of human behaviour on the spread of infectious diseases: A review
,”
J. R. Soc. Interface
7
,
1247
1256
(
2010
).
20.
F.
Verelst
,
L.
Willem
, and
P.
Beutels
, “
Behavioural change models for infectious disease transmission: A systematic review (2010-2015)
,”
J. R. Soc. Interface
13
,
20160820
(
2016
).
21.
S.
Meloni
,
N.
Perra
,
A.
Arenas
,
S.
Gómez
,
Y.
Moreno
, and
A.
Vespignani
, “
Modeling human mobility responses to the large-scale spreading of infectious diseases
,”
Sci. Rep.
1
,
62
(
2011
).
22.
H.
Nunner
,
V.
Buskens
, and
M.
Kretzschmar
, “
A model for the co-evolution of dynamic social networks and infectious disease dynamics
,”
Comput. Social Netw.
8
,
1
19
(
2021
).
23.
Z.-K.
Zhang
,
C.
Liu
,
X.-X.
Zhan
,
X.
Lu
,
C.-X.
Zhang
, and
Y.
Zhang
, “
Dynamics of information diffusion and its applications on complex networks
,”
Phys. Rep.
651
,
1
34
(
2016
).
24.
C.
Granell
,
S.
Gómez
, and
A.
Arenas
, “
Dynamical interplay between awareness and epidemic spreading in multiplex networks
,”
Phys. Rev. Lett.
111
,
128701
(
2013
).
25.
C.
Granell
,
S.
Gómez
, and
A.
Arenas
, “
Competing spreading processes on multiplex networks: Awareness and epidemics
,”
Phys. Rev. E
90
,
012808
(
2014
).
26.
C.
Zheng
,
C.
Xia
,
Q.
Guo
, and
M.
Dehmer
, “
Interplay between SIR-based disease spreading and awareness diffusion on multiplex networks
,”
J. Parallel Distrib. Comput.
115
,
20
28
(
2018
).
27.
Z.
Wang
,
Q.
Guo
,
S.
Sun
, and
C.
Xia
, “
The impact of awareness diffusion on SIR-like epidemics in multiplex networks
,”
Appl. Math. Comput.
349
,
134
147
(
2019
).
28.
C.
Xia
,
Z.
Wang
,
C.
Zheng
,
Q.
Guo
,
Y.
Shi
,
M.
Dehmer
, and
Z.
Chen
, “
A new coupled disease-awareness spreading model with mass media on multiplex networks
,”
Inf. Sci.
471
,
185
200
(
2019
).
29.
Q.
Yin
,
Z.
Wang
,
C.
Xia
, and
C. T.
Bauch
, “
Impact of co-evolution of negative vaccine-related information, vaccination behavior and epidemic spreading in multilayer networks
,”
Commun. Nonlinear Sci. Numer. Simul.
109
,
106312
(
2022
).
30.
M.
Wieczorek
,
J.
Silka
, and
M.
Woźniak
, “
Neural network powered COVID-19 spread forecasting model
,”
Chaos Soliton. Fract.
140
,
110203
(
2020
).
31.
S.
Varela-Santos
and
P.
Melin
, “
A new approach for classifying coronavirus COVID-19 based on its manifestation on chest X-rays using texture features and neural networks
,”
Inf. Sci.
545
,
403
414
(
2021
).
32.
A.
Brodeur
,
D.
Gray
,
A.
Islam
, and
S.
Bhuiyan
, “
A literature review of the economics of COVID-19
,”
J. Econ. Surv.
35
,
1007
1044
(
2021
).
33.
A.
Rizzo
,
M.
Frasca
, and
M.
Porfiri
, “
Effect of individual behavior on epidemic spreading in activity-driven networks
,”
Phys. Rev. E
90
,
042801
(
2014
).
34.
A. I. E.
Hosni
,
K.
Li
, and
S.
Ahmad
, “
Minimizing rumor influence in multiplex online social networks based on human individual and social behaviors
,”
Inf. Sci.
512
,
1458
1480
(
2020
).
35.
Z.
Wang
,
C.
Xia
,
Z.
Chen
, and
G.
Chen
, “
Epidemic propagation with positive and negative preventive information in multiplex networks
,”
IEEE Trans. Cybernet.
51
,
1454
1462
(
2021
).
36.
C.
Castellano
and
R.
Pastor-Satorras
, “
Thresholds for epidemic spreading in networks
,”
Phys. Rev. Lett.
105
,
218701
(
2010
).
You do not currently have access to this content.