A complex networked system typically has a time-varying nature in interactions among its components, which is intrinsically complicated and therefore technically challenging for analysis and control. This paper investigates an epidemic process on a time-varying network with a time delay. First, an averaging theorem is established to approximate the delayed time-varying system using autonomous differential equations for the analysis of system evolution. On this basis, the critical time delay is determined, across which the endemic equilibrium becomes unstable and a phase transition to oscillation in time via Hopf bifurcation will appear. Then, numerical examples are examined, including a periodically time-varying network, a blinking network, and a quasi-periodically time-varying network, which are simulated to verify the theoretical results. Further, it is demonstrated that the existence of time delay can extend the network frequency range to generate Turing patterns, showing a facilitating effect on phase transitions.

1.
W.
Gou
,
Y.
Song
, and
Z.
Jin
, “
The steady state bifurcation for general network-organized reaction-diffusion systems and its application in a metapopulation epidemic model
,”
SIAM J. Appl. Dyn. Syst.
22
,
559
602
(
2023
).
2.
V.
Colizza
,
R.
Pastor-Satorras
, and
A.
Vespignani
, “
Reaction–diffusion processes and metapopulation models in heterogeneous networks
,”
Nat. Phys.
3
,
276
282
(
2007
).
3.
L.
Chang
,
M.
Duan
,
G.
Sun
, and
Z.
Jin
, “
Cross-diffusion-induced patterns in an SIR epidemic model on complex networks
,”
Chaos
30
,
013147
(
2020
).
4.
J.
Gómez-Gardeñes
,
D.
Soriano-Panos
, and
A.
Arenas
, “
Critical regimes driven by recurrent mobility patterns of reaction–diffusion processes in networks
,”
Nat. Phys.
14
,
391
395
(
2018
).
5.
R.
Pastor-Satorras
and
A.
Vespignani
, “
Epidemic spreading in scale-free networks
,”
Phys. Rev. Lett.
86
,
3200
(
2001
).
6.
A. S.
Zadorin
,
Y.
Rondelez
,
G.
Gines
,
V.
Dilhas
,
G.
Urtel
,
A.
Zambrano
,
J.-C.
Galas
, and
A.
Estevez-Torres
, “
Synthesis and materialization of a reaction–diffusion french flag pattern
,”
Nat. Chem.
9
,
990
996
(
2017
).
7.
J. T.
Davis
,
N.
Perra
,
Q.
Zhang
,
Y.
Moreno
, and
A.
Vespignani
, “
Phase transitions in information spreading on structured populations
,”
Nat. Phys.
16
,
590
596
(
2020
).
8.
H.
Nakao
and
A. S.
Mikhailov
, “
Turing patterns in network-organized activator–inhibitor systems
,”
Nat. Phys.
6
,
544
550
(
2010
).
9.
D.
Ghosh
,
M.
Frasca
,
A.
Rizzo
,
S.
Majhi
,
S.
Rakshit
,
K.
Alfaro-Bittner
, and
S.
Boccaletti
, “
The synchronized dynamics of time-varying networks
,”
Phys. Rep.
949
,
1
63
(
2022
).
10.
Y.
Zhang
and
S. H.
Strogatz
, “
Designing temporal networks that synchronize under resource constraints
,”
Nat. Commun.
12
,
3273
(
2021
).
11.
M.
Asllani
,
J. D.
Challenger
,
F. S.
Pavone
,
L.
Sacconi
, and
D.
Fanelli
, “
The theory of pattern formation on directed networks
,”
Nat. Commun.
5
,
4517
(
2014
).
12.
R.
Muolo
,
M.
Asllani
,
D.
Fanelli
,
P. K.
Maini
, and
T.
Carletti
, “
Patterns of non-normality in networked systems
,”
J. Theor. Biol.
480
,
81
91
(
2019
).
13.
S.
Gao
,
L.
Chang
,
M.
Perc
, and
Z.
Wang
, “
Turing patterns in simplicial complexes
,”
Phys. Rev. E
107
,
014216
(
2023
).
14.
R.
Muolo
,
L.
Gallo
,
V.
Latora
,
M.
Frasca
, and
T.
Carletti
, “
Turing patterns in systems with high-order interactions
,”
Chaos Solitons Fractals
166
,
112912
(
2023
).
15.
F.
Battiston
,
E.
Amico
,
A.
Barrat
,
G.
Bianconi
,
G.
Ferraz de Arruda
,
B.
Franceschiello
,
I.
Iacopini
,
S.
Kéfi
,
V.
Latora
,
Y.
Moreno
et al., “
The physics of higher-order interactions in complex systems
,”
Nat. Phys.
17
,
1093
1098
(
2021
).
16.
D. J.
Stilwell
,
E. M.
Bollt
, and
D. G.
Roberson
, “
Sufficient conditions for fast switching synchronization in time-varying network topologies
,”
SIAM J. Appl. Dynami. Syst.
5
,
140
156
(
2006
).
17.
D.
Liberzon
and
A. S.
Morse
, “
Basic problems in stability and design of switched systems
,”
IEEE Control Syst. Mag.
19
,
59
70
(
1999
).
18.
J.
Petit
,
B.
Lauwens
,
D.
Fanelli
, and
T.
Carletti
, “
Theory of Turing patterns on time varying networks
,”
Phys. Rev. Lett.
119
,
148301
(
2017
).
19.
R. A.
Van Gorder
, “
A theory of pattern formation for reaction–diffusion systems on temporal networks
,”
Proc. R. Soc. A
477
,
20200753
(
2021
).
20.
G.
Ren
and
X.
Wang
, “
Epidemic spreading in time-varying community networks
,”
Chaos
24
,
023116
(
2014
).
21.
J. P.
Onnela
,
J.
Saramäki
,
J.
Hyvönen
,
G.
Szabó
,
D.
Lazer
,
K.
Kaski
,
J.
Kertész
, and
A.-L.
Barabási
, “
Structure and tie strengths in mobile communication networks
,”
Proc. Natl. Acad. Sci. U.S.A.
104
,
7332
7336
(
2007
).
22.
P. W.
Anderson
,
The Economy As An Evolving Complex System
(
CRC Press
,
2018
).
23.
N.
Wang
,
Z.
Jin
,
Y.
Wang
, and
Z.
Di
, “
Epidemics spreading in periodic double layer networks with dwell time
,”
Phys. A
540
,
123226
(
2020
).
24.
J. K.
Hale
, “
Averaging methods for differential equations with retarded arguments and a small parameter
,”
J. Differ. Eq.
2
,
57
73
(
1966
).
25.
B.
Lehman
and
S. P.
Weibel
, “
Fundamental theorems of averaging for functional differential equations
,”
J. Differ. Eq.
152
,
160
190
(
1999
).
26.
B.
Lehman
, “
The influence of delays when averaging slow and fast oscillating systems: Overview
,”
IMA J. Math. Control Inform.
19
,
201
215
(
2002
).
27.
M.
Duan
,
L.
Chang
, and
Z.
Jin
, “
Turing patterns of an SI epidemic model with cross-diffusion on complex networks
,”
Phys. A
533
,
122023
(
2019
).
28.
P.
Erdős
and
A.
Rényi
, “
On random graphs
,”
Publ. Math. Debrecen
6
,
290
297
(
1959
).
29.
I. V.
Belykh
,
V. N.
Belykh
, and
M.
Hasler
, “
Blinking model and synchronization in small-world networks with a time-varying coupling
,”
Phys. D
195
,
188
206
(
2004
).
30.
J.
Ritchie
, “
Turing instability and pattern formation on directed networks
,”
Commun. Nonlinear Sci. Numer. Simul.
116
,
106892
(
2023
).
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