Epidemics spreading in periodic double layer networks with dwell time

Physica A xxx (xxxx) xxx

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Epidemics spreading in periodic double layer networks with dwell time ∗

Ning-Ning Wang a , Zhen Jin b,c , , Ya-Jing Wang d , Zeng-Ru Di a a

School of systems science, Beijing Normal University, Beijing 100875, China Complex Systems Research Center, Shanxi University, Taiyuan 030006, China c Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan 030006, China d Department of Mathematics, North University of China, Taiyuan 030051, China b

article

info

Article history: Received 24 January 2019 Received in revised form 7 October 2019 Available online xxxx Keywords: Multilayer networks Switched systems Dwell time Epidemics

a b s t r a c t Epidemics spread in human contact networks, which evolve with diurnal and night time periodically and can be regarded as multilayer networks with different dwell time. The topologies of their subnetworks are different from each other, and the stabilities of their dynamic systems are also different. The stabilities of these switched systems, containing stable and unstable subsystems, not only depend on the stability of each subsystem, but also the dwell time of each subsystem. Hence, their stabilities are generally associated with the dwell time, network structure, infection and recovery rate. What is the quantitative relationship among these variables in threshold condition? In this paper, we establish a periodic switched N-intertwined SIS model and obtain its threshold condition in double layer networks, then extend this threshold condition to multilayer networks. This threshold condition is vital to accurately understand the effect of dwell time on spreading result in network. Some simulation results also indicate the accuracy of our threshold condition. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In the previous two decades, complex network has become an effective theoretical tool for the study of epidemics spreading, and these studies include the epidemics localization, estimation of epidemics threshold, basic reproductive number and the evaluation of effectiveness about strategies of reconnecting edges, human behavioral responses, immunization and seasonal-influenza epidemics etc [1–8]. Since the network topology is time-varying or periodic, more and more network propagation problems can be researched in multilayer networks [9–14]. Meanwhile, human behavior is influenced by circadian rhythms. They have relatively constant working relationships during the daytime and fixed family relationships at night, which leads that the epidemics contact networks evolve periodically. In each layer network, the individual social attribute, population density and clustering coefficient are different, and even the connectivity of the networks will change significantly [15–18]. Generally, the duration of epidemics spreading is longer or much longer than diurnal and night time. In the process of long-term epidemics spreading, we can ignore the intermediate evolution between the diurnal and night networks and only consider the periodic alternation of these two networks. Therefore, a multilayer network with periodic switching can be simplified into a periodic double layer network. One layer is a dense public network, possibly containing different size communities; the other one is a sparse private network that ∗ Corresponding author at: Complex Systems Research Center, Shanxi University, Taiyuan 030006, China. E-mail address: [email protected] (Z. Jin). https://doi.org/10.1016/j.physa.2019.123226 0378-4371/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

2

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

may consist of a group of connected or disconnected clusters and cliques. In those periodic double layer networks, the spreading of epidemics not only depends on the threshold of each layer, but also the operating time of each layer, i.e. the dwell time. Naturally, people are curious about the effect of the dwell time in epidemics threshold condition, which may have theoretical guidance for epidemics prevention. The switched system of double layer networks needs an appropriate theoretical analysis method. There have been some researches on the stability of switched systems based on dwell time [19–22]. These approaches are suitable to the switched nonlinear systems with both stable and unstable subsystems. However, their Lyapunov functions Vi (x) should satisfy the constraint of K functions and the dwell time are also constrained by ln(Vi (x)). In complex networks, the epidemics dynamic models are always high-dimensional, nonlinear and non-homogeneous, and hard to obtain a suitable Lyapunov function V (x) to prove the stability of system.1 Even if a suitable Lyapunov function V (x) can be found sometimes, the stability requires that each subsystem’s Lyapunov functions must be monotonic decreasing, i.e. V1 (x) ≤ V2 (x) ≤ · · · ≤ Vn (x). The monotonicity is a strong stable condition and not suitable for periodic switched systems. Hence, it is important to find a general method that can not only prove the stability of switched system but also adapt to any unweighted and undirected networks. In addition, there have been some studies on spreading in switched networks [23,24], and their results and methods are valuable for reference. While, the switched period in those studies is unit time, which means the dwell time is same in each subnetwork. Therefore, it is valuable to study the spreading problem in switched networks with different dwell time, and obtain the quantitative relationship of the dwell time in threshold condition. In this paper, a periodic switched system is established based on the N-intertwined SIS epidemics network model. In the theoretical analysis, we find that the subsystems of this switched system have some good dynamic properties, such as the Poincaré-Lyapunov stability [25]. The original system and its linearization system are all quasi-monotonic systems and satisfy the comparison theorem within the dwell time of each layer network. The above properties provide a lot of convenience in our theoretical analysis. As a result, we obtain a threshold condition of the switched system for the periodic ∑ double-layer networks and extend it to periodic switched multilayer networks. In fact, our threshold condition ρ ( σi Ai ) < 1/δ is the weighted processing of every layer networks based on dwell time. Compared with the normal static network, the threshold condition is a highlight and can’t be obtained in equal time switched network. In application, this may help us to accurately understand the effect of dwell time on spreading results and researchers to refine some disease control strategies of nonpharmaceutical interventions. In the switched system, the infected number is an expected value under the certain infection, recovery rate and network structure. First, we need some statistical simulations to reflect the model’s feasibility. In the process, we compare the advantages and disadvantages of the gillespie and event-driven algorithm, then improve the event-driven algorithm to make it suitable for switched network. Through a series of numerical and stochastic simulations, we verify the correctness of our switched system in BA and ER double layer network. Finally, the accuracy of threshold condition is verified by numerical simulations. This paper is outlined as follows. In Section 2, we establish a switched system of double layer network, and obtained its stability threshold condition. The simulation results and conclusions are presented in Sections 3 and 4. Some necessary proofs are summarized in Appendices A–D. 2. Derivation of the epidemics threshold condition There are various models in the network dynamics research. The commonly used models include the pair approximation models based on network edges, mean field models based on network degrees, models based on network adjacency matrix and models of percolation and random walk, etc. The model based on adjacency matrix can be easily established and completely preserve the structural information of networks. Hence, we use it as the foundation of our switched model. In these epidemics models, such as SIS, SIR and SIRS models, the states of nodes are quantified by the individuals infected probabilities, which are related to the network structure, infected and recovery rate. From the classical continuous [26,27] and discrete [28–30] adjacent matrix models, we continue to use the SIS model of single layer network:

⎛ p˙ i = (1 − pi (t)) ⎝1 −

∏(

⎞ ) 1 − λAji pj (t) ⎠ − µpi (t),

(1)

j

where pi (t) is the infected probability of node i at time t, naturally p = (pi )n×1 is the vector of infected probabilities. Parameter λ and µ are the infection of each edge and recovery rate of each node respectively, A = (Aij )n×n is the network adjacency matrix where n is network scale. Evidently, we can obtain that the model satisfies the Lipschitz condition and its detailed proof is shown in Appendix A. The linearization of model Eq. (1) is p˙ = (−µE + λA) p,

(2)

where E is identity matrix, then we obtain the stability condition of model Eq. (1) [27]:

ρ (A) <

1

δ

,

(3)

1 In this paper, the stabilities of systems all refer to stabilities of the origin. Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

3

Fig. 1. The diagram from the multilayer to double-layer network. Obviously, there exit sharp differences of network connectivity and edge density between the public network and private network. In addition, the neighbors of the same node in the each layer network are also different.

where ρ (A) is the spectral radius of A and δ = λ/µ, its detailed proof is in Appendix B. In a realistic multilayer network, the density of each contact network is different as well as the connectivity. For example, most networks are dense during the day and become sparse at night [31]. For constant infected and recovery rate, the subnetworks of multilayer network have different thresholds and its switched system has both stable and unstable subsystems. Compared with the duration of epidemics spreading, the evolution of network structure is a short process, which leads that the intermediate evolution between the diurnal and night networks can be ignored. Hence, we can use the periodic double layer network to establish the switched system instead of the multilayer network. Every layer networks have the same network scale without the individual birth and death. While, the density and connectivity of each network is different, then we call the diurnal network public network and the night one private network. Their adjacency matrices are Ac , Ap and network structure diagram is shown in Fig. 1. For individuals, their public and private relations always have different friends, so we assume that the public network and private network have this relationship Ac ◦ Ap = O in this paper, where ‘‘O’’ is zero matrix and ‘‘◦’’ is the Hadamard product of matrices [32]. In each period, these two networks alternately appear, their dwell time is Tc > 0, Tp > 0 and the period length is T = Tc + Tp . The period is less than average epidemics durations, T < 1/µ. We establish the periodic switched system

(

p˙ i = (1 − pi (t)) 1 −

∏(

(t)

))

1 − λAji pj (t)

− µpi (t),

(4)

where (t)

A

Ac , t ∈ [mT , mT + Tc ),

{ =

Ap , t ∈ [mT + Tc , (m + 1)T ),

and m is the integer periodic number. Similarly, the linear switched system is: p˙ = −µE + λA(t) p.

(

)

(5)

To prove the switched system stability, its linear switched system acts as a comparative system, and the comparison theorem and the property of quasi monotone system are the theoretical basis. If lim p(t) = 0,

t →∞

(6)

holds for the linear switched system, the switched system will also be stable, its detailed proof in Appendix C. Now, the key is to prove the stability of the linear switched system. Obviously, the solution of the linear switched system at time t after m periods can be written as p(t) = p(s + mT ) = ϕ m (T )p(s) = (ϕ (Tc )ϕ (Tp ))m p(s),

(7)

where 0 ≤ s < T , ϕ (Tc ) = exp((−µE +λAc )Tc ), ϕ (Tp ) = exp((−µE +λAp )Tp ) and p(s) is system initial value. For description convenience, switched system Eq. (4) consisting of stable subsystems are called as the first switched systems and consisting of both stable and unstable subsystems are the second switched systems. For the first switched systems, since all subsystems are stable, its linear switched system will eventually converge to zero, when the time is enough long. So the first switched systems are stable. In the analysis of the second switched systems, we introduce the relative dwell time σc = Tc /T , σp = Tp /T and σc + σp = 1, then can obtain

ϕ (Tc )ϕ (Tp ) ≤ exp((−µE + λ(σp Ap + σc Ac ))ξ T ),

(8)

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

4

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

where the parameter ξ ≥ 1. When the matrix Ac and Ap are commutative, the ‘‘=’’ is established and ξ = 1. The detailed analysis of this inequality is in the Appendix D. This condition means that when the spectral radius ρ (−µE + λ(σp Ap + σc Ac )) < 0 establishes, the linear switched system meets limm→∞ p(s + mT ) = 0, i.e. limt →∞ p(t) = 0 holds. The above inequation gives the threshold condition of the second switched systems

ρ (σc Ac + σp Ap ) <

1

δ

.

(9)

In order to reduce the number of variables, we introduce the ratio σ of the relative dwell time σc and σp , σ = threshold condition Eq. (9) can be equal to

ρ (Ac + σ Ap ) <

1+σ

δ

.

σp . σc

The

(10)

In multilayer networks, the switched system and its linear switched system still satisfy the comparison theorem in each layer network’s dwell time, as shown in the Appendix C. Meanwhile, its linear switched system also satisfies ϕ (T1 )ϕ (T2 ) · · · ϕ (Tn ) ≤ exp((−µE + λΣi σi Ai )ξ T ) just like the Eq. (8). Hence, the threshold condition with dwell time is also applicable to periodic multilayer networks. Similarly, the stability of the first switched systems in multilayer networks is same with the former; for the second switched systems in multilayer networks, the threshold condition is

ρ (Σi σi Ai ) <

1

δ

,

where Ai is the ith subnetwork adjacency matrix of multilayer networks and σi = is the dwell time of the ith network and Σi σi = 1.

(11) Ti T

is its relative dwell time, where Ti

3. Comparison with stochastic simulations and validation In this section, we only verify the threshold condition in the double-layer network by the experimental results between switched model Eq. (4) and stochastic simulations, then use the numerical simulation of switched model to confirm accuracy of condition Eq. (10) about the infected, recovery rate and dwell time. We take the Barabási–Albert (BA) and Erdős–Rényi (ER) double layer networks as the test networks. Since the communities are ubiquitous in real networks and cities [33,34], we use the LFR benchmark network with communities as the public network in the BA double layer networks, and the isolated cliques(complete graph) as the private network, where cliques’ scales satisfy the power-law distribution. For the ER double layer networks, the probability of edges connection in public network is larger than the probability in private network. Here, we use the subcritical network and connected network to construct the ER double-layer network. This method can ensure that the public network is connected and the private network is disconnected. First, we need explain whether the switched model Eq. (4) can reflect the real epidemic propagation, i.e. the infected numbers in equilibrium. Hence, we use these two kinds of double-layer networks to conduct the numerical and stochastic simulations. For convenience, each layer network with the same relative dwell time in all the simulations. The numerical simulation uses the Euler method. For stochastic simulation, there are two general algorithms: the Gillespie algorithm and event-driven algorithm [35]. The Gillespie algorithm is a well-known algorithm for simulating Markovian processes, while the event-driven approach is good at non-Markovian processes. After some experiments, we find that the event-driven algorithm is more suitable for switched networks. By improving this algorithm, we design the switched event-driven algorithm in Table 1. In the simulation, the initial infected numbers in stochastic simulations and infected probabilities are all equivalent. When the switched system is unstable, the initial infected probabilities of each node are all 0.002 in numerical simulations and accordingly the initial infected numbers is 2, randomly selected in each stochastic simulation. Similarly, the initial infected probabilities are 0.1 in numerical simulations and the initial infected numbers is 100 randomly selected in each stochastic simulation, when the switched system is stable. All comparison results show that switched model Eq. (4) can accurately estimate the expected number of the infected, no matter whether the threshold condition Eq. (9) is satisfied. Experiment results are shown in Fig. 2 and detailed parameters are as follows:

• Example 1. The experimental network is BA double-layer network and subnetworks’ node numbers are all 1000. The public network is generated by LFR benchmark. Its average degree is 20, and mixed parameter is 0.1. For the private network, every cliques consist of 2 to 7 nodes, the scales of cliques obey the power law and its power law exponent is 2. The spectral radius of each network is ρ (Ac ) = 24.1, ρ (Ap ) = 7 and ρ (Ac + Ap ) = 24.6543, and 1/δ is 8. Finally, the ratio of dwell time σ is 1. Its results are shown in Fig. 2(a). • Example 2. All parameters are the same as Example 1 except for parameter 1/δ , and its value is 15. Its results are shown in Fig. 2(b). • Example 3. The experimental network is ER double-layer network and subnetworks’ node numbers are all 1000. For the public and private network, the probabilities of edges connection are q1 = 0.03, q2 = 0.006, the spectral radiuses are ρ (Ac ) = 31.2428, ρ (Ap ) = 6.9369 and ρ (Ac + Ap ) = 36.9455, and 1/δ is 10. The ratio of dwell time is same as Example 1. Its results are shown in Fig. 2(c). • Example 4. All parameters are the same as Example 3 except for parameter 1/δ , and its value is 20. Its results are shown in Fig. 2(d). Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

5

Table 1 Event-driven algorithm simulating continuous-time SIS epidemics in periodic switched networks. Input: Public network Gc and Private network Gp , transmission rate per edge λ, recovery rate µ, index set of initial infected node(s), dwell time of each network Tc , Tp and maximum time tmax . Output: t: list of times and I: list containing number infected nodes at each time. function EventDriven_SIS(Gc , Gp , λ, µ, initial infected nodes, Tc , Tp , tmax ) times, S, I ← [0], [∥G∥], [0] source infected nodes with recovery time 0 Qa ← empty priority queue for u in Gc (Gp ) nodes do u.status ← susceptible u.pred_rec_time ← -1 for u in initial infected nodes do Event ← {node: u,time: 0,action: transmit} add Event to Q while Q is not empty do Event ← earliest remaining event in Q if Event.action is transmit then if Event.node.status is susceptible then process_trans_SIS(Gc , Gp , Event.node, Event.time, λ, µ, times, S, I, Q, Tc , Tp , tmax ) find_next_trans_SIS(Q, Event.source, Event.node, Event.t, λ, Tc , Tp , tmax ) else process_rec_SIS(Event.node, Event.t, S, I) return time, S, I function process_trans_SIS(Gc , Gp , Event.node, t, λ, µ, times, S, I, Q, Tc , Tp , tmax ) append times, S, and I with t, S.last-1, and I.last+1 u.status ← infected u.rec time ← t+exponential_variate(µ) if u.rec time < tmax then newEvent ← {node: u, time: u.rec time, action: recover} add newEvent to Q for v in Gc .neighbors(u) and Gp .neighbors(p) do find_next_trans_SIS(Q, t, λ, u, v , Tc , Tp , tmax ) function find_next_trans_SIS(Q, source, target, t, λ, Tc , Tp , tmax ) if target.rec time < source.rec time then if the edge [target,source] belongs to Gc then while transmission_time ∈ / [nT , nT + Tc ), n = 0, 1, 2, . . . do transmission_time = max(t, target.rec time)+△t, △t ∼ p1 (λ, ∆t)b else if the edge [target,source] belongs to Gp then while transmission_time ∈ / [nT + Tc , (n + 1)T ), n = 0, 1, 2, . . . do transmission_time = max(t, target.rec time)+△t, △t ∼ p2 (λ, ∆t)c if transmission_time < source.rec time then newEvent ← {node: target, time: transmission_time, action: transmit, source: source} push(Q, newEvent) function process_rec_SIS(u, times, S, I) append times, S, and I with t, S.last+1, and I.last-1 u.status ← susceptible a

Q is a priority queue and pops Event which has the earliest time.

b

λe The probability density function p1 (λ, ∆t) = ∑ ∞ ∫ nT +Tc

−λ∆t

n=0

c

nT

−λ∆t

λe The probability density function p2 (λ, ∆t) = ∑ ∞ ∫ (n+1)T n=0

nT +Tc

.

e−λt dt

.

e−λt dt

Last, we examine the accuracy of threshold condition Eq. (10) by numerical simulations of switched model. The doublelayer networks are same with networks used in Fig. 2, and the spectral radius ρ (Ac ) and ρ (Ap ) are constants. So the threshold condition is only related to the infected, recovery rate and relative dwell time, i.e. the parameter 1/δ and the ratio of dwell time σ . Based on the results in Fig. 3, we find that the threshold condition can accurately describe the stability of switched systems and estimate critical value of 1/δ and σ , especially for the second switched systems. 4. Conclusion and discussion Being able to provide a reliable and accurate epidemics threshold condition for a spreading process in multilayer networks is especially important. In this paper, we have obtained a threshold condition in double layer network, then ∑ extend it to periodic multilayer networks. Actually, the threshold condition ρ ( σi Ai ) < 1/δ reveals the quantitative relationship among the dwell time of each subnetwork and the traditional parameters, such as the network spectral radius (or average degree), infection and recovery rate. Meanwhile, it also shows that the dwell time of each layer network has a linear effect on the spreading ability of multilayer networks, because the spectral radius satisfies the homogeneity of norm. Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

6

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

Fig. 2. Simulation results between switched model and event-driven algorithm.

This can’t be obtained from researches in static networks and equal-interval multilayer networks. And it may provide a theoretical guidance for disease prevention strategies about time factor, which is the practical significance of our study. Moreover, our theoretical result may have some applications. For instance, close contacts between individuals in the public network are sometimes unavoidable, such as the respiratory epidemics. In various public transports, cinemas, stations, airports and other places with high density of people, crowded environment and narrow space make its spectral radius much larger than the epidemic threshold. If we can prevent the disease from spreading to a significant numbers, there are three key points (in no particular order) : the first one is to thin the network structure, the second one is to increase the private network effect time, the last one is to increase the recovery rate and decrease the infection rate. The first two are nonpharmaceutical interventions, while the last one needs to improve medical means. Medical research is what humans have been doing for a long time, but reducing travel times is also a time-honored emergency strategy to avoid catching some epidemics, which is consistent with the role of dwell time. Recently, many studies have tried to change the network structure to affect the spreading characteristics, but the network dwell time may be an important factor behind these measures. From the perspective of statistical ensemble, the dwell time of each subnetwork corresponds to its occurrence probability. Naturally, reducing the probability of some dense networks in the ensemble helps the spreading of disease. In addition, It will be valuable researches about the minimum spectral radius ) (∑ to prevent ρ σi Ai of multilayer networks and finding out the threshold condition with dwell time under different infected rates in each network. Similarly, whether we can get a threshold condition in the N-intertwined SIR or SIRS model to study the scale of spreading is also anticipated. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by National Natural Science Foundation of China under Grant Nos. 61873154. We thank the anonymous reviewers for their useful comments. Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

7

Fig. 3. The critical property of the threshold condition is studied in simulations, and the basic parameters of each type networks are same with the ones in Fig. 2. The initial infected probabilities of every nodes are all 0.002. (a)(c) when the ratio of dwell time σ is 1, we study the change of infected number with 1/δ in each type double layer networks. (b)(d) we do similar researches with the ratio of dwell time σ , when 1/δ is 20. All results show that the threshold condition Eq. (10) can accurately predict the epidemics outbreak in switched networks.

Appendix A. The proof of Lipschitz condition To prove the Lipschitz condition of the model Eq. (1), we need to give its matrix form p˙ = (E − M ) (u − p) − µp,

(A.1)

where u is a column vector, its elements are all 1 and dimension is n. The matrix M is

⎛

n ∏ (

1 − λAj1 pj (t) ⎜ ⎜ j=1 ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎝ 0

⎞ )

0

··· ..

.

.. .

⎟ ⎟ ⎟ ⎟ .. ⎟ . ⎟ n ∏( ) ⎟ ⎠ 1 − λAjn pj (t) .

(A.2)

j=1

The equivalent of model (A.1) is

⎛ p˙ = ⎝E −

⎞ ( )) E − diag λA·j ◦ p ⎠ (u − p) − µp,

n ∏ (

(A.3)

j=1

where ◦ is the Hadamard product for matrices [32]. Let f (p) = p˙ , if f (p) satisfies the Lipschitz condition, we only need to prove

∥f (p1 ) − f (p2 )∥ ≤ L∥p1 − p2 ∥.

(A.4)

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

8

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

The brief proof is as follows, we substitute p1 , p2 into the left of Eq. (A.4), then obtain 1

2

∥f (p1 ) − f (p2 )∥ = ∥(1 + µ)(p2 − p1 ) + Π p (u − p2 ) − Π p (u − p1 )∥, ( )) ( )) ∏n ( ∏n ( 2 1 where Π p = j=1 E − diag λA·j ◦ p1 , Π p = j=1 E − diag λA·j ◦ p2 . Then the inequation Eq. (A.4) becomes 2

1

∥f (p1 ) − f (p2 )∥ ≤ ∥(1 + µ)(p2 − p1 )∥ + ∥Π p (u − p2 ) − Π p (u − p1 )∥ 2

2

2

1

= ∥(1 + µ)(p2 − p1 )∥ + ∥Π p (u − p2 ) − Π p (u − p1 ) + Π p (u − p1 ) − Π p (u − p1 )∥ p2

p2

p2

1

≤ (1 + µ)∥p2 − p1 ∥ + ∥Π (u − p2 ) − Π (u − p1 )∥ + ∥Π (u − p1 ) − Π p (u − p1 )∥ p2

≤ (1 + µ)∥p2 − p1 ∥ + ∥Π ∥∥p2 − p1 ∥ + ∥(Π

p2

p1

− Π )(u − p1)∥.

(A.5)

Since 2

1

2

1

∥(Π p − Π p )(u − p1)∥ ≤ ∥(Π p − Π p )u∥ ≤ ∥p2 − p1 ∥,

(A.6)

the Eq. (A.5) meets

∥f (p1 ) − f (p2 )∥ ≤ (1 + µ)∥p2 − p1 ∥ + ∥p2 − p1 ∥ + ∥p2 − p1 ∥,

(A.7)

so there is a L = 3 + µ, satisfying the inequation Eq. (A.4). Appendix B. The epidemic threshold in single-layer network The threshold Eq. (3) of linear system is also effective to model Eq. (1). To prove the same stability of these two models, the model Eq. (1) need be rewritten in the expansion form by: p˙i = −µpi − (1 − pi )

ki ∑

(l)

(−λ)l Σi , ki ≥ 1,

(B.1)

l=1 (l)

where Σi =

∑

i1 ̸ =i2 ̸ =···̸ =il ∈N(i)

pi1 pi2 · · · pil is the sum of products, consisting of

(k ) i

l

products of l different neighbors infected

probabilities, and N(i) is the set of neighbors of node i, where l = 1, 2, . . . , ki . Thus the model Eq. (1) can be regarded as the disturbed system of linear system: p˙ = (−µE + λA) p + g(p),

(B.2)

where

⎛

⎞

g1 (p)

⎜ . ⎟

g(p) = ⎝ .. ⎠ , gn (p)

(B.3)

and

⎧ ki ∑ ⎪ ⎨ (1) −λpi Σi(1) − (1 − pi ) (−λ)l Σi gi (p) = ⎪ l=2 ⎩

, ki ≥ 2

(B.4)

.ki = 1

0

Now we just need to prove the Poincaré-Lyapunov theory [25] lim

p→0

∥g(p)∥ = 0. ∥p∥

(B.5)

By the positivity and triangle inequality, we have

√ g12 + g22 + · · · + gn2 ∥g(p, t)∥ 0 ≤ = √ ∥p∥ p21 + p22 + · · · + p2n ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ n ⏐ ⏐ g1 + g2 + · · · + gn ⏐ ∑ ⏐ ⏐ gi ⏐ ⏐ ⏐ ⏐, √ ≤ ⏐√ ≤ ⏐ ⏐ ⏐ ⏐ p2 + p2 + · · · + p2n ⏐ i=1 ⏐ p2 + p2 + · · · + p2n ⏐ 1

2

1

2

then if ki ≥ 2, we can get the inequality Eq. (B.7). ⏐ ⏐ ⏐ ⏐ ∑ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ −λpi Σi(1) − (1 − pi ) kl=i 2 (−λ)l Σi(1) ⏐ gi ⏐ =⏐ ⏐ ⏐√ √ ⏐ 2 ⏐ ⏐ ⏐ 2 2 2 ⏐ p + p2 + · · · + p2n ⏐ ⏐ ⏐ p + p + · · · + pn 1

2

1

(B.6)

(B.7)

2

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

9

⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∑ ∑ki l (1) ⏐ ⏐ ⏐ ⏐ ⏐ ki ⏐ (1) ⏐ ⏐ ⏐ pi kl=i 2 (−λ)l Σi(1) ⏐ ∑ ⏐ ⏐ λ Σ λ pi Σi i l = 2 ⏐√ ⏐ + ⏐√ ⏐+ ⏐ ≤ ⏐⏐ √ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ p2 + p2 + · · · + p2n ⏐ ⏐ p2 + p2 + · · · + p2n ⏐ l=2 ⏐ p2 + p2 + · · · + p2n ⏐ 1 2 1 2 1 2 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ki ⏐ ki ⏐ (l) l ⏐ λpi Σi(1) ⏐ ∑ ⏐ pi λl Σi(l) ⏐ ∑ ⏐ ⏐ λ Σ i ⏐ √ ⏐√ ⏐ ⏐+ ⏐+ ≤ ⏐⏐ √ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ p2i ⏐ p2i ⏐ p21 + p22 + · · · + p2n ⏐ l=2 ⏐ l=2 ⏐ ⏐ ⏐ ⏐ ki ⏐ ki ⏐ ⏐ ⏐ ⏐ ⏐ λl Σi(l) ⏐ (1) ⏐ ∑ ⏐ l (l) ⏐ ∑ ⏐⏐ ⏐. √ = ⏐λΣi ⏐ + ⏐λ Σi ⏐ + ⏐ 2 ⏐ p1 + p22 + · · · + p2n ⏐ l=2 ⏐ l=2 When p → 0, it tends to zero too. Consequently, Eq. (B.5) holds by squeeze rule. Appendix C. The stability of switched system The key of this proof is to illustrate that if limt →∞ pˆ (t) = 0 holds, we can also obtain limt →∞ p(t) = 0, where pˆ (t) and p(t) are the state variables of linear and nonlinear switched systems after m periods, t = s + mT and 0 ≤ s < T . The linear switched system is p˙i = −µpi + λ

∑

(t)

Aij pj ,

(C.1)

j

and the nonlinear switched system is

⎛ ∏(

p˙i = (1 − pi (t)) ⎝1 −

⎞ ) (t) 1 − λAji pj (t) ⎠ − µpi (t),

(C.2)

j

Now we just need to prove that the system Eqs. (C.1) and (C.2) are quasi monotone systems and satisfy the comparison theorem in the public or private network dwell time for m = 0, 1, 2, . . ., which ensures p(t) ≤ pˆ (t) is established when the initial value p(t0 ) ≤ pˆ (t0 ) in the beginning of the public or private network working. Obviously, the system Eq. (C.1) is a quasi monotone system. For the system Eq. (C.2), we take the public network as an example, and s + mT ≤ t < s + mT + Tc . Let

⎛

⎞ ) (t) 1 − λAji pj (t) ⎠ − µpi (t),

∏(

fi = (1 − pi (t)) ⎝1 −

(C.3)

j

then

( ) ∂ fi −p = (1 − pi ) 1 + λΠi j ≥ 0, ∂ pj

(C.4)

where

Πi =

∏

(1 − λpj ) = 1 +

j∈N(i)

ki ∑

(l)

(−λ)l Σi

(C.5)

l=1

= 1 − λΣi(1) +

ki ∑

(l)

(−λ)l Σi , ki ≥ 2,

l=2

−pj

−pj

and Πi meets Πi = (1 −λpj )Πi . Based on the Eq. (C.4), the system Eqs. (C.1) and (C.2) are all quasi monotone systems. Then, it just need to prove that they satisfy the comparison theorem, namely the following inequality Eq. (C.6) is established based on the nonlinear terms of Eq. (B.1)

− λpi Σi(1) − (1 − pi )

ki ∑

(l)

(−λ)l Σi ≤ 0,

(C.6)

l=2

which is equivalent to

λpi Σi(1) + (1 − pi )

ki ∑

(l)

(−λ)l Σi ≥ 0.

(C.7)

l=2

Since the 0 ≤ Πi ≤ 1, the following inequality relation can be derived from Eq. (C.5)

λ

(1) Σi

−

ki ∑

(l)

(−λ)l Σi = 1 − Πi ≥ 0,

(C.8)

l=2

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

10

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

thus the Eq. (C.7) can be converted to

λpi Σi(1) + (1 − pi )

ki ∑

(l)

(−λ)l Σi

l=2

≥

ki ∑

(l)

(−λ)l Σi pi + (1 − pi )

l=2

ki ∑

(l)

(−λ)l Σi

(C.9)

l=2 ki ∑

(l)

(−λ)l Σi ,

=

l=2

∑ki

l

and l=2 (−λ) function

(l) Σi

(1)

hi = λΣi

= λΣi(1) − 1 + Πi , so the proof of Eq. (C.6) is translated into the positivity of the following multivariate

− 1 + Πi ,

(C.10)

and its partial derivative of node j ∈ N(i) is

∂ hi −p = λ − λΠi j ≥ 0, ∂ pj Clearly, hi ≥ hi (p)|p=0 = 0, and the Eq. (C.6) holds. So in the public network working dwell time, if the initial value p(t0 ) ≤ pˆ (t0 ), p(t) ≤ pˆ (t) will always hold. This conclusion will also hold for the private network with s + mT + Tc ≤ t < s + (m + 1)T . Therefore, if the solution of linear switched system limt →∞ pˆ (t) = 0, we can also obtain limt →∞ p(t) = 0. Appendix D. The threshold condition of second switched systems In Section 2, we have ϕ (Tc ) = exp((−µE + λAc )Tc ), ϕ (Tp ) = exp((−µE + λAp )Tp ) and

ϕ (Tc )ϕ (Tp ) = exp((−µE + λAc )Tc ) exp((−µE + λAc )Tc ), ( ) = exp(−µE(Tc + Tp )) exp(Ac Tc ) exp(Ap Tp ) .

(D.1)

Since the network corresponding to adjacency matrix Ac + Ap is connected, so exp(Ac Tc + Ap Tp ) > 0. Obviously, for the exp(Ac Tc ) exp(Ap Tp ), we can always find a suitable parameter ξ ≥ 1 and obtain exp(Ac Tc ) exp(Ap Tp ) ≤ exp(ξ (Ac Tc + Ap Tp )).

(D.2)

If Ac and Ap are commutative, the ‘‘=’’ is established and parameter ξ = 1. Thus

ϕ (Tc )ϕ (Tp ) ≤ exp(−µE(Tc + Tp )) exp(ξ (Ac Tc + Ap Tp )), ( )ξ ≤ exp(−µE(Tc + Tp )) exp(Ac Tc + Ap Tp ) , ( ) = exp ξ (−µE(Tc + Tp ) + Ac Tc + Ap Tp ) .

(D.3)

We take the relative dwell time σc = Tc /T , σp = Tp /T into the former formula, then can obtain

ϕ (Tc )ϕ (Tp ) ≤ exp((−µE + λ(σp Ap + σc Ac ))ξ T ).

(D.4)

References [1] G.F. De Arruda, E. Cozzo, T.P. Peixoto, F.A. Rodrigues, Y. Moreno, Disease localization in multilayer networks, Phys. Rev. X 7 (1) (2017) 011014. [2] J. Jiang, T. Zhou, Resource control of epidemic spreading through a multilayer network, Sci. Rep. 8 (1) (2018) 1629. [3] Z. Wang, D.W. Zhao, L. Wang, G.Q. Sun, Z. Jin, Immunity of multiplex networks via acquaintance vaccination, Europhys. Lett. 112 (4) (2015) 48002. [4] Z. Jin, S. Li, X. Zhang, J. Zhang, X.L. Peng, Epidemiological modeling on complex networks, Underst. Complex Syst. 73 (2016) 51–77. [5] X.L. Peng, X.J. Xu, M. Small, X. Fu, Z. Jin, Prevention of infectious diseases by public vaccination and individual protection, J. Math. Biol. 73 (6–7) (2016) 1–34. [6] H.F. Zhang, P.P. Shu, Z. Wang, M. Tang, M. Small, Preferential imitation can invalidate targeted subsidy policieson seasonal - influenza diseases, Appl. Math. Comput. 294 (C) (2017) 332–342. [7] R. Li, W. Wang, Z. Di, Effects of human dynamics on epidemic spreading in côte divoire, Physica A 467 (2017) 30–40. [8] R. Li, W. Wang, P. Shu, H. Yang, L. Pan, A. Cui, M. Tang, Review of threshold theoretical analysis about epidemic spreading dynamics on complex networks, Complex Syst. Complex. Sci. 13 (1) (2016) 1–39. [9] S. Boccaletti, G. Bianconi, R. Criado, C.I.D. Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña Nadal, Z. Wang, M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep. 544 (1) (2014) 1–122. [10] S.F. Darabi, C. Scoglio, Competitive epidemic spreading over arbitrary multilayer networks., Phys. Rev. E 89 (6) (2014) 062817. [11] L.G. Zuzek, H.E. Stanley, L.A. Braunstein, Epidemic model with isolation in multilayer networks, Sci. Rep. 5 (2015) 12151. [12] C. Buono, L.A. Braunstein, Immunization strategy for epidemic spreading on multilayer networks, Europhys. Lett. 109 (2) (2015) 26001.

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.

N.-N. Wang, Z. Jin, Y.-J. Wang et al. / Physica A xxx (xxxx) xxx

11

[13] J.Q. Kan, H.F. Zhang, Effects of awareness diffusion and self-initiated awareness behavior on epidemic spreading - an approach based on multiplex networks, Commun. Nonlinear Sci. Numer. Simul. 44 (2017) 193–203. [14] R. Li, M. Tang, P.-M. Hui, Epidemic spreading on multi-relational networks, Acta Phys. Sin. 62 (16) (2013) 168903. [15] J.M. Read, M.J. Keeling, Disease evolution on networks: the role of contact structure, Proc. Biol. Sci. 270 (1516) (2003) 699–708. [16] C.O. Buckee, K. Koelle, M.J. Mustard, S. Gupta, The effects of host contact network structure on pathogen diversity and strain structure, Proc. Natl. Acad. Sci. USA 101 (29) (2004) 10839. [17] M.C. Gates, M.E.J. Woolhouse, Controlling infectious disease through the targeted manipulation of contact network structure, Epidemics 12 (2015) 11–19. [18] R. Li, P. Richmond, B.M. Roehner, Effect of population density on epidemics, Physica A 510 (2018) 713–724. [19] Z. Xiang, W. Xiang, Stability analysis of switched nonlinear systems based on dynamical dwell time approach, in: Proceedings of 2008 IEEE International Conference on Networking, Sensing and Control, in: IEEE International Conference on Networking Sensing and Control, vols. 1 and 2, IEEE Syst, Man & Cybernet Soc, 2008, p. 1758+. [20] G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach, Internat. J. Systems Sci. 32 (8) (2001) 1055–1061. [21] X. Wu, L. Yan, W. Zhang, Y. Tang, Stability of stochastic nonlinear switched systems with average dwell time, J. Phys. A 45 (8) (2012). [22] R. Guo, Stability analysis of a class of switched nonlinear systems with an improved average dwell time method, Abstr. Appl. Anal. 2014 (1) (2014) 1–8, 2014, (2014-3-5). [23] M.R. Sanatkar, Epidemic threshold of an sis model in dynamic switching networks, IEEE Trans. Syst. Man Cybern. Syst. 46 (3) (2016) 345–355. [24] B.A. Prakash, H. Tong, N. Valler, M. Faloutsos, C. Faloutsos, Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms, in: Machine Learning and Knowledge Discovery in Databases, PT III, vol. 6323, 2010, pp. 99–114. [25] F. Verhulst, Nonlinear differential equations and dynamical systems, Universitext 1 (5) (1990) 256–264. [26] P.V. Mieghem, The n-intertwined sis epidemic network model, Computing 93 (2–4) (2011) 147–169. [27] S. Xu, W. Lu, L. Xu, Z. Zhan, Adaptive epidemic dynamics in networks:thresholds and control, ACM Trans. Auton. Adapt. Syst. 8 (4) (2014) 1–19. [28] P. Van Mieghem, J. Omic, R. Kooij, Virus spread in networks, IEEE ACM Trans. Netw. 17 (1) (2009) 1–14. [29] E. Valdano, L. Ferreri, C. Poletto, V. Colizza, Analytical computation of the epidemic threshold on temporal networks, Phys. Rev. X 18 (3) (2015) 503–512. [30] E. Valdano, M.R. Fiorentin, C. Poletto, V. Colizza, Epidemic threshold in continuous-time evolving networks, Phys. Rev. Lett. 120 (6) (2018) 068302. [31] S. Pei, S. Kandula, W. Yang, J. Shaman, Forecasting the spatial transmission of influenza in the united states., Proc. Natl. Acad. Sci. USA 115 (11) (2017) 2752. [32] G.P.H. Styan, Hadamard products and multivariate statistical analysis, Linear Algebra Appl. 6 (1973) 217–240. [33] R. Li, L. Dong, J. Zhang, X. Wang, W.X. Wang, Z. Di, H.E. Stanley, Simple spatial scaling rules behind complex cities, Nature Commun. 8 (1) (2017) 1841. [34] N.N. Wang, Z. Jin, X.L. Peng, Community detection with self-adapting switching based on affinity, Complexity 2019 (6946189) (2019) 1–16. [35] I.Z. Kiss, J.C. Miller, P.L. Simon, Mathematics of Epidemics on Networks, Vol. 598, Springer, 2017.

Please cite this article as: N.-N. Wang, Z. Jin, Y.-J. Wang et al., Epidemics spreading in periodic double layer networks with dwell time, Physica A (2019) 123226, https://doi.org/10.1016/j.physa.2019.123226.