arX

iv:2

109.

1113

2v1

[ph

ysic

s.so

c-ph

] 2

3 Se

p 20

21

Simple and realistic models for efficient epidemic control on multiplex networks

Minsuk Kim and Soon-Hyung Yook∗

Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea

(Dated: September 24, 2021)

SUMMARY PARAGRAPH

The real epidemic spreading has generally two different types of transmission routes. One is the random anonymousinfection [1] and the other one is the transmission through regular and fixed contacts [2]. If the infectious diseasehas high mortality and there is no available vaccine or medicine, then many health authorities rely on the non-pharmaceutical interventions (NPIs) along the traceable fixed contacts, such as isolation of the infected [3]. In ourstudy, such realistic situations are implemented by the susceptible-infected-recovered (SIR) model [4] with isolationon multiplex networks [5]. The multiplex networks are composed of random interaction layer and fixed interactionlayer. We quantitatively compare the efficiency of two isolation protocols imposed on the SIR dynamics. One of themis the most popular protocol adopted by many health organizations over the globe. From the numerical simulationswe find that the isolation of the second nearest neighbors of the hospitalized individuals on the layer for regular andfixed contacts significantly reduces both the final epidemic size and the number of the isolated per unit time. Ourfinding suggests a better NPI for any type of epidemic even though the contact tracing is only partially available.

INTRODUCTION

An outbreak of a new disease, such as the bubonic plague pandemic in the 14th century [6], the 1918 influenzapandemic [7], and the recent outbreak of severe acute respiratory syndrome [8], has been a large threat throughouthuman history. Despite great advances in medical science and pharmacology, immediate use of an effective vaccineor antiviral drug is not always possible when new infectious diseases emerge. For example, due to the absence ofvaccines or antiviral drugs for new severe acute respiratory coronavirus 2 (SARS-CoV-2) during the early stage ofthe coronavirus disease 2019 (COVID-19) pandemic, more than 172 million people have been infected and has causedmore than 3.7 million deaths until March 2021 [9]. Thus, finding an efficient NPI is crucial to mitigate the pandemicsituation for new emerging diseases.The best NPI for a new disease is a perfect lockdown, under which all individuals are strictly isolated. For example,

during the early stage of the COVID-19 pandemic, strict lockdown measures had been successfully applied in mainlandChina and many European countries [10, 11]. However, the strict lockdown policy is not sustainable if the pandemicperiod continues long enough to cause a severe recession of economic activity and to increase social fatigue [12, 13].Thus, it is necessary to find NPIs that minimize the impact on the social and economic systems. The efficacy ofvarious NPIs has been intensively studied based on real data and theoretical models to alleviate the recent pandemicsituations [1, 14–28].Among the various NPIs, the quarantine of the infected individuals and their contacts is one of the most intuitive

measures and commonly shared by many health authorities over the world [3]. Thus, the isolation of the infected[23, 28] and tracing the contacts [14, 29, 30] are two important factors. However, if the infection from asymptomaticand pre-symptomatic patients are potential transmission routes like the COVID-19 case [31, 32], then finding thecontacts with such patients are not trivial. Furthermore, when airborne transmission is another important routeof spreading [33], the tracing becomes much harder due to the random anonymous contacts through the publiclyopened environments [1]. In this study, such random anonymous transmission is implemented by the double-layeredmultiplex networks (DLMNs) [5]. To model the situation with the pre-symptomatic and asymptomatic transmissions,we assume that individuals in our models have one of the following disease states, susceptible (S), infected (I), andrecovered (R) [4]. Furthermore, the contact tracing probability and isolation states are introduced to account formore realistic situations in our model. As we will show we first model the most popular protocol for NPI adoptedby many health authorities, and also introduce a reinforced protocol model. From the quantitative comparison of thetwo models, we suggest a simple and efficient NPI strategy for epidemic controls of any emerging infectious diseaseby using only the known topology of the fixed interaction layer.

∗ syook@khu.ac.kr

2

MODEL

Individuals are denoted by nodes and the interactions between them are represented by links in the DLMN. LetF and W be the two layers in the DLMN (see Fig. 1(a)). On F each node is connected with randomly chosen knodes drawn from a given degree distribution PF (k). The topology of networks on F does not change in time. At thesame time, each node interacts with k′(t) random nodes on W , where k′(t) is drawn from another degree distributionPW (k′) at each time t. Thus, the interaction topology on W changes at each t. Under severe epidemic situations,the government tries to cordon people off public facilities, and each individual refrains from social activities. Thus,the number of contacts of each individual is significantly restricted and homogeneous. To generate such homogeneousinteraction structures, we use the Poisson distribution for both PF (k) and PW (k′) [5, 34, 35] (see ’Underlying networks’in Methods).The state of each node at time t in the DLMN is described by a two-component variable σ = (σ1, σ2). σ1 has

one of the three disease states: S, I, and R. σ2 denotes the state for isolation measure. When the disease causessevere symptoms, two isolated states are possible: i) self-isolation at home when an individual feels mild symptoms orrecognizes a suspicious contact but has not been confirmed yet, and ii) hospitalization by the health authority whenthe patient is confirmed to be infected. If an individual is not isolated then it is in the unisolated state. Thus, σ2

can be one of the following states: self-isolated (X), hospitalized (H), and unisolated (U) (see Fig. 1(b)). Since wecannot trace the contacts on W due to the random anonymity, the self-isolation for the suspicious contacts can beapplied only to F .Depending on the range of the self-isolation, we introduce two intervention protocols, the basic isolation protocol

(BIP) and the reinforced isolation protocol (RIP). Under the BIP only the confirmed patients and the one who hasdirect contact with the confirmed patient are isolated. This is the most popular quarantine protocol adopted by manyhealth authorities [36]. However, the fraction of the household or workplace infection cannot be ignored in someinfectious diseases, for example, the household infection is more than 15% for COVID-19 [2]. Such household andworkplace infection can be caused by a self-isolated node. Thus controlling such local contacts is another importantfactor to mitigate the transmission. For the preemptive protection of the susceptible, in the RIP model if a node ishospitalized, then its first and also second nearest neighbors on F are isolated.In our models, each infected node transmits the disease to the connected susceptible nodes with the probability βF

(βW ) on F (W ). With the probability θX (θH) the nodes are self-isolated (hospitalized) for the isolation (hospitaliza-tion) period tX (tH). The infected nodes are recovered after the recovery time tR (see ’Basic Isolation Protocol’ and’Reinforced Isolation Protocol’ in Methods). We set the size of each layer as N = 100, 000. Since we are interestedin the control of the severe epidemic outbreak, we use βF = βW ≡ β(= 0.2) and tR = 6 to guarantee that thewhole system becomes infected without any intervention. In our model, the strength of the intervention measures iscontrolled by four parameters, θH , tH , θX , and tX . For simplicity, we assume that θH = θX ≡ θ∗ and tH = tX ≡ t∗.Thus, we use only two control parameters θ∗ and t∗ in the following simulations.

RESULTS

The fraction of nodes in each state

Let ρm(t) (m ∈ {S, I, R}) be the fraction of nodes whose disease states is σ1 = m at time t, regardless of σ2. Ifm ∈ {U,H,X}, then ρm(t) represents the fraction of nodes with σ2 = m. The peak of ρm(t) for each state m isdenoted by ρpeakm . In Figs. 2(a)-(d) we show {ρm(t)}’s under the BIP for the various values of parameters (θ∗, t∗). For

small θ∗(= 0.3), we find that ρpeakI (t) > 0.9 followed by ρpeakH (& 0.8), regardless of t∗ (Figs. 2(a), (b)). ρS(t) rapidly

decreases and reaches ρS ≈ 0 for t > 11. The value of ρpeakX (< 0.2) is relatively small. Thus, ρR(t → ∞) ≃ 1 when

θ∗ is small. On the other hand, as θ∗ increases, ρpeakI is drastically suppressed as well as ρpeakH (Figs. 2(c), (d)). Asa result, ρR(t → ∞) is reduced to ρR(t → ∞) ≃ 0.6 ∼ 0.7 for θ∗ = 0.9. For both values of θ∗ displayed in Fig. 2, t∗

only affects the behavior of ρH and ρX (the population of the isolated nodes), i.e., ρpeakH (t; t∗ = 4) < ρpeakH (t; t∗ = 10)

and ρpeakX (t; t∗ = 4) < ρpeakX (t; t∗ = 10). Note that ρpeakH is comparable with ρpeakI for all values of (θ∗, t∗), and ρH(t)becomes wider as t∗ increases. This means that the hospitalized period becomes longer without any significant changein the final epidemic size, ρR(t → ∞), as t∗ increases for all θ∗. Thus, increasing t∗ without the improvement oftraceability causes an overload on the medical system by making patients be hospitalized for a longer period.Figs. 2(e)-(h) show {ρm(t)}’s for the RIP model. When θ∗ . 0.3, {ρm(t)}’s for the RIP model show almost the

similar behavior with those for the BIP, but ρS(t → ∞) for the RIP is slightly larger than that for the BIP. Since

additional nodes are self-isolated under the RIP, ρpeakX increases compared with that for the BIP with the same (θ∗,t∗). However, we find that ρH for the RIP becomes much smaller than that for the BIP. This effect becomes more

3

drastic for θ∗ > 0.3. For example, ρR(t → ∞) and ρpeakI (t) significantly decrease to ρR(t → ∞) = 0.5 ∼ 0.6 and

ρpeakI ≈ 0.1 for the RIP with θ∗ = 0.9. {ρm(t)}’s for other values of (θ∗, t∗) are presented in the Extended Data Fig.2 and Extended Data Fig. 3. The results indicate that the collapse of the medical systems can be avoided underthe RIP if we trace the contacts with sufficiently high accuracy. In addition, we find ρI , ρH , and ρX oscillate withdecreasing amplitude under the RIP as θ∗ increases. This suggests that even though there is a rapid decrease in ρI(t)after its first peak when θ∗ is sufficiently large, it is still possible to be followed by successive multiple peaks of ρI(see also Extended Data Fig. 8).

The effective reproduction number

To quantify the efficacy of intervention measures, we estimate the instantaneous effective reproduction number,Re(t), at t. For a practical purpose, we define Re(t) as

Re(t) =Nnew

I (t)

NI(t− 1), (1)

where NnewI (t) is the number of new infected nodes at t and NI(t) is the number of infected nodes at t [37, 38]. Thus

Re(t) represents a metric to quantify how many nodes are newly infected by the existing infected nodes at each t.In Figs. 3(a),(b), we show Re(t) for the BIP and RIP with θ∗ = 0.9 and t∗ = 2 ∼ 12. The dashed line denotes

Re(t) without intervention. As shown in Fig. 3(a), Re(t) for the BIP rapidly decreases when t . 4 and shows aplateau followed by another rapid drop, regardless of t∗. Re(t) = 1 at t ≈ 11 and Re(t) < 1 for t > 11. When t > 15Re(t) approaches to Re(t) ≈ 0. On the other hand, Re(t) for θ

∗ = 0.9 under the RIP decreases more drastically andRe(t) < 1 for t & 5 as shown in Fig. 3(b). When t > 20, Re(t) oscillates with decreasing amplitudes and approachesRe ≈ 0 under the RIP.To investigate how θ∗ affects the epidemic spreading, we also measure Re(t)’s for various θ∗ when t∗ is fixed. In

Figs. 3(c),(d), as an example, we display Re(t)’s for t∗ = 10. Since θ∗ denotes traceability, Re(t) should decrease asθ∗ increases for both protocols as shown in Figs. 3(c), (d). Note that when θ∗ < 0.3, the difference between the BIPand RIP is not noticeable. However, if θ∗ > 0.3, then Re(t) for the RIP becomes much smaller than those for theBIP. From the data in Fig. 3, we find that increasing θ∗ is more important than increasing t∗ (See also ExtendedData Figs. 4 and 5).The rapid drop of Re(t) under both protocols has two different origins depending on θ∗. For θ∗ < 0.3 due to the

large infection of the early stage, there does not remain a sufficient number of susceptible nodes for t > 11 (see Figs.2(a), (b), (e), (f)). On the other hand, if θ∗ > 0.3 then a significant amount of the susceptible is self-isolated, whichprotects the susceptible nodes before contact with the patients for t > 10 (see Figs. 2(c), (d), (g), (h)).

The final epidemic size and the number of isolated nodes per unit time

For a direct comparison between the two intervention protocols, we measure the difference, ∆R, of the final epidemicsizes between the BIP and RIP (see ‘Final epidemic size’ in Methods). Here, if ∆R > 0 then ρR for the BIP is largerthan that for the RIP. The data in Fig. 4(a) clearly shows that ∆R rarely depends on t∗. However, ∆R stronglydepends on θ∗. ∆R ≤ 0.05 for all t∗ when θ∗ < 0.3, while ∆R > 0.1 for θ∗ & 0.4 and ∆R increases as θ∗ increases.This means that the RIP significantly reduces the final epidemic size compared to the BIP when θ∗ ≥ 0.3. For themaximal traceability, θ∗ = 1, we find that ρR(t → ∞) for the RIP is reduced by 67% compared to that for the BIP.This corresponds to the 100% increase of ρS(t → ∞) under the RIP compared to the BIP (see Extended Data Fig.6). Thus, isolation of the possible suspicious contacts in advance by applying the RIP significantly reduces the finalepidemic size when θ∗ ≥ 0.3.In epidemic control, reducing the number of isolated individuals at each time step becomes another crucial factor

to minimize social and economic recession. For the quantitative analysis, we define the difference in the fractions ofthe isolated nodes per unit time, ∆XH (see ’Fraction of isolated nodes per unit time’ in Methods). By definition, if∆XH > 0 then more nodes are isolated under the BIP than the RIP. As shown in Fig. 4(b), for all values of θ∗, wefind that ∆XH ≈ 0 for t∗ ≤ 6. However, we find that ∆XH > 0 when t∗ > 6 and ∆XH increases as t∗ increases. Thus,t∗ affects only ρX and ρH per unit time for both models. Note that, even though ∆XH ≤ 0 for t∗ ≤ 6, ∆R increaseswith θ∗ and ∆R ≥ 0 as shown in Fig. 4(a)(see also Fig. 2). Therefore, the RIP more effectively controls the diseasespreading through the preemptive isolation of suspicious contacts with fewer isolated nodes per unit time than theBIP.

4

DISCUSSION

In summary, we model the NPI adopted by many health authorities over the world, and introduce a model forreinforced NPI. In these models the state of each individual is characterized by three disease states with additionalisolation states. Two different types of transmission routes observed in real world are implemented by the multiplexnetworks. By using numerical simulations, we compare the efficacy of the two models, BIP and RIP models, and findthat the RIP controls the spreading of disease more efficiently by reducing both the epidemic size and the averagenumber of isolated individuals per unit time, despite its simplicity. Especially, when the traceability is maximal, thefinal fraction of the susceptible nodes under the RIP increases by almost 100% (almost doubled) compared to thatunder the BIP. This indicates that the RIP significantly and efficiently protect the susceptible nodes through thepreemptive isolation of the possible contacts. Furthermore, since we do not assume any characteristic property of aspecific disease, we expect that the suggested models can be used as a general framework for modeling disease controlfor any real disease outbreak.

[1] Sneppen, K., Nielsen, B. F., Taylor, R. J., Simonsen, L. Overdispersion in COVID-19 increases the effectiveness of limitingnonrepetitive contacts for transmission control. Proc. Nat. Acad. Sci. 118 (2021).

[2] Park, Y. J. et al. Contact tracing during coronavirus disease outbreak, South Korea, 2020. Emerg. Infect. Dis. 26, 2465(2020).

[3] Parmet, W. E., Sinha, M. S. Covid-19—the law and limits of quarantine. N. Engl. J. Med. 382, e28 (2020).[4] Hethcote, H. W. The mathematics of infectious diseases. SIAM Rev. 42, 599 (2000).[5] Newman, M. E. Networks (Oxford university press, 2018).[6] Gould, G. M. Anomalies and curiosities of medicine (Blacksleet River, 1966).[7] Taubenberger, J. K., Morens, D. M. 1918 Influenza: the mother of all pandemics. Rev. Biomed. 17, 69 (2006).[8] Chan-Yeung, M.,Xu, R.-H. SARS: epidemiology. Respirology 8, S9 (2003).[9] WHO, WHO Coronavirus(COVID-19) Dashboard, https://covid19.who.int/ (2020).

[10] Ren, X. Pandemic and lockdown: a territorial approach to COVID-19 in China, Italy and the United States. Eurasian.Geogr. Econ. 61, 423 (2020).

[11] Lu, G. et al. COVID-19 in Germany and China: mitigation versus elimination strategy. Global health action 14, 1875601(2021).

[12] Harvey, N. Behavioral fatigue: real phenomenon, naıve construct, or policy contrivance?. Front. Psychol. 11, 2960 (2020).[13] Nicola, M. et al. The socio-economic implications of the coronavirus pandemic (COVID-19): A review. Int. J. Surg. 78,

185 (2020).[14] Ferretti, L. et al. Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing. Science

368 (2020).[15] Flaxman, S. et al. Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature 584, 257

(2020).[16] Perra, N. Non-pharmaceutical interventions during the COVID-19 pandemic: A review. Phys. Rep. (2021).[17] Maier, B. F., Brockmann, D. Effective containment explains subexponential growth in recent confirmed COVID-19 cases

in China. Science 368, 742 (2020).[18] Chan, L. Y. H., Yuan, B., Convertino, M. COVID-19 non-pharmaceutical intervention portfolio effectiveness and risk

communication predominance. Sci. Rep. 11, 1 (2021).[19] Thurner, S., Klimek, P., Hanel, R. A network-based explanation of why most COVID-19 infection curves are linear. Proc.

Nat. Acad. Sci. 117, 22684 (2020).[20] Nimmagadda, V., Kogan, O., Khain, E. Path-dependent course of epidemic: Are two phases of quarantine better than

one?. Europhys. Lett 132, 28003 (2020).[21] Mukhamadiarov, R. I. et al. Social distancing and epidemic resurgence in agent-based Susceptible-Infectious-Recovered

models, Sci. Rep. 11, 1 (2021).[22] Choi, K., Choi, H., Kahng, B. Covid-19 epidemic under the K-quarantine model: Network approach. Preprint at

https://arxiv.org/abs/2010.07157 (2020).[23] Arenas, A. et al. Modeling the spatiotemporal epidemic spreading of COVID-19 and the impact of mobility and social

distancing interventions. Phys. Rev. X 10, 041055 (2020).[24] Schlosser, F. et al. COVID-19 lockdown induces disease-mitigating structural changes in mobility networks. Proc. Nat.

Acad. Sci. 117, 32883 (2020).[25] Pastor-Satorras, R., Vespignani, A. Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63, 066117

(2001).[26] Newman, M. E. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002).[27] Balcan, D. et al. Multiscale mobility networks and the spatial spreading of infectious diseases. Proc. Nat. Acad. Sci. 106,

21484 (2009).[28] Zuzek, L. A., Stanley, H. E., Braunstein, L. A. Epidemic model with isolation in multilayer networks. Sci. Rep. 5, 1 (2015).

5

[29] Fraser, C., Riley, S., Anderson, R. M., Ferguson, N. M. Factors that make an infectious disease outbreak controllable.Proc. Nat. Acad. Sci. 101, 6146 (2004).

[30] Peak, C. M., Childs, L. M., Grad, Y. H., Buckee, C. O. Comparing nonpharmaceutical interventions for containing emergingepidemics. Proc. Nat. Acad. Sci. 114, 4023 (2017).

[31] Tong, Z.-D. et al. Potential presymptomatic transmission of SARS-CoV-2, Zhejiang province, China, 2020. Emerg. Infect.

Dis. 26, 1052 (2020).[32] Bai, Y. et al. Presumed asymptomatic carrier transmission of COVID-19. Jama 323, 1406 (2020).[33] Zhang, R. et al. Identifying airborne transmission as the dominant route for the spread of COVID-19. Proc. Nat. Acad.

Sci. 117, 14857 (2020).[34] Solomonoff, R., Rapoport, A. Connectivity of random nets. Bull. Math. Biol. 13, 107 (1951).[35] Erdos, P., Renyi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5, 17 (1960).[36] Webster, R. K. et al. How to improve adherence with quarantine: rapid review of the evidence. Public Health 182, 163

(2020).[37] Fraser, C. Estimating individual and household reproduction numbers in an emerging epidemic. PloS One 2, e758 (2007).[38] Cori, A., Ferguson, N. M., Fraser, C., Cauchemez, S. A new framework and software to estimate time-varying reproduction

numbers during epidemics. Am. J. Epidemiol. 178, 1505 (2013).

6

Fig. 1 | Schematic diagram for the SIR model with isolation on DLMNs. (a) The schematic diagram forinteraction on the DLMN. W and F layers are composed of the same set of nodes, i.e., the nodes at the ends of eachdotted line are identical. The yellow infected node chooses random partners on W at each time step (dashed arrows).Thus, if t1 6= t2 then it interacts with different nodes on W , while its interacting partner does not change on F (blacksolid lines). (b) The change of states under the BIP or RIP. Red, yellow, and green boxes denote the states for σ1,and white boxes represent the states for σ2.

7

� �� �� ��t

���

���

���

��

��

���

ρ(t)

���

ρSρIρRρHρX

� �� �� ��t

���

���

���

��

��

���

ρ(t)

���

� �� �� ��t

���

���

���

��

��

���

ρ(t)

� �

� �� �� ��t

���

���

���

��

��

���

ρ(t)

���

� �� �� �� ��t

���

���

���

��

��

���

ρ(t)

���

� �� �� �� ��t

���

���

���

��

��

���ρ(t)

���

� �� ��� ��� ���t

���

���

���

��

��

���

ρ(t)

���

� ��� ��� ���t

���

���

���

��

��

���

ρ(t)

���

Fig. 2 | The fraction of nodes in each state.(a)-(d) are {ρm(t)}’s (m ∈ {S, I, R,H,X}) under the BIP with (a) θ∗ = 0.3, t∗ = 4, (b) θ∗ = 0.3, t∗ = 10, (c)

θ∗ = 0.9, t∗ = 4, (d) θ∗ = 0.9, t∗ = 10. (e)-(h) shows {ρm(t)}’s under the RIP with (e) θ∗ = 0.3, t∗ = 4, (f) θ∗ = 0.3,t∗ = 10, (g) θ∗ = 0.9, t∗ = 4, (h) θ∗ = 0.9, t∗ = 10. β = 0.2 and tR = 6 are used in common. Each data is obtainedfrom 500 independent simulations by averaging the surviving samples at time t.

8

� �� �� ��t

�

�

�

�

Re(t)

�������� �� �����

� �� �� ��t

�

�

�

�

Re(t)

��t∗ =2

t∗ =4

t∗ =6

t∗ =8

t∗ =10

t∗ =12

� �� �� ��t

�

�

�

�

Re(t)

���

� �� �� ��t

�

�

�

�

Re(t)

���θ ∗ =0.3

θ ∗ =0.6

θ ∗ =0.9

θ ∗ =1.0

Fig. 3 | The effective reproduction number. Plot of Re(t) under the (a) BIP and (b) RIP when t∗ = 2 ∼ 12and θ∗ = 0.9 with β = 0.2, tR = 6. Plot of Re(t) for the (c) BIP and (d) RIP when t∗ = 10 and θ∗ = 0, 3 ∼ 1.0with β = 0.2, tR = 6. The black dashed curve denotes the case when the isolation protocol is absent. The red dottedhorizontal line depicts Re(t) = 1.

9

��� ��� ��� ��� �� ���θ ∗

���

���

���

���

���

∆R

��

t ∗ =2

t ∗ =4

t ∗ =6

t ∗ =8

t ∗ =10

t ∗ =12

��� ��� ��� ��� �� ���θ ∗

����

���

���

���

���

���

∆XH

���

Fig. 4 | Difference in the final epidemic size and the number of isolated nodes per unit time. Plot of(a) ∆R and (b) ∆XH against θ∗ for t∗ = 2 ∼ 12 when β = 0.2 and tR = 6. The black dashed horizontal line depicts(a) ∆R = 0 and (b)∆XH = 0.

10

METHODS

UNDERLYING NETWORKS

Let N be the number of nodes in a network. Each node belongs to the networks defined on both layers, F and W atthe same time. Thus we investigate the epidemics on multiplex networks. To construct a fixed random network on F ,we randomly select two nodes among N nodes and connect them if they are not linked. This process continues untilwe have L = N 〈k〉 /2 links on the F layer. Here 〈k〉 is the mean degree of the network on F . The degree distribution

of the obtained network on F , PF (k), is known to be the Poisson distribution, P (k) = 〈k〉ke−〈k〉/k! [5, 34, 35]. The

links on F do not change in time.

On the other hand, the topology of the network on W changes in time. Therefore, at each time t, a node i with

σi,1 = I randomly chooses k′i neighbors on W . k′i is drawn from the Poisson distribution, PW (k′) = 〈k′〉k′

e−〈k′〉/k′!.

We use the mean degrees 〈k〉 = 〈k′〉 = 8. The value of the mean degree only affects the epidemic threshold[5], anddoes not change the main conclusion of our study.

BASIC ISOLATION PROTOCOL (BIP)

To specify the update rule for each protocol, we introduce additional parameters T I , TX , and TH which correspondto the time of infection, self-isolation, and hospitalization, respectively. The state of each node at time t in the DLMN isdescribed by a two-component variable σ = (σ1, σ2). σ1(∈ {S, I, R}) represents the disease state and σ2(∈ {U,X,H})denotes the isolation state.

In the BIP model, all nodes are initially in the state σ = (S,U). Then a node i is randomly selected and set tobe σi = (I, U) and T I

i = 0. At each time step t, three processes are repeated for all infected nodes whose time ofinfection is T I < t: (1) infection, (2) isolation, and (3) unisolation and recovery. Each process is composed of thefollowing sub-processes. Infection: (1-i) Each node i with σi = (I, U) or (I,X) transmits the disease to ki connectednodes on F with the probability βF if the state of the connected node j is σj = (S,U) or (S,X). (1-ii) If σi,2 = U ,then it randomly chooses k′i nodes on W and infects with probability βW when the randomly chosen node j is in thestate σj = (S,U). T I

j for all new infected nodes j is set to be T Ij = t. Isolation: (2-i) Each infected node i with

T Ii < t is hospitalized with the probability θH , i.e., σi,2 = H and TH

i = t. (2-ii) Let Γi be the set of nodes connectedto the hospitalized node i on F . Then we set σj,2 = X and TX

j = t for all j ∈ Γi with the probability θX , if σj,2 = Uat t− 1. This corresponds to the self-isolation. (2-iii) If the state of node i was σi = (I,X) at t− 1, then it becomesσi = (I,H) and TH

i = t for all i(= 1, 2, · · · , N). Unisolation and recovery: (3-i) For all nodes i with σi,1 = I becomesσi,1 = R, if t > T I

i + tR. Here tR is a constant representing a recovery time from the infection. (3-ii) For all nodesi with σi,1 ∈ {S,R} and σi,2 = X are unisolated if t > TX

i + tX , where tX is the duration time for self-isolation.(3-iii) For all nodes i with σi = (R,H), if t > TH

i + tH then the node i is released from hospitalization, and itsstate becomes σi = (R,U). Here tH denote the duration time for hospitalization. These processes are repeated untilthere left no infected node. Under the BIP only the confirmed patients and the one who has direct contact withthe confirmed patient are isolated as shown in Extended Data Fig. 1. This is the most popular quarantine protocoladopted by many health authorities over the globe [36].

REINFORCED ISOLATION PROTOCOL (RIP)

However, household or workplace infection is significant in some infectious diseases, for example, it is more than15% for COVID-19 [2]. Such transmission frequently occurs during when the patients are self-isolated or refrainedfrom the unnecessary social activities by themselves. Furthermore, such transmissible paths can become a part ofsuper-spreading events if the pre-symptomatic or asymptomatic infection is possible. Thus controlling such localcontacts is another important factor to mitigate the transmission. For this purpose, we introduce the RIP by addingthe sub-process (2-iv) to the end of the isolation process of the BIP: (2-iv) Let j be the node whose state is changedinto σj,2 = X at t and Γj be the set of nodes connected to j on F . Then the state of node n(∈ Γj) with σn,2 = Ualso becomes σn,2 = X with the same probability θX for all n. Thus, in the RIP model if a node is hospitalized, thenits first and second nearest neighbors on F are isolated with the given probability (see Extended Data Fig. 1).

11

FINAL EPIDEMIC SIZE

The final epidemic size under each protocol Y (= BIP or RIP ) is defined as ρ∗R,Y ≡ ρR,Y (t → ∞). The difference

in final epidemic size under each protocol is defined as ∆R ≡⟨

ρ∗R,BIP

⟩

s−⟨

ρ∗R,RIP

⟩

s. Here 〈...〉s denotes the sample

average over independent runs. In order to obtain the average final epidemic size, we used 10, 000 samples. Theaverage final epidemic sizes,

⟨

ρ∗R,BIP

⟩

sand

⟨

ρ∗R,RIP

⟩

swith various parameter sets, (θ∗, t∗), are plotted in Extended

data Fig. 6.

FRACTION OF ISOLATED NODES PER UNIT TIME

To compare the number of isolated nodes we define the fractions of the isolated nodes per unit time (with σ2 = Xor H) for protocol Y as,

〈ρXH,Y 〉t =

∫ Tfinal

0(ρX(t) + ρH(t))dt

Tfinal

, (2)

where Tfinal represents the time at which ρI(t) becomes zero. Then we define the difference between two protocols as∆XH ≡

⟨

〈ρXH,BIP 〉t⟩

s−⟨

〈ρXH,RIP 〉t⟩

s. Here 〈...〉s denotes the sample average over independent runs. The fraction

of isolated nodes per unit time,⟨

〈ρXH,BIP 〉t⟩

sand

⟨

〈ρXH,RIP 〉t⟩

s, are obtained from 500 independent trajectories

and plotted in Extended data Fig. 7.

CODE AVAILABILITY

All code used in this work is available from the corresponding authors upon reasonable request.

ACKNOWLEDGMENTS

This research was supported by Basic Science Research Program through the National Research Foundation ofKorea (NRF) funded by the Ministry of Education (Republic of Korea) (grant number: NRF-2019R1F1A1058549).

12

Extended Data Fig. 1 | Schematic diagram of the intervention protocol on F layer. (a) On a network withsize N = 8 the node A is infected while nodes B ∼ H are susceptible. (b) Node A is hospitalized with probabilityθH . (c) BIP and RIP: Node B and C are self-isolated with probability θX while node D remains unisolated withprobability 1 − θX . (d) RIP: In addition to (c) node E and G are self-isolated with probability θX while node Fremains unisolated with probability 1 − θX . Node H are not included in the self-isolation candidate since node D isnot self-isolated in step (c).

13

� �� �������������������

t ∗ =2

� �� �������������������

t ∗ =4

� �� �������������������

t ∗ =6

� �� �������������������

t ∗ =8

� �� �������������������

t ∗ =10

� �� �������������������

θ ∗ =0.1

t ∗ =12

� �� �������������������

� �� �������������������

� �� �������������������

� �� �������������������

� �� �������������������

� �� �������������������

θ ∗ =0.2

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �������������������

θ ∗ =0.3

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

θ ∗ =0.4

� �� �� �������������������

ρ(t)

� �� �������������������

� �� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

θ ∗ =0.5

� �� �������������������

� �� �� �������������������

� �� �������������������

� �� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

θ ∗ =0.6

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �������������������

� �� �� �������������������

� �� �������������������

θ ∗ =0.7

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �������������������

� �� �������������������

� �� �������������������

θ ∗ =0.8

� �� �� �������������������

� �� �� �������������������

� �� �� ������������������

� �� �������������������

� �� �������������������

� �� �������������������

θ ∗ =0.9

� �� �� �������������������

� �� �� ������������������

� �� ��� ��������������������

tρS ρI ρR ρH ρX

� �� �������������������

� �� �������������������

� �� �������������������

θ ∗ =1.0

Extended Data Fig. 2 | Plot of {ρm(t)}’s (m ∈ {S, I, R,X,H}) under the BIP. t∗ increases from t∗ = 2 tot∗ = 12 (from left to right). θ∗ changes from θ∗ = 0.1 to θ∗ = 1.0 (from top to bottom). Each data is obtained from500 independent simulations by averaging the surviving samples at time t.

14

� �� �������������������

t ∗ =2

� �� �������������������

t ∗ =4

� �� �������������������

t ∗ =6

� �� �������������������

t ∗ =8

� �� �������������������

t ∗ =10

� � �� � �������������������

θ ∗ =0.1

t ∗ =12

� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �������������������

� �� �������������������

θ ∗ =0.2

� �� �� �������������������

� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� �������������������

θ ∗ =0.3

� �� �� �������������������

� �� �� �������������������

� �� �� �������������������

� �� � ��������������������

� �� ��� ��������������������

� �� ��� �������������������

θ ∗ =0.4

� �� �� �������������������

ρ(t)

� �� �� �������������������

� �� � ��������������������

� �� � ��������������������

� �� ��� ��������������������

� �� ��� �������������������

θ ∗ =0.5

� �� �� �������������������

� �� � ��������������������

� �� ��� ��������������������

� �� ��� ��������������������

� �� ��� �������������������

� � ��� ��������������������

θ ∗ =0.6

� �� �� �������������������

� �� � ��� ��������������������

� �� ��� ��������������������

� �� ��� �������������������

� � ��� ��������������������

� � ��� ��������������������

θ ∗ =0.7

� �� �� ������������������

� �� ��� ��������������������

� �� ��� �������������������

� � ��� ��������������������

� � ��� ��������������������

� � ��� ��������������������

θ ∗ =0.8

� �� �� ������������������

� �� ��� ��������������������

� �� ��� �������������������

� � ��� ��������������������

� � ��� ��������������������

� � ��� ��������������������

θ ∗ =0.9

� �� �� �������������������

� �� � ��������������������

� �� ��� ��������������������

tρS ρI ρR ρH ρX

� �� ��� �������������������

� � ��� ��������������������

� � ��� ��������������������

θ ∗ =1.0

Extended Data Fig. 3 | Plot of {ρm(t)}’s (m ∈ {S, I, R,X,H}) under the RIP. t∗ increases from t∗ = 2 tot∗ = 12 (from left to right). θ∗ changes from θ∗ = 0.1 to θ∗ = 1.0 (from top to bottom). When both θ∗ and t∗ arelarge, oscillatory behaviors are observed.

15

� � �� �� ���

�

�

�

�

Re(t)

θ ∗ =0.1

� � �� ���

�

�

�

�θ ∗ =0.2

� �� �� ���

�

�

�

�θ ∗ =0.3

� �� �� ���

�

�

�

�θ ∗ =0.4

� �� �� ���

�

�

�

�θ ∗ =0.5

� �� ���

�

�

�

�θ ∗ =0.6

� �� �� ���

�

�

�

�θ ∗ =0.7

� �� �� ���

�

�

�

�

t

θ ∗ =0.8

� �� �� ���

�

�

�

�θ ∗ =0.9

� �� ��� ����

�

�

�

�θ ∗ =1.0

t ∗ =2

t ∗ =4

t ∗ =6

t ∗ =8

t ∗ =10

t ∗ =12

Extended Data Fig. 4 | Effective reproduction number under the BIP. Each plot shows the obtained Re(t)’sfor various t∗(= 2 ∼ 12) under the BIP when θ∗ is fixed. θ∗ increases by 0.1 from θ∗ = 0.1 to θ∗ = 1.0 (from top-leftto bottom-right). As shown in the data, t∗ rarely affect the behavior of Re(t). However, the increase of θ∗ leads to adrastic drop of Re(t).

16

� � �� �� ���

�

�

�

�

Re(t)

θ ∗ =0.1

� � �� ���

�

�

�

�θ ∗ =0.2

� �� �� ���

�

�

�

�θ ∗ =0.3

� �� ��� ����

�

�

�

�θ ∗ =0.4

� �� ��� ����

�

�

�

�θ ∗ =0.5

� �� ��� ����

�

�

�

�θ ∗ =0.6

� �� ��� ����

�

�

�

�θ ∗ =0.7

� �� ��� ����

�

�

�

�

t

θ ∗ =0.8

� �� ��� ����

�

�

�

�θ ∗ =0.9

� �� ��� ����

�

�

�

�θ ∗ =1.0

t ∗ =2

t ∗ =4

t ∗ =6

t ∗ =8

t ∗ =10

t ∗ =12

Extended Data Fig. 5 | Effective reproduction number under the RIP. Plot of Re(t)’s under the RIP fort∗ = 2 ∼ 12 and θ∗ = 0.1 ∼ 1.0. Like Re(t)’s under the BIP in the Extended Data Fig. 4, t∗ does affect the behavior ofRe(t), while θ

∗ significantly changes the behavior of Re(t). In addition, we find that there are multiple peaks in Re(t)for θ∗ > 0.3 when we the RIP is applied. This implies that, even though the preemptive isolation of the susceptiblenodes effectively reduces the infection rate, the instantaneous effective reproduction number can increase again whenthe isolated susceptible nodes are released to be free.

17

��� ��� ��� ��� ��� ���θ ∗

������������������

〈ρ∗ R,B

IP〉 s

��

t ∗ =2

t ∗ =4

t ∗ =6

t ∗ =8

t ∗ =10

t ∗ =12

��� ��� ��� ��� ��� ���θ ∗

������������������

〈ρ∗ R,R

IP〉 s

��

Extended Data Fig. 6 | Final epidemic size. We plot the average final epidemic size under the (a) BIP and (b)RIP for t∗ = 2 ∼ 12 and θ∗ = 0.1 ∼ 1.0. The increment of the isolation period (t∗) has negligible effect on the finalepidemic size while the traceability (θ∗) significantly changes the final epidemic size.

18

��� ��� ��� ��� ��� ���θ ∗

���

���

���

���

〈〈ρXH,B

IP〉 t

〉 s

��

��� ��� ��� ��� ��� ���θ ∗

���

���

���

���

〈〈ρXH,R

IP〉 t

〉 s

��t ∗ =2

t ∗ =4

t ∗ =6

t ∗ =8

t ∗ =10

t ∗ =12

Extended Data Fig. 7 | The fraction of the isolated nodes per unit time. We plot the average fractionof the isolated nodes per unit time under the (a) BIP and (b) RIP for t∗ = 2 ∼ 12 and θ∗ = 0.1 ∼ 1.0. Under theBIP,

⟨

〈ρXH,BIP 〉t⟩

shardly change as θ∗ increases for a fixed t∗. However, when the RIP is applied,

⟨

〈ρXH,RIP 〉t⟩

sdrastically decrease when θ∗ > 0.3. and t∗ > 6. This suggests that with high traceability, the RIP can effectivelymitigate the epidemic spreading while minimizing the damage on the social and economic system.

19

� �� ��� ��� ���t

����

����

����

����

����

ρ(t) τ≈ 21

ρH

ρI

ρf

Extended Data Fig. 8 | Oscillatory behaviors under the RIP. Plot of the fractions ρH , ρI and ρf under theRIP. ρf (t) is defined as the fraction of the susceptible nodes which are just released from self-isolation at time t. Theepidemic parameters are β = 0.2 and tR = 6, and the intervention parameters are θ∗ = 0.9 and t∗ = 12. The periodof oscillation τ for each fraction with the given parameters is estimated as τ ≈ 21. The peak position of each curveindicates that the increase of the infected causes an increase of the hospitalized individuals. Due to the hospitalizationand self-isolation, the number of the infected rapidly decreases. However, after t∗ the isolated nodes are set to befree which increases the number of unisolated susceptible nodes. Thus it increases the number of infected individualsagain. This pattern is repeated with decreasing amplitude due to the depletion of the susceptible nodes until there isno more infected node left. This oscillatory behavior is observed only for the case of large θ∗ and t∗ in the RIP.