COVID-19 epidemic under the K-quarantine model: Network approach.

The COVID-19 pandemic is still ongoing worldwide, and the damage it has caused is unprecedented. For prevention, South Korea has adopted a local quarantine strategy rather than a global lockdown. This approach not only minimizes economic damage but also efficiently prevents the spread of the disease. In this work, the spread of COVID-19 under local quarantine measures is modeled using the Susceptible-Exposed-Infected-Recovered model on complex networks. In this network approach, the links connected to infected and so isolated people are disconnected and then reinstated when they are released. These link dynamics leads to time-dependent reproduction number. Numerical simulations are performed on networks with reaction rates estimated from empirical data. The temporal pattern of the accumulated number of confirmed cases is then reproduced. The results show that a large number of asymptomatic infected patients are detected as they are quarantined together with infected patients. Additionally, possible consequences of the breakdowns of local quarantine measures and social distancing are considered.

Introduction

The COVID-19 pandemic has changed various aspects of our societies, ranging from public health and economic conditions to human rights. Two other recent coronavirus pandemics, Severe Acute Respiratory Syndrome (SARS) in 2002 and Middle East Respiratory Syndrome (MERS) in 2013, have produced 8437 and 2519 cases, respectively. On the other hand, within two years, there have been about 0.35 billion cases of COVID-19 and 4 million resulting deaths. This is mainly due to an abnormally high transmission rate and asymptotic spreading [1], [2]. Many countries are trying to vaccinate people, but it is difficult to produce a sufficient amount of vaccines in a short time to supply the whole world aside from their safety. Moreover, the highly contagious mutant viruses have been continuously detected. In case the vaccine has not been developed yet or is insufficient, non-pharmaceutical interventions such as social distancing among individuals, masking, and reinforcing personal hygiene are alternative approaches to prevention.

Beginning with Wuhan [3], [4] in China, the majority of countries implemented a lockdown policy as an initial response to face COVID-19. The policy restricts travel from other countries and prevents people from participating in non-essential social activities. Much research has been done on the efficient implementation of the policy or its effectiveness with mobility data and metapopulation model [5], [6]. However, such lockdown prevention is not sustainable, because it drastically reduces economic activities [7], [8]. Indeed, the majority of countries that adopted the lockdown policy failed to sustain it for more than two to three months; they gradually returned to their former policies. Thus, related issues, such as the possibility of second waves and an exit strategy from the lockdown prevention, were addressed and studied [8], [9], [10].

The Korean Center for Disease Control and Prevention (KCDC) has achieved great success in the early stage of the epidemic using the so-called K-quarantine measure, which enforces local quarantine around confirmed patients rather than implementing a global lockdown. This approach, implemented by contact tracing and aggressive quarantine, efficiently prevents the spread of disease without critical economic damage.

Contact tracing is a primitive but efficient strategy to select people who require treatment or isolation. However, owing to high social costs and privacy invasions, this strategy has been applied to limited cases with low infection rates, such as sexually transmitted diseases [11] or early-stage SARS [12]. A simple theoretical model [13] was proposed and numerically investigated. For the simple model, the relationship between the infection rate of the target disease and the frequency of contact tracing was studied. This study was extended to more practical quantities, such as latent time and the rate of asymptomatic infection [14] for the spread of the SARS-CoV-2 virus [15].

In this paper, a K-quarantine model is proposed based on the basic principles of the K-quarantine measure (contact tracing and quarantine). The model was simulated on several types of complex networks, where the links of a node were disconnected when the node was quarantined and reinstated when it was released. The model control parameters were estimated based on the empirical data of the spread of the SARS-CoV-2 virus during its early stages in South Korea. The accumulated number of confirmed cases and the time-dependent reproduction number were successfully reproduced from simulations in the early stage. Further, the model was applied to an empirical network and a synthetic network with a modular structure, and the patterns of the number of daily confirmed cases were investigated.

The remainder of this paper is organized as follows. Sec. II describes the complex networks in which the model is simulated. Sec. III describes the -quarantine model constructed using various reaction parameters. Sec. IV presents the estimation of the numerical values of the reaction parameters, which was achieved by comparing them with empirical data and simulating the model. Macroscopic measurable quantities, such as the accumulated number of confirmed cases and daily confirmed cases, were obtained from the model and compared with the empirical data. Sec. V discusses the consideration of temporally changing cases of reaction rates and network structure to reflect the empirical situations in which the virus species was changed and a social gathering in a street demonstration, respectively. A synthetic lockdown situation, which can be realized by deleting a fraction of links, was also considered. A summary of this study is presented in Sec. VI.

Networks

It is worth examining how the epidemic contagion spreads under the K-quarantine model as compared to its spread under global lockdown without local restrictions. To achieve this, a mathematical model is considered in this work. The conventional epidemiological model is a compartmental model in which each person is considered to be in one of the following possible states: susceptible (), latent (), infected (), or recovered/deceased (). The proportions of people in each state are regarded as continuous variables, and their rate equations (time derivatives) are set up as a function of these proportions with appropriate rate constants. By solving these differential equations, the fraction of each state as a function of time is obtained. In the past, this approach has successfully predicted the evolution of the fraction of infected populations. However, it may not be useful when considering the local quarantine effect under the K-quarantine measures stated above.

Here, the epidemic reactions are simulated on networks. A network is composed of nodes and links, which represent people and contact between a pair of connected people, respectively. The numbers of nodes and links that are simulated are taken as and , respectively. This implies that a society composed of people is being considered, and the average number of people in contact with each person (called the ‘degree’ in graph theory) is given as . Note that these links are static and do not change temporally with people’s movement. One may suppose that these are essential links, representing close contact with family members and colleagues in the workplace every day. In contrast, contact with people who met occasionally or only once (such as in street demonstrations or Jazz bars) is called loose contact. The contagion through loose contact is implemented as follows. Once extra links are connected between susceptible and infectious nodes, random selection is performed, and it is then investigated whether the susceptible nodes are infected by the infectious nodes with a given infection rate. The infected nodes change their states to latent states. Subsequently, the extra links are disconnected. In simulations, this process is simply implemented as follows. A proportion of susceptible nodes is selected, and their state is changed to the latent state. The contagion through a large street demonstration around is realized in this manner.

The K-quarantine process is realized by locally disconnecting the links to an infected node. As soon as the quarantine is completed and the patient is released, these links are reinstated. In the K-quarantine model, once a person is quarantined, they are required to take a diagnostic test. If the result is positive, then the people in contact with the patient are quarantined. Thus, links connected to the neighbors of the confirmed patient also need to be disconnected.

Networks are classified into two types based on their connection configurations: random networks and scale-free networks. For random networks, each link is added between two randomly selected nodes. Thus, the degree of each node has a Poisson distribution. Because this model was first proposed by Erdős and Rényi, it is often called the ER model [16]. For scale-free networks, following the power law, the degree of each node is heterogeneous. This implies that a few nodes have large degrees, but the remaining nodes have small ones. The nodes with large numbers of neighbors are called hubs. When a hub is infected, a large number of susceptible neighbors are exposed to the contagion. This may result in a spike in contagion. Scale-free networks were constructed using the models proposed by Goh et al. [17] and Chung and Lu [18].

Models

The epidemic reactions proceed as per Markovian dynamics, which are realized by the Gillespie algorithm [19], [20]. Each node is in one of the following states [21], [22]: susceptible (), latent (), asymptomatic infectious (), symptomatic infectious (), asymptomatic in quarantine(), symptomatic in quarantine (), or recovered (, , or ) [23], [24], [25]. The states of susceptible in quarantine () and latent in quarantine () also exist. The dynamic begins with one infected person, with all the others being in a susceptible state. When nodes in states , , and are absent, the dynamic falls into an absorbing state, and the nodes in state or remain. The detailed dynamics are as follows:

A susceptible individual in contact with an infectious individual and enters the latent state () at the rate . These reactions are expressed asWhen the latency period ends, the individual becomes infectious, that is, they can transmit the infection with or without symptoms. These states are denoted as or , respectively. These processes occur at rates and , respectively. Here, represents the fraction of asymptomatic infectious patients. These reactions are expressed as

When symptoms develop, the infected individual must go to the hospital and take a diagnostic test. If the result is positive, they are quarantined. This process occurs at the rate and is expressed aswhere the prime indicates that the individual is quarantined. On the other hand, an asymptomatic individual may recover naturally without any treatment. This process occurs at the rate and is expressed asThe isolated individual in state may be recovered through treatment or succumb to the disease. This recovered individual is counted as a confirmed case of recovery, denoted by . This process occurs at the rate and is expressed as

In the K-quarantine model, confirmed cases () and their neighbors are self-quarantined as potential infectious people even if they are asymptomatic. Regardless of their state being , , , , or , they are quarantined at the rate . This process is expressed aswhere is the quarantine rate. Because quarantined individuals must undergo a diagnostic test, isolated asymptomatic carriers are identified as confirmed cases. Accordingly, the neighbors of the identified asymptotic carrier are also quarantined at the rate :This trace process is repeated until no further confirmed cases are identified. [25], [26]. During the quarantine period, identified asymptomatic infected individuals recover at the rate , expressed asHere, it is assumed that asymptomatic patients have the same recovery rate regardless of isolation. Individuals in the states , , or with negative diagnostic test results are released from quarantine. They then return to their original states.where is the quarantine period (not the rate).

The reproduction number (denoted as ), the number of individuals who are susceptible and become infectious by contacting an infected individual, is calculated as , where is the mean number of neighbors on a given network. Herd immunity is the level of immunity in a population that prevents the spread of a disease over the entire system. The herd immunity threshold is described as  [27], [28].

Reaction rates

To explore the effect of the self-quarantine measure on the transmission of COVID-19, the rates and were estimated based on empirical data on COVID-19 provided by the Korea Center for Disease Control (KCDC) and others: First, to find the rate and , the period between exposure and the onset of symptoms is used. This interval was estimated to be 6 (mean) days [5], [31]. The infected individual can transmit the disease 1 - 3 days before the onset of symptoms [1]. Thus, the interval between exposure and becoming infectious is estimated to be 4 (mean) days [6], [32], [33], [34], [35]. We take . The resulting data show that the percentage of asymptomatic infections is estimated to be 15%–40% [36], [37], [38]. We take . Thus, and .

In South Korea, a potential symptomatic infectious individual develops symptoms and is then quarantined approximately in three days. Thus, was set. Further, it takes approximately 9 and 12 days for an asymptotic carrier and a confirmed infected individual, respectively, to recover. Thus, and were set. was taken to be 14 days.

The infection rate is estimated using the relation at the beginning of epidemic. Using the rate and the mean degree , is obtained when is taken [31]. Using these parameter values, it is observed that the simulation result fits the empirical data from the early stages of the COVID-19 outbreak in South Korea (March 2020) to the end of August. For the same outbreak, the value of directly measured from the empirical data is  [39]. Note that when the spread of an epidemic is simulated on a scale-free network, the reproduction number is expressed as  [40]. This formula is reduced to when the degree distribution follows a Poisson distribution.

Here, the K-quarantine model was simulated with fixed rates () and , and a controllable quarantine rate () on several types of networks. These included random networks (Fig. 2 (a)–(d)) scale-free networks (Fig. 2(e)), an empirical social network (Fig. 2(f)), and random networks with modules (Fig. 2(g)–(h)) [41], [42]. It should be noted that all the rates are fixed throughout the epidemic spreading process unless otherwise specified. The proportions of nodes in each state are measured as a function of time in days. Note that the quarantine rate is taken as . This value was obtained using a simulated annealing method. This method enables the calculation of an optimal parameter value that minimizes the mean-square error between the accumulated numbers of confirmed cases obtained by simulations and empirical values. The details of the algorithm are presented in Appendix B.

Fig. 2

Plot of the densities , , , and , where denotes as a function of for the Susceptible-Exposed-Infected-Recovered (SEIR) model. These represent the proportions of susceptible individuals, recovered individuals without noticing, newly confirmed cases, and accumulated confirmed cases, respectively. The rates are taken as , , , , , and . (a), Simulations are performed on ER random networks without the K-quarantine measures. System size , the mean degree , and are set. (b), Similar plot to (a), but under the K-quarantine strategy . (c), Similar plot to (b), but the rate changes suddenly at to . This change is caused by a new type of coronavirus, GH clade [29]. (d), Similar plot to (b), but the rate at . This change is considered to occur because the quarantine system no longer functions owing to overloading. (e) and (f), Similar plots to (b), but simulations are performed on scale-free networks with degree exponent  [18] and on an empirical social network [30], respectively. For (f), and . Owing to this smaller mean degree, the contagion rate is lower. (g) and (h), Similar plots to (b), but on modular networks. The network is composed of modules and each module contains nodes and has the mean degree of intra-module edge . Those modules are connected through inter-modular links. For (g), , , and are set. For (h), , , and are set.

Flowchart for the K-quarantine model. The states under quarantine are represented by squares, while others are represented by stadiums.

Temporal behaviors of several quantities

In Fig. 2(a), the SEIR model [44], [45], [46] is considered without any quarantine on random networks. Thus, was set. Initially, one node is assumed to be infected, while the other nodes are susceptible. The fractions , , , and , are obtained, where , and the dot represents the time derivative. and represent the proportions of newly confirmed cases and the accumulated confirmed cases, respectively. The three densities are shown in Fig. 2(a). The contagion spreads rapidly during the early stage and eventually reaches a steady state. As shown in Fig 2(b) with , when the quarantine system is functioning, the fraction initially increases rapidly, then slowly increases with some fluctuations, and finally reaches a steady state. Resurgent behavior is observed in . Further, it is noted that for the system with no quarantine strategy, the absorbing state of the infectious node completely disappears on reaching the 150th day, whereas for the K-quarantine system, it reaches the 400th day. The proportions of the accumulated confirmed cases for (a) and (b) are close; however, the proportion of remaining susceptible people is extremely small for (a), but it is more than 20% for (b). On the other hand, the fraction of asymptomatic infected patients appears to be about 30% for (a), but it is approximately 10% for (b). This is because asymptomatic patients can be detected when they are in quarantine owing to the infection of their neighbors.

Figure 2 (c) depicts the case in which the infection rate suddenly increases to at , owing to the change of virus species from S or V to GH clade [29]. There exists another significant peak of around , and the infection rate increases dramatically. Following this, the system reaches a steady state. The density of in the steady state increased by 22.72% compared to that of the case (b). However, no such dramatic change is observed in the empirical data. Figure 2(d) depicts the case in which the quarantine system is overloaded and does not act at a certain time (e.g., ). Then, instantaneously exhibits resurgent behavior, and rapidly increases and reaches a steady state, as in the SIR model.

Next, the K-quarantine model is simulated on a scale-free network in Fig. 2(e) and on an empirical social network in Fig. 2(f). It seems that the contagion pattern is insensitive to network structural type, because the overall patterns of (e) and (f) are similar to that of (b), even though the absolute values of the accumulated number of confirmed cases are different. This seems to be counterintuitive because a scale-free network contains a few super-spreaders. Thus, if they are infected, many susceptible nodes linked to them could be infected, and the proportion of infectious nodes would increase drastically. However, the pattern of increase is similar because the K-quarantine measure is relatively effective in contact tracing [14]. Information related to the times and places of visit by an infectious person is collected, and the people at these places and times are traced within a short period using various methods. These people self-quarantine themselves or are quarantined at an isolated place, even for the contagion by a super-spreader. This containment still works effectively. Therefore, the overall contagion pattern is slightly sensitive to the type of network structure (either heterogeneous or homogeneous).

The scale of the second wave, however, is moderately dependent on the type of network structure. Comparing (b) and (e) in Fig. 2, we find that the peak of the second wave is slightly higher for a scale-free network (e) than that of (b) for a random network, which may be caused by the epidemic of super-spreaders. The ratio of the peak height of the second wave to that of the first wave is approximately 27% for a scale-free network, which is comparable to approximately 11% for a random network. Note that for the empirical network (coauthorship network) (f), the mean degree is smaller than that of (a)-(e). Thus, the accumulated fraction of confirmed cases in the steady state was considerably smaller. Nevertheless, the peak of the second wave is higher.

For (g)-(h), simulations are performed on modular networks [41], [47]. The modular networks are composed of modules, each of which contains nodes. Thus, the total number of nodes in the system is . Nodes within each module are connected to each other randomly with mean degree . To make the modules connected, pairs of modules are selected randomly, each of the pairs is connected by links by selecting nodes from each module. Thus, the total number of inter-modular edges is . Specific those numbers are listed in the caption of Fig. 2(g)-(h).

Figure 3(a) and (b) depict the cases in which the system is lockdown for 60 days. The lockdown can be realized by either social distancing or restriction of transportation [5], [6], [8]. In (a), the lockdown is implemented by deleting the fractions of links (indicated in the legend) randomly selected on the 30th day. After 60 days, those links are recovered. When the fractions are below 50%, the lockdown effect is almost negligible. On the other hand, when the fraction is 90%, then the epidemic spread is highly suppressed. In the intermediate range, a resurgent behavior appears. However, such behaviors fluctuate depending on the density of infectious nodes on the 30th day. Thus, in (b), we consider the case that the starting day of lockdown measure is determined by the fraction of accumulated confirmed cases, called the lockdown threshold. Once the lockdown comes into force, 70% of links are deleted and they are recovered after 60 days. Depending on the threshold value, the time of the resurgent peak is determined. In short, while the lockdown measure during the 60 days is effective during some intervals, the outbreak eventually occurs. Note that the fraction, for instance, 50%, does not indicate 50% of a real system. The value is meaningful only in the simulated model network.

Fig. 3

Similar plots to Fig. 2(b). (a) But, at , some fractions of links (indicated in legend) are artificially deleted. This change is considered to occur when a global lockdown is functioning. Depending on the fraction, diverse temporal patterns of appear. (See the details in the main text.) (b), Similar plot to (a) but the lockdown is functioning when the fraction of accumulated confirmed cases reaches a threshold value given in the legend. Then 70% links are deleted at random. (c), Similar plot to Fig. 2(b), but a small fraction of the nodes in state randomly selected every day are forced to change their state to the latent state from the specified day. This is caused by the transmission of the disease by people from abroad. (d), Similar plot to Fig. 2(b), but a large fraction of the nodes in state instantaneously change state to on a single occasion [43]. This change reflects the transmission of the disease by close contact among people participating in a large street demonstration.

In Fig. 3(c), a small number of the nodes in state every day change their state to . This change is considered to occur when individuals from abroad become new sources of the epidemic. Because in this case, no root is found explicitly and implicitly in the trail of disease transmission, the pattern of spread may somewhat differ from the previous patterns. In Fig. 3(d), a large fraction of the nodes in state instantaneously change state to . This change is considered to occur by the transmission of disease among people participating in a large street demonstration at owing to their close contact and shouting.

In Fig. 4 , the simulation results are compared with the empirical data of South Korea, starting from January 20, 2020, and accumulated as of September 9th, 2020. It is observed that the increasing behavior of the accumulated number of confirmed cases from the model during the early stage is well-fitted to the empirical data with the rates assumed herein. However, there is some difference during the intermediate stage, which may be due to the unexpected social event (a festival opening in a club) that was held shortly after reducing the level of social distancing. In the latter region, the number of confirmed cases abruptly increases owing to the large demonstration on the main street near the city hall in Seoul. Among over 10,000 people participating in the demonstration, a non-negligible portion of them did not wear masks. Therefore, the disease transmission would be high. The model proposed herein cannot reproduce the output of such a large-scale perturbation. Instead, some portion (80%) of the remaining susceptible nodes were changed to nodes in the latent state, under the assumption that those portions of people are infected in high-risk areas. With the passage of time, the surge decreases owing to the K-quarantine measures.

Fig. 4

(Red solid curve) Plot of the accumulated number of confirmed cases that occur in South Korea versus time in days. (Black dashed curve) Plot of the same quantity obtained from the K-quarantine model on an ER random network. The arrows indicate the dates of the three surges. With the choice of reaction rates , the increasing behavior of the accumulated number of confirmed cases from the model is well-fitted to the empirical data up to . Beyond this point, owing to an explosive epidemic contagion by a large number of street demonstrations, the theoretical curve no longer matches the empirical curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In Fig. 5 , three time-dependent reproduction numbers are plotted from the empirical and simulation data. can be measured using two methods. Method (i), used in (a), utilizes the number of daily confirmed cases [48]. is obtained as the ratio of the number of new infectious patients generated at time step to the total number of infectious patients during all preceding time steps, that is, , weighted with an infectivity function . Here, is the weight that a person infected at a given time remains in the infectious state after time steps. Moreover, the ratio is averaged over a time window of size (days) ending at time . Accordingly, the curve exhibits low noise.

Fig. 5

Plots of time-dependent reproduction number obtained by different methods. (a) is estimated by method (i) from the empirical data (orange), and from the simulation data with and without the quarantine process (green, brown), respectively. (b) Plot of by method (i) from the empirical data (orange) in a shorter interval. Plot of by method (ii) (green and brown) from the simulation data with (orange) and without (brown) quarantine. Owing to the different methodologies of (i) and (ii), there exists a time delay between the two curves of . The curve of (i) is shifted by 12 days to the left to overlap the two curves of (i) and (ii) in the early stage. On March 21st (), the Korean government increased the level of social distancing. This enhanced prevention was maintained until April 19th (). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

In (b), method (ii) is used to estimate from simulation data with and without a quarantine process. The formula is used to obtain . While and are fixed, the mean number of susceptible neighbors of each infectious node at a given time is variable. Owing to the different methodologies, there exists a time delay between the estimated curves. To make them overlap, obtained by method (ii) is shifted by 12 days to the left in the early stage.

While curves calculated from two different methods are close to that from empirical data in the early stage and region, they are not in agreement with each other in the intermediate region . This deviation may be caused by an increase in the level of social distancing by the Korean government. Note that decreases to zero as the system reaches a steady state, in which a new infection rarely occurs, even though there exist a nonzero fraction of nodes in the infectious state. This case occurs when the infectious nodes are surrounded by the recovered nodes. This state may be called a herd-immunized state.

Discussion and conclusion

This paper proposed an epidemic contagion model on complex networks to investigate the effects of contact tracing and local quarantine on the spread of an epidemic disease, COVID-19, in South Korea. Contact tracing and quarantine are essential factors of the “K-quarantine measure” in South Korea. Under this measure, information on the spatial and temporal trajectories of infected individuals is collected not only by self-statement but also by using mobile phone data. This information is used to find other individuals who were present at the same place and time, who are then requested to undergo a diagnostic test. If infected, they are quarantined. The process is then repeated. Thus, contact tracing and quarantine are effectively implemented in South Korea, and a large outbreak is prevented.

In this study, the contact tracing and quarantine were modeled by disconnecting links to an infected node. As the node was released from quarantine following a given period, the links were reinstated. This process was repeated as new individuals became infected. The model contained seven parameters, which were determined based on empirical data. It was demonstrated that after determining these parameter values, the proposed model reasonably produced the empirical data of the accumulated and daily numbers of confirmed cases. However, owing to the limitation of the system size, the proportion of accumulated confirmed cases already reached a steady state at days in the performed simulations. Thus, it can be concluded that this simple network model is useful for predicting the pattern of epidemic spread for the near future. Moreover, a large number of asymptomatic infected patients were detected, as they were quarantined with infected people. In contrast, in the K-quarantine measure, the level of social distancing is adjusted on a timely basis, depending on the number of daily confirmed cases. Accordingly, the infection rate and other parameter values must be updated in the model. However, establishing a general relationship between the level of social distance and the infection rate remains an important future task.

Simulations were performed on several types of networks, including random, scale-free, and modular structures. The patterns of epidemic spreading on each type of network were compared, and it was concluded that the overall pattern of the accumulated number of confirmed cases was insensitive to the network type. This is because, regardless of the number of susceptible nodes infected by an infectious node, they are quarantined. The network structure of the proposed model is static because the links do not consider human mobile behavior. This is because the links represent close contact among family members and people in the same workplace. To represent the contagion through loose contacts, a proportion of susceptible nodes were randomly selected. These were then regarded as newly infected nodes through loose contacts. This is a simple approach. Further, the possible consequences of the breakdown of local quarantine measures and social distancing were also investigated.

Notably, the time-dependent reproduction number was considered. It was found that the theoretically obtained value is in good agreement with the empirical data in the early epidemic stage. It decreases to zero as the system reaches a steady state in a finite system. However, still changes in the real system, because the pandemic is still ongoing as of January 2022.

Vaccination strategy is indeed an important issue, particularly when vaccines cannot be sufficiently supplied to the majority of population. Many researches have been performed to improve the vaccination strategy, for instance, [49]. In the K-quarantine measure, because the infected individuals are traced, at first glance, it would be reasonable to vaccinate the susceptible people around infected people in the network. However, the model we considered in the paper is constructed on static (not temporal) networks. Thus, this vaccination strategy would not be sufficiently effective in real systems.

In summary, the K-quarantine measure was modeled on complex networks. In this model, infection and quarantine processes are implemented locally and stochastically, in contrast to the homogeneous method over the system in the numerical method using the compartment equation. With the appropriate choice of parameter values from the empirical data, the model successfully reproduced the patterns of the accumulated and daily numbers of confirmed cases. Thus, it can be used to predict the near future pandemic patterns.

CRediT authorship contribution statement

K. Choi: Conceptualization, Methodology, Software, Investigation, Writing – original draft. Hoyun Choi: Conceptualization, Methodology, Software, Investigation, Writing – original draft. B. Kahng: Conceptualization, Investigation, Writing – review & editing, Funding acquisition, Supervision.

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.