Study on population behavior under home quarantine policies of COVID-19 in China based on double-layer network evolutionary games

1Introduction

With the gradual and effective control of the COVID-19, China has entered the regular stage of epidemic prevention and control. The domestic epidemic situation is generally stable, while clusters of epidemics caused by sporadic cases have appeared in some areas. The effective identification and control of sporadic cases has become an important link in preventing the further spread of the epidemic. In this stage, instead of nationwide strict prevention and control policies, more refined non-therapeutic and targeted measures have efficiently contained several COVID-19 clusters [1].

China has mainly established seven protection and prevention networks in different layers, as shown in Fig. 1, including hospital treatment, Fangcang hospitals, vaccination, centralized isolation, nucleic acid test, home quarantine and mask wearing, while hospital treatment, medical isolation and vaccination are hospital-level measures, the first two layers correspond to different types of infected people: for patients with severe syndrome, they are sent into hospitals for admission and treatment; as for mild and asymptomatic patients, Fangcang hospitals have been set up to ensure their recovery. Centralized isolation and nucleic acid test are measures taken by the government, nucleic acid test is used to screen all vulnerable people, while suspected patients, patients with fever who cannot rule out the possibility of infection and close contacts need to be isolated collectively under medical observation. Home quarantine belongs to the community layer, which can effectively prevent the spread of the disease among communities. Mask wearing is the most common and extensive protection measure in the resident layer. The protection and prevention system takes measures to different groups of people from top to bottom in the aspects of hospitals, governments, communities and residents, which has largely controlled the spread of the epidemic.

Fig. 1

Epidemic prevention and control network system in China.

According to the Guidelines on the Novel Coronavirus-Infected Pneumonia Diagnosis and Treatment (version 8), there are four categories of people for home quarantine and medical observation: close contacts and special groups of people in close contacts; people entering China under the “7 + 7” policy; patients discharged from hospital and asymptomatic infected people released from quarantine; others who are assessed as unable to be placed under centralized isolation for medical observation by professionals. Due to the large scale of residents and high management costs, the home quarantine rule is often actually supervised and executed by the local communities. Marston, C et al. emphasized the necessity of the participation of communities in a pandemic as they are well placed to work with others to design collective response [2]. In China, the community played an important role in controlling the transmission of COVID-19. The National Health and Construction Commission of China issued the “Notice on Strengthening Community Prevention and Control of Pneumonia Epidemics Caused by Novel Coronavirus Infection”[3], claiming that comprehensive prevention and control measures based on communities should be put forward in a scientific and orderly manner, detect cases as early as possible and contain the spread of the epidemic effectively. However, at present, quarantine policies implemented in each area are different, and there is no unified definition, which brings confusion in practice. Currently, due to a lack of personnel and awareness of epidemic prevention, it is difficult to enforce home quarantine, which is one of problems that the grassroots epidemic prevention and control faces.

In recent years, with the reform of community governance, the interests of administrative forces and social forces in community governance are intertwined and complicated, so there is a long-term game relationship. In managing public health emergencies, there exist conflicts of interest among governments, communities, and residents, so it is important to analyze the interaction of each party’s behavior strategy [4]. In the process of the epidemic quarantine policies, the community replaces the government to supervise and manage residents who need to be quarantined; moreover, due to the system of community-level self-governance, there exists initiatives in the actual execution of the work. Therefore, there will be an interactive game between the community and residents, and the decisions of each subject will affect the effect of the quarantine policies.

Since the outbreak of COVID-19 epidemic, scholars from various countries have carried out researches from different aspects based on the classical epidemiological transmission model (Susceptible Infected Recovered Model, SIR) proposed by Kermack and McKendrick [5]. Some scholars have improved the model according to the characteristics of COVID-19, and the effect of epidemic prevention and control measures on controlling the spread of the epidemic was studied. Tang et al. explored the impact of quarantine and isolation measures on the final stage of the development of the epidemic in China based on the SEIR model [6]. Zhang et al. utilized the SICRD model to find that the time and intensity of intervention play an important role in controlling the scale of the epidemic [7]. The study by Hellewell et al. proved that in most cases, highly effective contact tracing and case isolation are sufficient to control a new outbreak of COVID-19 within 3 months [8].

These studies mainly analyzed and predicted the transmission dynamics of the COVID-19 based on a homogeneous network, that is, they assumed that the contact probabilities of different individuals in the network were the same, without considering the impact of the topology of the social network on the spread of the epidemic. However, the actual network is often not homogeneous, with the characteristics of small world and scale-free [9]. Considering that, other scholars supplemented relevant studies based on the complex network structure.

Newman proved that the SIR model could be accurately solved on a variety of networks [10]. Subsequently, Pastor-Satorras and Vespignani simulated the spread of infectious diseases on scale-free networks and discovered infectious diseases were ubiquitous in scale-free networks, and the topology of the network had a great influence on the spread of infectious diseases [11]. In recent years, with the popularity of complex network research, more scholars have considered the impact of network topology when studying the spread of COVID-19 epidemic. Wang et al. considered the heterogeneity of the number of contacts on the basis of asymptomatic infections in the SLIAR model [12] and studied the spreading dynamics of influenza-like diseases in the two non-uniform networks which are quenching network and annealing network [13]. In addition, other scholars have considered other realistic factors on the basis of complex networks and infectious disease models. For example, Zhang and Li constructed a random SEIR epidemic dynamic model that considers migration and human consciousness [14].

Compared with the study of transmission dynamics in a single-layer network, the real disease transmission process can be described more accurately in a double-layer or a multi-layer network. Zheng et al. constructed a two-layer network UAU-SIR model to characterize the coupling between the spread of infectious diseases and disease-related information or consciousness, and used the Markov chain method to work out the epidemic threshold [15]. This is an advanced method focusing on epidemic dynamics on multiple-layer coupling complex networks, and can describe the real transmission process than single-layer and homogeneous networks network transmission dynamics more accurately.

In addition, there are some scholars also used evolutionary game methods to study on major emergency public health events. Liu et al. combined the evolutionary game between the government and the public with the SI model, taking the large-scale spread of the epidemic, the control of the epidemic, and the active governance and control into account and fitted the stage of epidemic spread in these three situations of management and control [16]. Fan et al. used evolutionary game and system dynamics methods to study the evolutionary strategic stability points of the government, the community and the residents under dynamic and static reward and punishment mechanisms in public health emergency management [4].

The traditional evolutionary game describes the relationship between individual and group behavior from the perspective of system, and introduces the factors that affect individual behavior in the evolutionary model, thus affecting the group’s decision-making behavior. Based on complex network theory, network evolutionary game describes the relationship between individual and group behavior more accurately, and provides a new perspective of group evolution, which can be used to analyze the evolution process of complex multi-agent system. Existing researches on infectious diseases mainly focus on combining complex networks with infectious disease models, or using evolutionary games for analysis. Few combine evolutionary games with multi-layer complex networks to study how different subjects interact with each other in the study on COVID-19. Considering this, in addition to the advanced method mentioned above [15], this study adopts the method combining evolutionary game theory and double-layer coupling network to model public health event policy, hoping to carry out research from micro and macro perspectives and draw novel conclusions.

Therefore, based on the perspective of the game relationships between the community and the quarantined residents on a double-layer complex network, this paper analyses the impact of the implementation of the quarantine rules under the different strategic choices of the two parties, explores the evolutionary behavior and learning mechanism of the two game subjects of the community and the residents, and simulates evolution mechanism in the conditions of different parameters using Agent-Based Modeling(ABM) method. This paper aims to evaluate the home quarantine policy of COVID-19 prevention in China, and to figure out whether and to what extent residents will comply with the quarantine policy under different situations. According to the results of analysis, this paper is committed to shedding light on optimization suggestions during the process of developing and implementing policies and provide effective decision-making basis for the formulation of effective epidemic prevention measures.

The novelty of this paper can be divided into practical contributions and theoretical contributions. As for practical contributions, this research is of great significance to clarify the interest relationship among communities and residents for the improvement of the emergency prevention and control system and the implementation of prevention and control policies. In major public health emergencies, the community allocates resources, contacts the masses and mediates conflicts from the micro level to ensure the effective implementation of policies. While residents are the most basic nodes in the social network, and their extensive participation and positive response have an important impact on the effectiveness of the prevention and control. The strategies of two subjects complement and influence each other, and neither is indispensable. Based on realistic scenarios, this paper analyzes the roles and functions of communities and residents respectively in public health emergencies. It contributes to studying the influence of interest game and strategy learning on the implementation of prevention and control policies among them, which can better promote the construction of emergency prevention and control system.

In terms of theoretical contributions, firstly, this paper concludes the epidemic prevention and control network system in China, where the community layer and the resident layer play important roles in the protection and prevention networks. It shows the construction of the collaborative management system of public health emergencies is a social system project that requires the cooperation of multiple subjects. Therefore, the mechanism and platform of multi-subjects’ collaborative participation in the emergency management is of great significance for the improvement of the emergency prevention and control system. Secondly, a two-layer complex coupling network is established to simulate the evolutionary behavior and learning mechanism of the two game players, community and residents, from a micro perspective, and the effectiveness of epidemic prevention policies was fitted from multiple perspectives. This paper gives contributions to combining the evolutionary game theory and two-layer complex coupling network in the field of public health emergencies, which may provide a new methodological perspective for follow-up relative research.

The rest of the paper is organized as follows: Section 2 establishes the evolutionary game model and double-layer complex network. Section 3 simulates the results of evolutionary game under different conditions of related parameters, and the analysis of the effect of rules and implications for optimization are given in Section 4.

2Materials and Methods

2.2Construction of a double-layer complex network

2.2.1The construction logic of a double-layer network

The relationship between two groups of the community and the residents is supervision and supervised, namely the local community supervises residents who need to be quarantined and residents are supervised by the local community. And the relationship within each group is the general homogeneity relationship, which means they can learn from each other’s strategies. This article regards the community network as the upper-level network, in which each node in the network represents each community in the reality and each edge represents a neighbor relationship; while the social network of residents is recognized as the lower-level network, in which each node represents each resident in the society and each edge represents a neighbor relationship.

2.2.2The construction of sub-networks of the community and residents

Related studies have found that real social networks have the characteristics of a small world, that is, the network has a large clustering coefficient and a small average distance [18]. Therefore, this article uses an undirected and unauthorized small-world network to describe the social network of the residents, which means some people who have few social relationships in the network can be connected through a few short acquaintance relationships. A connected and undirected edge in this network means that two nodes have a social relationship, which includes online and offline connections, and the nodes are closely connected.

The number of nodes in the community network is relatively small, and the distribution is relatively independent. To simplify the analysis, it is assumed that the distribution of nodes on the community network has the characteristics of the ER network.

2.2.3The construction of a double-layer network of the community and the residents

The community nodes in the upper network establish a directed connection with the residents nodes in the lower network according to the principle of distance priority, that is, a community node will establish a supervision and supervised directed connection with residents nodes that are close to them. And a community node can establish multiple connections with different residents nodes in the lower-level network, while each residents node can only have one connection with the closest community node in the upper-level network. The double-layer network is shown in Fig. 2:

Fig. 2

double-layer network visualization of communities and residents.

2.3Evolutionary game model on the double-layer network

2.3.1Hypothesis

Hypothesis 1: Both communities and residents are bounded rationality, they choose strategies according to their expected return, and there is a certain probability that the optimal strategy is not selected by two parties;

Hypothesis 2: Communities and residents have game relationships with neighbors in another layer of the network who have established a directed connection with each other;

Hypothesis 3: Two groups can learn from neighbor nodes in the same layer of the network, and their strategic update is based on the comparison with their neighbors on the result of payoffs in the previous game.

2.3.2The establishment of game model

The community network and the residents’ social network are represented as G_a = (V, E), (a = 1,2), V represents the collection of all nodes in the network, E = e_ij represents the collection of all edges. G₁ = (V₁,E₁)represents the community network (the upper-level network), G₂ = (V₂,E₂) represents the residents’ social network(the lower-level network). i represents a node in the network G₁, j represents a node in the network G₂, l_i and l_j represents the degree of node i in the network G₁ and node j in the network G₂ respectively, s_i represents the strategy of node i selection in network G₁, s_j indicates the strategy selected by node j in the network G₂. The nodes in the community network will play games with their neighbor nodes in the residents’ social network according to the game matrix in Table 2.

The game evolution rules of the double-layer network are as shown in Fig. 3:

Fig. 3

The process of the game model.

Each node participating in the game may adjust its strategy selection at time t according to the comparison of the payoff of its neighbors and its own at time t-1. This paper solves the stable solution based on the evolutionary game payoff matrix, and introduces an evolutionary rule to explore the influence of individual heterogeneity and network structure on the stability of the game.

(1) Game

Each node in the G₁ network plays a game with its neighbor nodes in the G₂ network, and counts its own payoff. Therefore, the total payoff of the node after a round of the game is counted as follows:

(1)

Pi=∑j∈Ωi2siAijsjT

where Ωi2 represents the collection of all the neighbors of node i in the network G_2, s_i represents the strategy of node i, represents the transposition of the strategy of node j, A_ij represents the game matrix of node i and j.

The node j₂ in the G₂ network plays a game with his only neighbor node i in the G₁ network, and the total payoff of the node after a round of the game is counted as follows:

(2)

Pi=sjAijsiT

where s_j represents the strategy of node j in the G₁ network, represents the transposition of the strategy of node i, A_ij represents the game matrix of members i and j.

(2) Learning

After a round of the game, community nodes in the upper network will randomly select a neighboring node of the same network to compare their payoffs, so will residents nodes in the lower network. There is a certain probability for each node to decide whether to learn the neighbor’s strategy to change its own strategy in this round. This article adopts Fermi rule [19] as the strategy updating rule:

(3)

Wi=11+e(Ui1-Ui2k)

where W_i represents the probability that the individual i₁ learns from the strategy of one of its neighbors in the G₁ network, U_i1 and U_i2 respectively represent their payoffs, and k depicts the noise effect, which indicates that individuals may make irrational choices [20]. Similarly, the probability that individual j₁ in the G₂ network learns from the same-level neighbor is:

(4)

Wj=11+e(Uj1-Uj2k)

(3) Repeat the above steps, after a certain period of time, when network and the game have become stabilized, and calculate the average stable cooperation level of each network.

3Results

3.1Basic variable settings of NetLogo simulation platform

ABM is an approach simulates the dynamics of complex systems based on describing the continuous interaction of agents who are heterogeneous in terms of information and decision rules, while the agent stands for each component of the system or each individual in the specific situation. It models each agent from a microscopic perspective from the bottom up and allows spontaneous emergence at the macroscopic level [21]. ABMs polymerize interactions at the micro level to macro level, which describe direct interactions among agents in a group accurately and contribute to more comprehensive policy assessments. Therefore, this paper uses the NetLogo6.2.0 simulation platform to conduct evolutionary game simulation research on complex networks through the ABM method.

According to the steps of game simulation on the double-layer network, WS small world network is first generated, and all nodes on the network are divided into two types: the yellow house-shaped nodes represent communities with strict supervision and the blue ones represent communities with inaction, while the gray person-shaped nodes stand for residents choosing compliance and the red ones stand for those with violation. In the initial NetLogo interface, the initial parameters of the network can be determined by adjusting the slider of each parameter, as shown in Fig. 4.

Fig. 4

Initial state of WS small world network.

In Fig. 4, the left side shows the relevant initial parameters of the model. For example, n-residents represents the scale of residents, that is, the total number of people who need to be quarantined at one time, n-communities represents the number of communities, and k represents the value of the environmental noise parameter in the formula (3) and (4). Payoff is the payoff matrix obtained in Table 2. The initial parameters of the variables in the payoff matrix are shown by each slider such as a, b, c, v, E, w, q, p, etc. in the figure. x₀ represents the proportion of residents who choose to comply with the quarantine rules in the initial state, and y₀ represents the initial proportion of the community’s choice of active supervision. The graph on the bottom left shows the change in the proportion of the number of residents and communities who choose different strategies over time.

In Fig. 4, the right side is the view interface of the model. The human-shaped node represents residents, and the gray ones represent those who comply with the quarantined strategy and are quarantined at home, while the red ones represent people who violate the rules; the house-shaped nodes represent communities, and the yellow nodes represent communities with the choice of active supervision, while the blue nodes represent communities that choose inaction. The rewiring-probability at the bottom of the view interface represents the random reconnection probability p of the WS small-world network; clustering-coefficient and average-path-length reflect the basic characteristics of the social small-world network of residents: there is a relatively small average path length and a relatively high clustering coefficient.

Table 3 shows the corresponding initial parameter settings in the initial state:

Table 3

Simulation parameter values

Parameters	a	b	c	V	E	w	q	p	x0	y0	k	Rewiring-probability
Values	5	10	10	50	20	0.5	0.8	0.05	0.5	0.5	0.1	0.2

3.2The impact of related parameters on evolutionary stability strategies

(1) Simulation results of different initial strategy ratios of communities

When x₀ = 0.5, let y₀ = 0.2, 0.5, 0.8 respectively, keep other parameters unchanged and take T = 200 iterations, the result of evolutionary stability is shown in Fig. 5:

Fig. 5

two-party strategy evolution results when y₀ = 0.2, 0.5, 0.8.

It can be seen from Fig. 5 that the initial strategy ratio of the community has an impact on the strategy choice ratio of the residents. When y₀ = 0.5, that is, half of the communities choose active supervision initially, the proportion of residents who choose to comply with the quarantine requirements (“X” in the graph) fluctuates as the increasing of iterations, but it is slightly higher than the proportion of people who choose violation(“1-X” in the graph); when y₀ = 0.2, however, the proportion of people who choose to violate is slightly higher than the proportion of those who choose to comply; when y₀ = 0.8, after several iterations, the proportion of residents who choose the “X” strategy stabilizes at 1 basically. In addition, the proportion of the community’s strategies always fluctuates, while residents can remain relatively stable after a certain period of time in any condition, indicating that the community’s choice of strategies will affect the residents. The higher the proportion of the community choosing active supervision, the more likely the residents will abide by the quarantine requirements; while 80% of the communities chose to supervise at the beginning, after a moment all the people will choose to follow the rules and nobody chooses to violate them.

(2) Simulation results of different reconnection probabilities in small world networks

When x₀ = y₀ = 0.5, keep other initial parameters unchanged, and let p = 0.2, p = 0.4 and p = 0.6, p = 0.8, the evolutionary stability results are shown in Fig. 6:

Fig. 6

Evolution result when p = 0.2, 0.4, 0.6, 0.8.

We can see from Fig. 6 that under the initial condition of x₀ = y₀ = 0.5, as the reconnection probability p increases, the clustering coefficient and average path length of the small world network gradually decrease, and the small world network gradually evolves to a completely random network. When the probability of reconnection is 0.2, the proportion of residents who choose to comply with the strategy is slightly higher than the proportion of those who choose to violate. When the probability of reconnection reaches 0.8, the proportion of residents who choose to comply with the strategy is much higher than the proportion of the other strategy, and the difference in the proportion of different strategic choices is getting larger. In addition, with the increase in the probability of reconnection, the proportion of the residents with different strategies reaches a relatively stable state at a faster rate, which shows that different network structures have an impact not only on the proportion of residents’ strategy selection but also the stabilization speed. This is because in the small-world network, as the probability of network reconnection increases, the average path length and clustering coefficient decrease accordingly. On the other hand, the reduction of the clustering coefficient makes the lower-level residents network tend to present a decentralized state, and the heterogeneity of the network is more obvious. People with larger average degrees have greater influence and are more likely to influence neighboring nodes to learn from their strategies, thereby reducing the time for all nodes to evolve to a relatively equilibrium state.

(3) Simulation results of the government reward distribution parameter w

When x₀ = y₀ = 0.5 and other initial parameters are kept unchanged, let w = 0.2, w = 0.5, w = 0.8 respectively, the evolutionary stability results are shown in Fig. 7:

Fig. 7

The evolution result of w = 0.2, 0.5, 0.8.

When w = 0.2 and w = 0.5, the proportion of residents who comply with isolation is greater than the proportion of violations, and it is obvious that w = 0.2, that is, when the government’s rewards for residents account for 80 percent of the total reward, the proportion of residents who choose to quarantine is higher than when w = 0.5, that is, the reward from the government when the reward is equally divided; and when w = 0.8, that is, when the residents can only get 20% of the government’s reward, the proportion of choosing to violate the quarantine rules is much higher than that choosing complying with the quarantine requirements. As for the community, when the distribution parameter w takes different values, there is no obvious difference, and the proportion of their choice always fluctuates around 0.5, which shows that people are more sensitive to w than the community group.

(4) Simulation results of environmental noise k

k is the environmental noise variable in Equation (3) and Equation (4). When k is very large, such as close to 1, the probability of node’s learning strategy W is approximately 0.5, which means that the probability of residents imitating their neighbor’s strategy is 0.5 and their selections are more random. On the other hand, when k is very small, for example, close to 0, the value of W will be close to 1, indicating that external factors will not interfere with the learning strategy of the node, and there is a high probability that the node will imitate its neighbors. When x₀ = y₀ = 0.5 and other initial parameters are kept unchanged, k = 0.1 and k = 0.5 are taken respectively, the evolutionary stability results are shown in Fig. 8:

Fig. 8

The evolution result when k = 0.1, k = 0.5.

When other conditions remain unchanged, let k = 0.5, the ratio of residents who choose to violate the strategy is higher than the ratio of compliance, which is contrary to the previous default situation when k = 0.1. It shows that when the environmental noise becomes larger, the residents are more likely to be affected by various information in the surrounding environment rather than choose strategies in the next round based on the difference between their payoffs and those of their neighbors, thus changing their original choices of choosing the strategy.

4Discussion

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.

Funding

This work was supported by the Major Projects of National Social Science Foundation of China (Grant No. 20&ZD155); the Philosophy and Social Sciences Research Project of Chinese Ministry of Education (Grant No. 19JHQ091).