Emerging infectious diseases (EIDs) are a class of diseases with a wide range of transmission, multiple modes of transmission, incidence rates much higher than the annual incidence level, difficulty of control, ease of infection of the population, lack of specific treatment and prevention methods (1) and other characteristics. Once an EIDs occurs, it will spread faster and cover a wider area than traditional infectious diseases, causing social problems, affecting the global economy, and even endangering human lives. However, we note that the risk of an epidemic varies from region to region, and people's concerns and social sentiments in different regions also differ. Therefore, a practical classification of the risk level of an epidemic area and the implementation of targeted prevention and control measures according to the risk level are effective means of controlling the rapid spread of the epidemic.

There is a wealth of literature on analyzing the transmission patterns of infectious disease outbreaks and preventative and control measures from various perspectives. On the one hand, studies focus on analyzing and grasping the causes of the occurrence and spread of emerging infectious diseases. For example, Lashley (2) analyzed the factors influencing the occurrence of common emerging infectious diseases (EIDs) and concluded that microbial characteristics are essential in the emergence of infectious diseases. However, humans' behaviors and lifestyle choices are major factors in the emergence and spread of many EIDs. Sabin et al. (3) show how several factors related to human activities play a role in spreading infectious diseases and discuss the main factors contributing to the global spread of the COVID-19 pandemic. Yang and Zhang (4) summarized the uncertainty and complexity of EID and provided the preventive measures for dealing with EIDs.

On the other hand, the model is an effective mathematical tool to analyze and predict infectious disease transmission modes and laws. For example, Chu et al. (5) studied the epidemic spreading in weighted scale-free networks with community structure based on the SI disease model and showed the hierarchical dynamics of the epidemic spreading in the weighted scale-free networks with communities. Sun et al. (6) introduced three modified SIS models on scale-free networks that take into account variable population size, non-linear infectivity, adaptive weights, behavior inertia, and time delay, which could better characterize the actual spread of epidemics. Li et al. (7) using China's prefecture-level high-speed rail network and based on a probabilistic risk model, assessed the risk of COVID-19 infection in 19 provincial-level regions from Wuhan to the whole country in the early stage of domestic transmission, and found that the probability and impact could play different roles in the risk ranking of different regions. Shi et al. (8) developed a comprehensive model for simulating and predicting emerging infectious diseases, based on transmission dynamics and a statistical model driven by public health data. Chowell et al. (9) used the early exponential growth rate method to propose a simple susceptibility-exposure-infection-recovery (SEIR) model, a more complex SEIR model with asymptomatic and hospitalized cases and a stochastic susceptibility-infection-removal (SIR) model with Bayesian estimation to estimate the reproduction number of Spanish influenza, respectively. Bentout et al. (10) predict the peak time and the number of infectious cases at the peak before and after the implementation of non-pharmaceutical interventions for the COVID-19 based on an age-structured model. Reema et al. (11) compares the effectiveness of the SIR and SEIR models in analyzing epidemic data and also discusses how measures such as social distance and vaccination affect virus transmission in the SIR model.

Mathematical models based on differential equations provide statistical results consistent with the situation in practice, which is an effective tool for studying EIDs. However, as mentioned by Yu and Xue (12), most of the models based on differential dynamical systems are generally computationally cumbersome, and the solutions of the equations are extremely sensitive to the initial conditions, so they cannot deal with the unexpected and random events in the actual process well. In addition, the system formed by the communication, contact, and linkage between people is complex, which leads to the complexity of the transmission process of infectious diseases.

Graph theory is an ideal tool for analyzing, modeling, predicting and forming opinions to formulate strategies to rapidly contain the epidemic and minimize the devastating effects of viral infections (13). Baagyere et al. (14) characterized several complex networks from different domains using concepts from graph theory, and the node degrees, graph spectral radius, degree assortativity, and the entire topological structure of selected complex networks are studied on the SIR epidemic model. It is a fact that graph theory and complex networks are inevitably related. For a complex network, if not considering its dynamic features, the complex network is a graph, and the relevant features of the graph, such as subgraph and complementary graph features, have great significance for the modeling of complex networks (15). In addition, the classical infectious disease model combined with the complex network structure is important for investigating and analyzing the EIDs. Note that a social network (16, 17) is a complex network system based on the relationship between people and established according to certain rules. Zhang et al. (18) constructed an interpersonal network model, and the experimental results indicate that it is feasible and valuable to study virtual social simulation. Moore and Newman (19) studied the transmission characteristics of infectious diseases through the small-world network model and found that the nature of the small world would accelerate the transmission of infectious diseases. Huang et al. (20) found that the scale-free network nodes (21–25) in the complex network satisfy the power law distribution, which is more consistent with the real social network and the transmission law of infectious diseases in the entire society.

In addition, the effective division of regional risk levels is significant for preventing and controlling the rapid spread of EIDs. At present, relevant research has focused on studying regional risk classification methods. For example, Jia et al. (26) constructed an epidemic risk assessment model based on the analysis of population flow data and evaluated the degree of risk for each city using the collected population flow data related to Wuhan, China. Li et al. (27) evaluated the risk level of 38 districts in Chongqing, China, using the single index evaluation method, the analytical hierarchy process, and the systematic clustering method, respectively. Based on the population migration during COVID-19, Feng et al. (28) constructed a migratory imported risk index by using the number of accumulated cases and the number of new cases and showed that the index could better evaluate the epidemic risk in different places. Tu et al. (29) designed the scoring system with expert consultation and calculated the import, spread, and combined risk scores of regions using quantitative analysis methods to determine the risk level. Using unsupervised machine learning techniques, Fidan et al. (30) applied two clustering methods to classify COVID-19 risk degree.

In this paper, we consider the whole epidemic area as a scale-free network and classify the risk level for each area according to the association between regions. Meanwhile, combining the complex networks with risk assessment models to analyze the risk transmission of infectious diseases process and provide a theoretical basis for evaluating the prevention and control of epidemics and the risk classification of the region. The paper is organized as follows. Section 2 presents the steps for constructing social networks in new epidemic regions. Section 3 presents the method for building the epidemic classification model. Section 4 Simulation of the effectiveness of the method. Section 5 concludes with a brief discussion.

Since infectious disease transmission has been oriented, the directed weighted scale-free networks (31) are more suitable to describe the spread of epidemic risk in real social networks. In the following, we introduce the method of defining the social network as scale-free one and provide the main step for constructing the directed weighted scale-free networks in new epidemic regions.

Let E⊆V×V be the edge set, as introduced by Pastor-Satorras et al. (32), if V = {1, 2, ⋯ , N} is the node set, then the directed weighted network constructed by N nodes is denoted as G = (V, E).

Now, we introduce the necessary notions as follows. Let k_m be the number of edges connected to the node m. It can be divided into in-degree kmin and out-degree kmout for a directed graph, where kmin is the number of directed edges ending at node m and kmout is the number of directed edges starting from node m.

In addition, suppose that s_m is the strength of node m, and it can be divided into vertex in weight smin and vertex out weight smout for a directed graph, where smin is the sum of the weights of all edges reaching node m, smout is the sum of the weights of all edges starting from node m.

Moreover, let w_{m, s} be the weight of the connection between the node m and s, then the weight of a weighted network is divided into edge weight w_{m, s} and point weight w_{s, m}, and the w_{m, s}≠w_{s, m}, w_{m, s}≠ 0 if m→s are connected.

In the following, under the necessary notions above, we will introduce the main steps of the directed weighted scale-free network construction method provided by Barabási et al. (33).

Step 1 Starting from a network with only two nodes, a new node is added each time and connected to existing ones.

Step 2 Assuming that the node m is an existing node, the node with the higher degree is preferentially connected when a new node is added. Suppose that the node s is a new node, the probability of node s connecting to node m is

where k_s and k_m represent the degree of node s and m, respectively.

Step 3 Repeat steps 1 and 2 above until the target number of points and edges is reached and the directed weighted scale-free network is constructed.

In this section, we will present the methodology for determining the risk level of the epidemic regions.

Now, we assume that the division of regional units is consistent and relatively independent. Let v_m be a node of a new epidemic region. The edge of the network represents the social connection between the two regions, and the weight of the edge represents the correlation between the two regions m and s, usually represented by the correlation function denoted by L_{m, s}. Note that an outbreak in one region will affect all connected regions, and the size of L_{m, s} directly determines the size of the epidemic in the region.

In this section, we aim to verify the feasibility of the regional risk classification method proposed in this paper by simulation. Firstly, we use MATLAB software to construct a scale-free network with nodes that follow a power law distribution and have the characteristics of a real network. Secondly, we consider five necessary indicators that influence the correlation intensity of regional risk, namely the distance between two regions, personnel flows, economic traffic, transport convenience and logistics intensity, and use the MATLAB software to generate two sets of original data for each of the above five indicators, as shown in the Supplementary material. Moreover, using Equation (2), we calculate the correlation intensity of two sets of original data for these five indicators as shown in Tables 2, 3 below. Finally, we analyzed the two sets of experimental data using the presented classification method of regional risk levels.

This paper uses graph theory and the risk assessment method to construct the epidemic risk classification method. In addition, the rationality and effectiveness of the classification method are verified by simulation. Furthermore, we use the MATLAB software to construct a scale-free network and generate the original the original data of the five required indicators. Then the risk classification degree in each node for two cases of the epidemic occurring in one node and two nodes are discussed. The experiment shows that the number of medium and high risk nodes does not show a significant increasing trend, and the number of high risk regions is relatively small compared to the number of medium risk regions, and the number of low risk regions is the largest, which is consistent with the classification of regional risk levels in the real society.

The construction of the social network of risk classification in new epidemic regions by the directed weighted scale-free network is more suitable for the transmission law of epidemic occurrence risk. It describes the transmission status of epidemic occurrence risk in social network relations. Furthermore, the established regional risk classification method can well classify the risk levels of different regions in the new epidemic area by determining the correlation function between the two regions and the risk value of the regional node. The experiment verified the rationality of the method, and it can provide a theoretical basis for the government to quickly judge the risk levels of different regions in epidemic prevention.

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary material.

XW: formal analysis and writing—review and editing. YL: methodology. CZ: resources. All authors contributed to the article and approved the submitted version.