A medical resource allocation method based on a time-series complex network and a patch model

Abstract

The application discloses a medical resource allocation method based on a time sequence complex network and a patch model and belongs to the technical field of resource allocation optimization. In view of the problem that how to predict the medical resource carrying capacity in the population flow situation in the light of the current infectious disease transmission trend and the population flow between regions, the application carries out data cleaning and preprocessing by Python data crawling; the infectious disease evolution trend of each province in the research period is analyzed by using Arcgis software, a time sequence complex network based on a gravity model is constructed, and the infectious disease transmission is described; network density and clustering coefficient are used as global indexes to comprehensively evaluate the spatial topological structure of the infectious disease network and the evolution trend, and the 'key' region influencing the infectious disease transmission is obtained; then, a SEIQHDR patch model is used, and a Monte Carlo algorithm is combined to fit the daily new cumulative patient number and the inpatient number; and finally, a medical resource dynamic allocation model is used to analyze the relationship between the population flow and the medical resource situation between the 'key' regions, so that the resource allocation is optimized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of resource configuration optimization, and particularly relates to a medical resource configuration method based on a time series complex network and a patch model. BACKGROUND

[0002] Previously, an acute respiratory infectious disease has caused a global pandemic, which has had a huge impact on the social and economic production and life of each country. At present, the infectious disease has entered a lower level, but it is far from over, so the research on its transmission mechanism is still of great significance. Past research on infectious diseases has focused on the overall trend of a certain region, but there is less research on the internal transmission mechanism. During December 2022 to January 2023, each province in China suffered from the rapid spread of the infectious disease, and the medical system also suffered great pressure. Although no mandatory personnel flow restriction measures were taken, the public consciously implemented self-protection behaviors such as staying at home and reducing going out, which reduced the transmission speed of the infectious disease.

[0003] The domestic traffic network of each province is developed, including highways, railways and air transportation, and there is high population mobility within the province, which significantly increases the risk of infectious disease transmission. The close connection between regions within the province has become a potential infectious disease risk spillover network, and each region can become a source of diffusion. At the same time, population mobility has a great impact on cities with tight medical resources, and population out-migration usually reduces the pressure on the local medical system, but increases the pressure on the immigration destination, so resource allocation in areas with more immigration population should be strengthened during this period. Restricting population migration is not always beneficial, and in some provincial cities, the time of medical resource squeeze actually increases when there is no migration; while for third and fourth-tier cities, the change of migration rate has little effect on the duration of medical resource squeeze. Formulating medical resource allocation and population migration management methods needs to be customized according to specific circumstances. SUMMARY

[0004] In view of the current transmission trend of infectious diseases and the problem of how to predict the carrying capacity of regional medical resources under the condition of population flow, the application provides a medical resource configuration method based on a time series complex network and a patch model.

[0005] In order to achieve the above purpose, the application adopts the following technical scheme:

[0006] A medical resource configuration method based on a time series complex network and a patch model, comprising the following steps:

[0007] Step 1: data is crawled through Python, and then the data is cleaned and pretreated;

[0008] Step 2, analyze the evolution trend of infectious diseases in each province during the study period by Arcgis software, construct a time series complex network based on the gravity model to describe the spread of infectious diseases;

[0009] Step 3, use network density, clustering coefficient and centrality index as global index to comprehensively evaluate the spatial topological structure and evolution trend of infectious disease network, and obtain the "key" area affecting the spread of infectious diseases;

[0010] Step 4, based on the "key" area, use SEIQHDR patch model and combine with Monte Carlo algorithm to fit the daily new cumulative patient number and hospitalization patient number; then use medical resource dynamic configuration model to analyze the relationship between population flow and medical resource in "key" area, and then optimize resource allocation.

[0011] Further, the specific operation of step 2 is:

[0012] The study period [T1, T2] is divided into N net time windows by using rolling window method, forming segmented time series network, which is expressed as follows:

[0013] {[T1, T1+Δ], [T1+Δ, T2+2Δ], …, [T1+(N net -1)Δ, T2]}

[0014] Where Δ represents the window width; T1 represents the date of starting data collection; T2 represents the date of ending data collection; infectious diseases last for a long time, and it is not suitable to describe the spatial characteristics of infectious disease spread with the network in the same time period, so the time series network is constructed in stages to describe the spatial characteristics of infectious disease spread.

[0015] According to the change of the change rate of the total number of daily new infections, the time series network is divided into N net sliced networks; each sliced network is recorded as: Where G t = (V t , E t )t=1,2,···,N net , V t represents the set of the tth network node, G t is the set of the tth network edge, and E t represents the edge between the tth network node and the node.

[0016] Time series network is a dynamic network; unlike static topological structure network, after adding time dimension, the edges in the network will appear and disappear intermittently, such network is called time series network (temporal networks).

[0017] Further, the specific method of constructing the slice network is:

[0018] The improved gravity model is used to describe the evolution process of the network structure and the spread of infectious diseases between cities, and the gravity model is constructed as follows:

[0019]

[0020] Wherein, b ij represents the spatial correlation between cities i and j, which is also the constituent element of the adjacency matrix; p ij represents the correlation coefficient of the number of infected persons between cities i and j in the selected time period; k ij represents the proportion of the number of infected persons in city i in the number of infected persons between cities i and j, which is used to describe the directionality and asymmetry of the spread of infectious diseases between cities; I i represents the number of infected persons in city i in a period of time; P i represents the number of permanent population in city i in a period of time; G i represents the number of highway toll stations in city i in a period of time; D ij represents the actual geographical distance between cities i and j;

[0021]

[0022] B t represents the adjacency matrix corresponding to the time period t.

[0023] Further, the network density in step 3 represents the ratio of the actual existing connection of the network node to the theoretically existing edge, which is used to measure the degree of mutual contact between the nodes in the network; The greater the network density represents the more mutual contact between the nodes, and the more dense the connection between the nodes; The calculation formula of the network density is as follows:

[0024]

[0025] Wherein, m t represents the actual edge of the tth slice network, and N represents the number of nodes;

[0026] The clustering coefficient in step 3 represents the ratio of the actual number of edges between the nodes and the nearby nodes to the maximum number of edges of these nodes, which is used to measure the degree of clustering of the connected nodes of a node; The greater the clustering coefficient indicates that the network has small world characteristics; The calculation formula of the clustering coefficient is as follows:

[0027]

[0028] Wherein, k i represents the number of nodes adjacent to node i, and E it represents the ki The number of edges connected between nodes.

[0029] Further, the centrality index specifically includes node degree, degree centrality, proximity centrality, intermediate centrality and eigenvector centrality; the centrality index is used to study the areas playing a key role in the spread and development of infectious diseases in the province and the relative importance of each city;

[0030] The node degree represents the number of actual edges connected between a single node and surrounding nodes; the number of incoming edges of a node is referred to as the in-degree of the node, i.e., point in-degree; the number of outgoing edges of a node is referred to as the out-degree of the node, i.e., point out-degree; in the infectious disease network, the point out-degree is used to measure the number of infectious disease outflows caused by a city to other cities, and the point in-degree is used to measure the degree of infectious disease spread from other cities to the city; the calculation formula of the node degree is as follows:

[0031] degree ti = |E ti |, t = 1, 2, ···, N net ; i = 1, 2, ···, N

[0032] Wherein, degree ti represents the degree of the i-th node in the t-th slice network, and |E ti | represents the number of edges connected between the node and surrounding nodes.

[0033] The degree centrality represents the ratio of the actual number of edges to the theoretical number of edges possessed by a node; it is used to measure the number of contacts possessed by a node in the network; the greater the degree centrality of a city, the stronger the correlation between the city and other cities, and the closer the contact; the calculation formula of the degree centrality is as follows:

[0034]

[0035] The proximity centrality represents the reciprocal of the average number of shortest paths between a node and all other nodes in the network; the shortest distance refers to the number of edges of the shortest path between two points; if a point is taken as the starting point, and another point is the k-step neighbor of the point, the shortest distance between the two points is k; the greater the proximity centrality value, the closer the node is to other nodes, and the node has the status of a "central actor" in the network; the calculation formula of the proximity centrality is as follows:

[0036]

[0037] Wherein, d ij represents the shortest path between the i city and the j city in the t-th slice network.

[0038] The intermediate center degree represents the number of all shortest paths in which a node appears between other nodes; the larger the intermediate center degree, the greater the role of the city as an intermediary in the process of infectious disease transmission; the calculation formula of the intermediate center degree is as follows:

[0039]

[0040] Wherein, g jk represents the number of shortest paths between j city and k city, and represents the number of paths passing through node i between j city and k city.

[0041] The eigenvector centrality is used to measure the influence or importance of nodes in the network; in the graph, nodes are mutually influenced, and a node receives influence from neighbor nodes through incoming edges, so the score of the eigenvector centrality of the node depends not only on the number of incoming edges, but also on the importance of the incoming edge neighbors; the greater the score of the node, the greater the influence; the calculation formula of the eigenvector centrality is as follows:

[0042] EC ti =x i ,t=1,2,···,N net ;i=1,2,···,N

[0043] Wherein, x i represents the ith component of the eigenvector corresponding to the maximum eigenvalue of the tth slice network adjacency matrix.

[0044] Further, the model assumptions of the SEIQHDR patch model in step 4 are as follows: (1) the recovered person will not be infected again during the study period; (2) the total number of people N i in each city does not change during the study period; (3) asymptomatic infections and their transmission to susceptible persons are not considered; (4) patients during home isolation are not infectious because they are not active; (5) as long as there are beds, there will be enough medical staff and medical equipment; even during the medical intensive period, although there may be challenges in accommodating all patients, hospitalized patients can still receive adequate treatment; (6) since most people do not undergo nucleic acid testing after control is lifted, people with similar symptoms are considered new cases; (7) the number of hospital beds is changed as a quantitative indicator to reflect the dynamic changes of medical resources; the increase and decrease of the hospital bed configuration are used as quantitative indicators to reflect the dynamic configuration of medical resources.

[0045] Further, the transfer relationship of the population is as follows: the susceptible person S i has a chance to become a latent infected person; the latent person E i has a chance to develop into a symptomatic infected person after the incubation period; the symptomatic infected person I iThere is a chance of recovery or death due to delayed treatment, most of which will enter the home; home Q i There is a chance of illness and hospitalization H i , or recovery R i There is also a certain risk of death; hospitalized patients H i There is a chance of death D i Or recovery and discharge; H ci (t) represents the hospital bed capacity of i area at time t, reflecting medical resources; i and j can be susceptible, latent, infected, hospitalized and recovered;

[0046] Further, the SEIQHDR patch model is represented as:

[0047]

[0048] Where i = 1 ~ n, and

[0049] The contact rate is a function of time t, which is represented as:

[0050]

[0051] Where c 0i represents the contact rate at the initial time, c 0i = c i (0); c bi represents the minimum contact rate under the given current control strategy δ 1i represents the contact rate in the form of exponential decline.

[0052] Further, the medical resource dynamic configuration model is: the change of bed number is simulated by logistic growth model, which is represented as:

[0053]

[0054] Where δ i represents the production and configuration capacity of medical resources, H mi represents the maximum number of beds available during the epidemic;

[0055] According to the different stages of disease development, the daily bed number is calculated by piecewise function, which is represented as:

[0056]

[0057] Where H 0i represents the initial bed capacity of novel coronavirus patients during the outbreak of infectious diseases, T iThe key moment when the city began to increase medical resources (including hospital beds); therefore, the daily potential number of empty beds is calculated as H ji (t) = max{H ci (t) - theta i Q i , 0}.

[0058] Further, the specific operation of step 4 is: first, model parameter estimation and data fitting are performed, Latin hypercube sampling and Markov chain Monte Carlo simulation are used to estimate unknown parameters, and SEIQHDR patch model is used to fit the daily new cases and hospitalization numbers; the ode45 function in MATLAB software is used to calculate the daily available beds using the network density formula and the clustering coefficient formula and to perform sensitivity analysis, and then the medical resource dynamic configuration model is used to analyze the relationship between population flow and medical resources in the "key" areas.

[0059] Compared with the prior art, the present application has the following advantages:

[0060] The present application tests the Shanxi Province by regarding the Shanxi Province as 11 independent patches (also called "nodes"), studies the restrictions of population flow between patches on infectious diseases, and the influence of these restrictions on the final number of infected people, the number of hospitalizations and medical resources (using the number of empty hospital beds as an indicator). In addition, the present application also discusses the influence of population flow on cities with tight medical resources, and how to shorten the time of medical resource squeeze through macro-control. The present application first takes the 11 prefecture-level cities in Shanxi Province as the research unit, identifies the areas that play a "key" role in the spread of infectious diseases through network structure and node centrality properties. The present application innovatively discusses how the population flow between key nodes affects the size of infectious diseases and the allocation of medical resources within the nodes. This research is of great significance for understanding and controlling the spread of infectious diseases, and also provides valuable reference for the rational allocation of medical resources. BRIEF DESCRIPTION OF DRAWINGS

[0061] Figure 1 is a schematic diagram of the state transition relationship of the population;

[0062] Figure 2 is a schematic diagram of the flow of the method of the present application;

[0063] Figure 3 is a schematic diagram of the daily new infection number curve of Shanxi Province;

[0064] Figure 4 is an evolution graph of the cumulative new COVID-19 infection number in five periods in the 11 cities of Shanxi Province;

[0065] Figure 5 is a graph of the degree centrality, closeness centrality, betweenness centrality and eigenvector centrality of the 11 cities of Shanxi Province changing with time.

[0066] Figure 6 are the parameter estimates for Taiyuan and Jinzhong; a, b represent the simulation results of the parameters β1, β2, c of the Markov chain realized with 10,000 samples 01 and c 02 are the simulation results of the parameters β1, β2, c of the Markov chain realized with 10,000 samples

[0067] Figure 7 are the fitting results of the cumulative new cases and new hospitalizations from December 9, 2022 to January 13, 2023 in Taiyuan and Jinzhong. The circles and triangles represent the cumulative number of newly reported human cases and hospitalized cases, and the curve represents the solution of equation (10).

[0068] Figure 8 are the parameter estimates for Linfen and Yuncheng; a, b represent the simulation results of the parameters β3, β4, β5, and β6 of the Markov chain realized with 10,000 samples; c, d are the bar charts.

[0069] Figure 9 are the fitting results of the cumulative new cases and new hospitalizations from December 9, 2022 to January 13, 2023 in Linfen and Yuncheng. The circles and triangles represent the cumulative number of newly reported human cases and hospitalized cases, and the curve represents the solution of equation (10).

[0070] Figure 10 are the daily bed vacancy number change graphs of Taiyuan and Jinzhong under different population migration situations; (a) represents the daily bed vacancy number change graph of Taiyuan and Jinzhong under different population migration situations; (b) represents the daily bed vacancy number change graph of Jinzhong under different population migration situations. The red, green, black, purple, and blue solid lines respectively represent the bed vacancy number change situations under the conditions of only considering population migration from Jinzhong to Taiyuan, actual population migration situation, actual population migration scale of 4 / 5, no population migration, and only considering population migration from Taiyuan to Jinzhong.

[0071] Figure 11 are the daily bed vacancy number change graphs of Linfen and Yuncheng; (a) represents the daily bed vacancy number change graph of Linfen under different population migration situations; (b) represents the daily bed vacancy number change graph of Yuncheng under different population migration situations. The blue, green, black, purple, and red solid lines respectively represent the bed vacancy number change situations under the conditions of only considering population migration from Yuncheng to Linfen, actual population migration situation, actual population migration scale of 4 / 5, no population migration, and only considering population migration from Linfen to Yuncheng. DETAILED DESCRIPTION

[0072] For a more complete understanding of the present application, a detailed description of the application will be made. However, the present application has various implementations and is not limited to the specific examples listed herein. The presentation of these examples is intended to deepen the overall understanding of the disclosure of the present application.

[0073] Table 1 Model symbol explanation table

[0074]

[0075]

[0076] Wherein, i, j represents the region

[0077] Example 1

[0078] This embodiment takes Shanxi Province as an example, a medical resource allocation method based on time series complex network and patch model, comprising the following steps:

[0079] Step 1, through Python to crawl data, and then clean and pretreat the data;

[0080] The daily new infection data is collected from the infection rate of each city provided by Shanxi Medical University and Shanxi Center for Disease Control and Prevention, and is calculated by "local total population x incidence rate". The hospitalization case data is provided by Shanxi Health Commission through the designated hospital monitoring system. The total number of hospital beds in each city of Shanxi Province is obtained from the planning of Shanxi medical and health service system. The population migration data is obtained by processing the Baidu migration scale index, and the processing process is as follows: first, use Python to crawl the Baidu migration scale index related data provided by Baidu migration big data platform; then, with the help of Baidu migration scale index structure inversion, the actual number of migrants in any direction (migration in / migration out) in a certain place per day is calculated.

[0081] Daily new infection data from December 1, 2022 to January 13, 2023 is collected and analyzed and processed by R language, and the analysis results are shown in Figure 3 We found that during the study period, the trend of infectious diseases in Shanxi Province can be divided into five stages: slow rising period, rapid rising period, stable period, rapid falling period and slow falling period.

[0082] Step 2, analyze the evolution trend of infectious diseases in each province during the study period by using Arcgis software, construct a time series complex network based on gravity model, and describe the spread of infectious diseases;

[0083] The evolution process of infectious diseases in each prefecture-level city of Shanxi Province in the five stages is simulated by using Arcgis software, and the specific situation is shown in Figure 4The analysis found that the second and third stages were the peak periods of infectious diseases in Shanxi Province, and the infectious diseases in Taiyuan, Jinzhong, Lüliang, Linfen, Jincheng, Changzhi, and Yuncheng were more serious. To further explore the role of each city in the evolution of infectious diseases and assess the trend of infectious diseases, the above five stages were further divided into five time slices, and an infectious disease correlation network was constructed accordingly.

[0084] The research period [T1, T2] is divided into N net time windows using the rolling window method, forming a segmented time series network, which is represented as follows:

[0085] {[T1, T1 + Δ], [T1 + Δ, T2 + 2Δ], …, [T1 + (N net -1)Δ, T2]}

[0086] Where Δ represents the window width, which is 5 in this embodiment; T1 represents the date of the start of data collection; T2 represents the date of the end of data collection; infectious diseases last for a long time, and it is not appropriate to use the network in the same time period to depict the spatial characteristics of infectious disease spread, so the time series network is constructed in stages to depict the spatial characteristics of infectious disease spread.

[0087] The time series network is divided into N net slice networks according to the change in the daily increase in the number of infected persons; each slice network is denoted as: Where G t = (V t , E t )t = 1, 2, …, N net , V t represents the set of nodes in the tth network, G t is the set of edges in the tth network, and E t represents the edges between the nodes in the tth network. The time series network is a dynamic network; unlike a static network with a fixed topology, the edges in the network appear and disappear intermittently over time, and such a network is called a time series network (temporal networks).

[0088] Further, the specific method of constructing the slice network is:

[0089] An improved gravity model is used to depict the evolution of infectious disease spread and network structure between cities, and the improved gravity model is constructed as follows:

[0090]

[0091] Where b ij represents the spatial correlation of infectious disease transmission between cities i and j, and is also an element of the adjacency matrix; ρ ijdenotes the correlation coefficient of the number of infected people between city i and j in the selected time period; k ij denotes the proportion of the number of infected people in city i in the number of infected people between city i and j, which is used to describe the directionality and asymmetry of the transmission of infectious diseases between cities; I i denotes the number of infected people in city i in a period of time; P i denotes the number of permanent residents in city i in a period of time; G i denotes the number of highway toll stations in city i in a period of time; D ij denotes the actual geographical distance between city i and j;

[0092]

[0093] B t denotes the adjacency matrix corresponding to time period t.

[0094] In the infectious disease network, the nodes represent prefecture-level cities, and the edges represent the correlation between the transmission of infectious diseases between two cities. The absence of edges represents the lack of direct correlation between the transmission of infectious diseases between two cities. By constructing the corresponding five network graphs from the adjacency matrices of the five sliced networks, the evolution of the infectious disease correlation network and the evolution of the infectious disease risk are studied. The greater the correlation between cities, the more edges there are in the network graph, and the more easily the infectious disease spreads.

[0095] In the first period, the growth rate of the daily increase in the number of infected people in each city is relatively small, but the number of edges in the sliced network is relatively large, indicating that the correlation between cities at this time is already high and that a large-scale outbreak of infectious diseases is imminent. In the second period, the daily increase in the number of infected people in each city rises sharply, and the edges of the sliced network become more dense, indicating that the correlation between cities increases rapidly. In the third period, which is the most severe period of infectious diseases, the number of edges in the sliced network is the most dense, forming a complex infectious disease transmission network. With the reform of the policy and the active cooperation and self-quarantine of residents, the growth rate of the number of newly infected people begins to appear negative in the fourth period; until the fifth period, it generally shows a downward trend, but due to the robustness of the infectious disease network, government units and residents still need to be highly vigilant to break the small-world property of the infectious disease network and thus slow down the spread of infectious diseases.

[0096] Step 3: Use network density and clustering coefficient as global indicators to comprehensively evaluate the spatial topology structure of the infectious disease network and its evolution trend, and obtain the "key" areas that affect the spread of infectious diseases;

[0097] Network density represents the ratio of actual existing connections to theoretically existing edges, which is used to measure the degree of mutual contact between nodes in the network; the greater the network density, the more the mutual contact between nodes, and the more the connection of edges between nodes; the calculation formula of network density is as follows:

[0098]

[0099] wherein, m t represents the actual edge of the t-th slice network, and N is the number of nodes;

[0100] The clustering coefficient represents the ratio of the actual number of edges between the node and the nearby nodes to the maximum number of edges of these nodes, which is used to measure the degree of aggregation of the connected nodes of a certain node; the greater the clustering coefficient, the more the small world characteristics of the network; the calculation formula of the clustering coefficient is as follows:

[0101]

[0102] wherein, k i represents the number of nodes adjacent to node i, and E it represents the number of edges between the k i nodes of the t-th slice network.

[0103] According to the network density formula and the clustering coefficient formula, the infectious disease transmission density and the average clustering coefficient of Shanxi Province from December 1, 2022 to January 13, 2023 are calculated, and the results are shown in Table 2.

[0104] Table 2 Overall analysis of infectious disease network in Shanxi Province

[0105]

[0106] Further, the centrality index is used to study the areas that play a key role in the transmission and development of infectious diseases in the province and the relative importance; the centrality index specifically includes node degree, degree centrality, closeness centrality, betweenness centrality and eigenvector centrality;

[0107] The node degree represents the actual number of edges between a single node and surrounding nodes; the number of incoming edges of a node is called the in-degree of the node, i.e. the point in-degree; the number of outgoing edges of a node is called the out-degree of the node, i.e. the point out-degree; in the infectious disease network, the point out-degree is used to measure the number of infectious disease outflows caused by a city to other cities, and the point in-degree is used to measure the degree of infectious disease transmission from other cities to a city; the calculation formula of the node degree is as follows:

[0108] degree ti =|E ti |,t=1,2,···,N net ;i=1,2,···,N

[0109] wherein, degree ti represents the degree of the i-th node of the t-th slice network, and is represented by |E ti | represents;

[0110] The degree centrality represents the ratio of the actual number of edges to the theoretical number of edges of a node; it is used to measure the number of contacts that a node has in the network; the greater the degree centrality of a city, the stronger the correlation between the city and other cities, and the closer the contact; the calculation formula of the degree centrality is as follows:

[0111]

[0112] The closeness centrality represents the reciprocal of the average number of shortest paths of a node to all other nodes in the network; the shortest distance refers to the number of edges of the shortest path between two points; if a point is taken as the starting point, and another point is the k-step neighbor of the point, the shortest distance between the two points is k; the greater the closeness centrality value, the closer the node is to other nodes, and the node has the status of “central actor” in the network; the calculation formula of the closeness centrality is as follows:

[0113]

[0114] wherein, d ij represents the shortest path from the i-th city to the j-th city in the t-th slice network;

[0115] The betweenness centrality represents the number of shortest paths that pass through a node between other nodes; the greater the betweenness centrality, the greater the role of the city as an intermediary, and the city plays the role of a bridge intermediary in the process of infectious disease transmission; the calculation formula of the betweenness centrality is as follows:

[0116]

[0117] wherein, g jk represents the number of shortest paths between the j-th city and the k-th city, and represents the number of paths passing through the i-th node between the j-th city and the k-th city;

[0118] The eigenvector centrality is used to measure the influence or importance of a node in the network; in a graph, nodes are mutually influenced, and a node receives influence from neighbor nodes through incoming edges, so the score of the eigenvector centrality of a node depends not only on the number of incoming edges, but also on the importance of the incoming edge neighbors; the greater the score of a node, the greater the influence; the calculation formula of the eigenvector centrality is as follows:

[0119] EC ti = x i , t = 1, 2, ···, N net; i = 1, 2, · · ·, N

[0120] where x i represents the i-th component of the eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the t-th slice network.

[0121] The network characteristics of individual nodes in 11 cities of Shanxi Province from December 1, 2022 to January 13, 2023 include: degree centrality, node degree (point out-degree, point in-degree), closeness centrality, closeness centrality, and eigenvector centrality. The results are shown in Table 3.

[0122] Table 3 Network centrality analysis of infectious diseases in Shanxi Province

[0123]

[0124] ① Degree centrality

[0125] The data in Table 3 shows that the average degree centrality of the 11 cities is 0.4982. Taiyuan and Jinzhong are located in the central city group of Shanxi Province. Taiyuan, as the capital, has a large population and is densely populated, while the Shanxi University City is located here. These factors together result in the transmission speed of infectious diseases in these two regions being significantly higher than in other regions. In the degree centrality ranking, Linfen and Yuncheng in the southeast city group of Shanxi Province rank high among the southern cities of Shanxi Province. These two cities have a large economy, a large population, and a large flow of people.

[0126] ② Point out-degree and point in-degree

[0127] The analysis in Table 3 shows that the average point out-degree and point in-degree of the 11 cities under study is 4.9818. The point out-degree of Jinzhong and Taiyuan during the infectious disease period far exceeds the average level, indicating that these cities play a "spillover" role in the spread of infectious diseases. The point in-degree of Shuozhou and Datong is nearly twice their point out-degree, indicating that these two cities mainly play the role of receiving the inflow of infectious diseases from other cities during this infectious disease. Yuncheng and Linfen in the southeast city group of Shanxi Province rank high in the region, indicating that these cities play a "spillover" role in the spread of infectious diseases in the southeast city group.

[0128] ③ Closeness centrality

[0129] As shown in Table 3, the average closeness centrality of the 11 cities is 0.6132. The closeness centrality of the six prefecture-level cities in the interior of Shanxi is higher than the average. While Datong and Shuozhou in the north, Yuncheng and Jincheng in the south, and Yangquan in the east have a closeness centrality lower than the average. This indicates that during this infectious disease period, the interior regions of Shanxi have more direct connections with other regions and play the role of "central actors" in the network, while the peripheral regions play the role of "peripheral actors".

[0130] IV. Betweenness Centrality

[0131] Betweenness Centrality describes the "hub" nature of the region node in the associated network, which is the intermediary of the infectious disease transmission network. As shown in Table 3, the average betweenness centrality of the 11 cities is 3.6751. In the central city group of Shanxi Province, the sum of the betweenness centralities of Taiyuan and Jinzhong is 15.79, so the two cities occupy an absolute position in the infectious disease transmission network in the central cities of Shanxi Province. In the southeast cities of Shanxi Province, the sum of the betweenness centralities of Linfen and Yuncheng is 3.89, so the two cities occupy an absolute position in the infectious disease transmission network in the southeast cities of Shanxi Province.

[0132] V. Eigenvector Centrality

[0133] Eigenvector Centrality indicates the importance of the region nodes connected to the node. As shown in Table 3, the average eigenvector centrality of the 11 cities is 0.2744. Taiyuan and Jinzhong have the highest eigenvector centrality in Shanxi Province, and these two cities also have the largest number of initial infections in this infectious disease. In the entire infectious disease risk network, they belong to the source of risk transmission and the traditional intermediaries and hubs. In the southeast of Shanxi, Linfen and Yuncheng have the highest eigenvector centrality, which belong to the source of regional risk transmission and the traditional intermediaries and hubs.

[0134] Step 4: Based on the "key" region, the SEIQHDR patch model is used, combined with the Monte Carlo algorithm, to fit the daily new cumulative patient number and the number of hospitalized patients; then the medical resource dynamic allocation model is used to analyze the relationship between population flow and medical resource situation between "key" regions, and then optimize the resource allocation;

[0135] The centrality analysis shows that the key areas of infectious disease transmission in Shanxi Province are Taiyuan, Jinzhong, Linfen and Yuncheng. The simulation results show that the change of population migration rate has a significant impact on medical resource squeeze. If the population only migrates from Jinzhong to Taiyuan, the medical resource squeeze in Taiyuan will occur earlier and be prolonged; if the population only migrates from Taiyuan to Jinzhong, the medical resource squeeze in Jinzhong will be prolonged. If the migration rate of the two places is reduced to 4 / 5 of the original level, the situation in Taiyuan remains unchanged, and the squeeze state in Jinzhong will end earlier. Without migration, the squeeze time in Taiyuan increases, while that in Jinzhong decreases. The situation in Linfen and Yuncheng is similar, which also shows the direct impact of population flow on medical resource squeeze.

[0136] The assumptions of the SEIQHDR patch model are: (1) recovered patients will not be infected again during the study period; (2) the total population N i(1) the number of new cases is the number of new symptomatic infections; (2) the number of new cases is the number of new infections; (3) asymptomatic infections and their transmission to susceptible persons are not considered; (4) patients at home are not infectious because they are not active; (5) there are enough medical staff and medical equipment as long as there are beds; even during a medical crisis, hospitalized patients can be adequately treated, although there may be challenges in accommodating all patients; (6) people with similar symptoms are considered new cases because most people are not tested for nucleic acids after control is lifted; (7) the number of hospital beds is a quantitative indicator reflecting the dynamic changes in medical resources; the increase and decrease in the allocation of hospital beds are quantitative indicators reflecting the dynamic allocation of medical resources.

[0137] The transfer relationship of the population is: susceptible persons S i have a chance to become latent infections; latent infections E i have a chance to develop into symptomatic infections after the incubation period; symptomatic infections I i have a chance to recover or die due to lack of timely treatment, most of which will be at home; persons at home Q i have a chance to worsen and need to be hospitalized H i , or have a chance to recover R i , also with a certain risk of death; hospitalized patients H i have a chance to die D i or recover and be discharged; H ci (t) represents the hospital bed capacity of i area at time t, reflecting medical resources; susceptible, latent, infected, hospitalized, and recovered persons can migrate between i and j areas;

[0138] The SEIQHDR patch model is represented as:

[0139]

[0140] where i = 1 ~ n, and

[0141] The contact rate is a function that decreases over time t, represented as:

[0142]

[0143] where c 0i represents the contact rate at the initial time, c 0i = c i (0); c bi represents the minimum contact rate under the given current control strategy δ 1i represents the contact rate decreases in an exponential manner.

[0144] The dynamic allocation model of medical resources is: the logistic growth model is used to simulate the change of the number of beds, which is expressed as:

[0145]

[0146] where δ i represents the production and allocation capacity of medical resources, H mi represents the maximum number of beds available during the infectious disease period;

[0147] According to the different stages of disease development, the daily bed number is calculated by using a piecewise function, which is expressed as:

[0148]

[0149] where H 0i represents the initial bed capacity for COVID-19 patients during the outbreak of infectious diseases, T i represents the critical moment when the city starts to increase medical resources (including hospital beds); therefore, the daily potential number of empty beds is calculated as H ji (t) = max{H ci (t) - θ i Q i , 0}.

[0150] Latin hypercube sampling and Markov chain Monte Carlo simulation are used to estimate unknown parameters, and the SEIQHDR patch model is used to fit the daily new cases and hospitalization numbers; the ode45 function in MATLAB software is used to calculate the daily available beds using the network density formula and clustering coefficient formula and to perform sensitivity analysis.

[0151] Empirical analysis of SEIQHDR patch model based on population flow

[0152] 1. Data fitting and prediction results

[0153] In this study, by analyzing the centrality of the time series network, we determined the four cities with the highest degree centrality and closeness centrality, namely Taiyuan, Jinzhong, Linfen and Yuncheng, as the "key" nodes of infectious disease transmission. Based on the processed Baidu migration index data, we observed that the proportion of population migration between Taiyuan and Jinzhong (Linfen and Yuncheng) was significantly higher than that between these cities and other cities during the study period. Therefore, this study selected Taiyuan City-Jinzhong City and Linfen City-Yuncheng City in Shanxi Province as two research cases, collected the number of infected people and the number of hospitalization data from December 10, 2022 to January 13, 2023, and fitted these data, while simulating and predicting the number of empty beds of medical resources.

[0154] The MCMC method is used to simulate formula (10) 4000 times, and the parameter estimates of Taiyuan-Jinzhong and Linfen-Yuncheng groups are obtained. The detailed results are shown in Figure 6 and Figure 8 In addition, the solutions of formula (10) and their confidence intervals for the two groups of cities are obtained, and the detailed results are shown in Figure 7 and Figure 9 All the parameter values related to the calculation are listed in Tables 4 and 5. The relationship between the actual population migration and the dynamic change of the spare beds of medical resources is shown in the green line of Figure 10 and Figure 11 In Taiyuan, the medical resource crunch began on December 21 and lasted for 4 days. In contrast, the same situation in Jinzhong began on December 19 and lasted for 13 days. The medical resource crunch in Taiyuan occurred later and lasted for a shorter period, indicating that its medical conditions are more adequate than those in Jinzhong. In Linfen, the medical resource crunch began on December 15 and lasted for 69 days. In contrast, the same situation in Yuncheng began on December 10 and lasted for 64 days. Compared with Taiyuan and Jinzhong, Linfen and Yuncheng have poorer medical resource reserves and weaker configuration capabilities.

[0155] Table 4 Parameter table of Taiyuan (i = 1) Jinzhong (j = 2)

[0156]

[0157] Table 5 Parameter table of Linfen (i = 3) Yuncheng (j = 4)

[0158]

[0159]

[0160] 2. Analysis of the relationship between population migration and medical resource allocation

[0161] The impact of population migration rate on medical resource allocation. The duration of medical resource crunch is used as an indicator to measure the status of medical resources, which is quantified by the number of spare beds and their available time. Medical resource crunch is defined as the state of no spare beds.

[0162] Then the number of spare beds and their duration in are analyzed, and the related results are shown in the changes under the scenario of different population migration rates, as shown in Figure 10 and Figure 11 The scenario is constructed as follows: one-way population migration, population migration scale reduced to 4 / 5 of the original, and no population migration at all. The research found that:

[0163] (1) Only when the population migration occurs from Jinzhong to Taiyuan, the end time of the medical resource squeeze in Taiyuan will be advanced to January 5, 2023, while the start time remains unchanged. The medical squeeze in Jinzhong will be delayed to December 16, 2022, and the end time will be delayed to January 20, 2023, which is 9 days longer than the actual situation. Conversely, if the population only migrates from Taiyuan to Jinzhong, the start time of the medical resource squeeze in Taiyuan will not change, and the end time will be delayed to January 11, 2023. The medical squeeze in Jinzhong will start on December 17, 2022, and end on December 26, 2022. If the population migration rate of both cities decreases to 4 / 5 of the original level, the medical resource squeeze in Taiyuan will remain unchanged, but the medical resource squeeze in Jinzhong will end two days earlier than the actual situation. Without population migration, the medical resource squeeze in Taiyuan will remain unchanged, and the end time will be delayed to January 9, 2023. The medical resource squeeze in Jinzhong will be delayed to December 17, 2022, and the end time will be advanced to December 28, 2022, which is 15 days shorter than the actual situation.

[0164] (2) Only when the population migration occurs from Yuncheng to Linfen, the end time of the medical resource squeeze in Linfen will be advanced to January 1, 2023, and the start time remains unchanged. The medical resource squeeze in Yuncheng will be delayed to January 12, 2023. If the population only migrates from Linfen to Yuncheng, the end time of the medical resource squeeze in Linfen will be delayed to January 6, 2023, and the medical resource squeeze in Yuncheng will end on January 1, 2023. If the population migration rate of both cities decreases to 4 / 5 of the original level, the medical resource squeeze in Linfen will remain unchanged, but the medical resource squeeze in Yuncheng will be delayed by one day. Without population migration, the end time of the medical resource squeeze in Linfen will be advanced to January 2, 2023, which is one day shorter than the actual situation. The medical resource squeeze in Yuncheng will be delayed by two days to January 6, 2023.

[0165] From the results, population migration can affect the duration of medical resource squeeze. Generally speaking, population migration out of the area can alleviate the pressure on the local medical system and increase the pressure on the medical system of the destination. Therefore, it is necessary to strengthen resource allocation in areas with more population migration during the outbreak of infectious diseases. However, restrictions on population migration may not always be beneficial, such as in the case of Taiyuan, where the medical squeeze time actually increases without population migration. It may also be ineffective, such as for Yuncheng, where the duration of the medical squeeze changes little regardless of the population migration rate. This also reflects the need for appropriate medical resource allocation and population migration restriction policies that are tailored to local conditions.

[0166] In summary, the present application firstly adopts an improved gravity model to construct the adjacency matrix of infectious disease transmission, and on this basis, a time slice threshold network is established. By analyzing the spatial topology evolution of the network, the following phenomena are observed: the change trend of graph density and clustering coefficient shows that after the residents voluntarily stay at home, these indicators first increase and then decrease, eventually reaching the inflection point. This trend reflects that the voluntary home behavior of residents helps to reduce the population flow between nodes, thereby reducing the connectivity of the network, effectively containing the spread of infectious diseases, and significantly reducing the demand for anti-epidemic resources. Through the analysis of the centrality index, it is found that Taiyuan City, Jinzhong City, Linfen City and Yuncheng City rank first in terms of closeness centrality and betweenness centrality, indicating that these cities play the key role of "central actors" and "bridges" in the infectious disease network. Therefore, the present application regards Taiyuan City and Jinzhong City, as well as Linfen City and Yuncheng City as key areas for infectious disease management, and quantitatively analyzes the impact of population flow on infectious diseases in these areas.

[0167] The results show that the medical resource squeeze in Taiyuan City started on December 17 and lasted for 22 days, while the medical resource squeeze in Jinzhong City started on December 10 and lasted for 27 days. Taiyuan City has stronger medical resources and allocation capacity compared to Jinzhong City. The medical resource squeeze in Linfen City and Yuncheng City started on December 14, 2022 and December 13, 2022, respectively, and lasted for 21 days and 23 days, respectively, reflecting the inadequacy of these regions in terms of medical resource reserves and allocation capacity. In addition, the present application also discusses the impact of population flow on medical resource allocation. The research results show that population outflow can alleviate the pressure on the local medical system, but at the same time, it will increase the burden on the medical system of the destination. Therefore, during the period of high incidence of infectious diseases, for regions with more population migration, it is necessary to strengthen the allocation of medical resources. However, according to the example analysis, it is found that the restriction of population migration is not always beneficial, and appropriate medical resource allocation and population migration restriction strategies should be developed according to specific circumstances.

[0168] The present application regards 11 cities of Shanxi Province as independent nodes respectively, constructs network topology structure, and discusses dynamic evolution process of infectious disease diffusion of 11 cities under the condition that only residents consciously implement home and reduce going out and other self-protection behaviors. Considering population density, infected number, correlation between cities of Shanxi Province and highway traffic and other factors influencing infectious disease diffusion, the classic gravity model is innovatively improved, the interaction between cities is more accurately described, each index of each city of Shanxi Province in infectious disease transmission is analyzed, and the region playing a key role in infectious disease transmission is determined. Considering complexity of actual infectious disease, the classic SEIR model is improved, the SEIQHDR patch model is established, the medical resource situation of each city of Shanxi Province is predicted and warned in combination with network topology structure model results, the relationship between population flow and medical resource allocation is analyzed, and corresponding optimization is made, so that medical squeeze duration is reduced, and medical resource allocation is optimized.

[0169] The contents not described in detail in the specification of the present application belong to the prior art known to those skilled in the art. Although the above describes the specific embodiments of the present application for the purpose of facilitating the understanding of the present application by those skilled in the art, it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, it is obvious that all kinds of changes within the spirit and scope of the present application defined and determined by the appended claims are obvious, and all kinds of inventions utilizing the concept of the present application are included in the protection.

Claims

1. A method for allocating medical resources based on temporal complex networks and patch models, characterized in that, Includes the following steps: Step 1: Scrape data using Python, then clean and preprocess the data; Step 2: Analyze the evolution trend of infectious diseases in each province during the research period using ArcGIS software, and construct a temporal complex network based on a gravity model to describe the spread of infectious diseases; Step 3: Using network density, clustering coefficient, and centrality as global indicators, we comprehensively evaluate the spatial topology and evolution trend of the infectious disease network to identify the "key" regions that affect the spread of infectious diseases. Step 4: Based on the "critical" areas, the SEIQHDR plaque model is used, combined with the Monte Carlo algorithm, to fit the daily cumulative number of new patients and the number of hospitalized patients; then, the dynamic allocation model of medical resources is used to analyze the relationship between population flow and medical resources in the "critical" areas, so as to optimize resource allocation. The SEIQHDR patch model is represented as follows: ； in, Indicates contact rate; Indicates susceptible population; Indicates an undercover agent; This indicates a symptomatic infected person; This indicates the infection rate when an infected person comes into contact with a susceptible person; This represents the migration rate from location j to location i; i and j represent regions. This represents the total population of location i. Indicates the average duration of the incubation period; Indicates the recovery time of infected individuals; Indicates the recovery time for those staying at home; Indicates the recovery time for hospitalized patients; This indicates the conversion rate from patients to home-based care. This indicates the conversion rate from home-based individuals to patients. Indicates someone who stays at home; Indicates the number of hospital beds; Indicates the case fatality rate among infected individuals; This indicates the mortality rate among those staying at home; This indicates the mortality rate among hospitalized patients; Indicates the initial death population; Indicates the initial recovery population; Indicates the initial hospitalization population; ,and ; The contact rate is a function that decreases with time t, expressed as: ； in, This indicates the contact rate at the initial time. ; This represents the minimum contact rate given the current control strategy. ; This indicates that the contact rate decreased exponentially.

2. The medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: The specific operation of step 2 is as follows: Using the rolling window method to divide the research period [ Divided into The time windows are divided into segmented time series networks, as shown below: ； Where Δ represents the window width; T1 is the date when data collection begins; and T2 is the date when data collection ends. The time-series network is divided into groups based on the change rate of the total number of new daily infections. There are 1 slice network; each slice network is denoted as: ,in , Indicates the first A set of network nodes For the first A set of network edges, E t Indicates the first The edges connecting network nodes.

3. The medical resource allocation method based on temporal complex networks and patch models according to claim 2, characterized in that: The specific method for constructing a slice network is as follows: An improved gravity model was used to characterize the spread of infectious diseases among cities and the evolution of the network structure. The improved gravity model is constructed as follows: ； in, Represents city and The spatial relationships of infectious disease transmission between individuals are also constituent elements of the adjacency matrix; Indicates the city within the selected time period and The correlation coefficient between the number of infections; Represents city The number of infections in the city and The proportion between them is used to describe the directionality and asymmetry of infectious disease transmission between cities; Indicates the city over a period of time The number of infections; Indicates the city over a period of time The number of permanent residents; Indicates the city over a period of time The number of highway toll stations; Represents city and The actual geographical distance between them; ； This represents the adjacency matrix corresponding to time period t.

4. The medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: In step 3, network density represents the ratio of actual connections to theoretically existing edges between network nodes, used to measure the degree of interconnection between nodes in the network; a higher network density indicates more interconnections between nodes and denser edge connections between them; the formula for calculating network density is as follows: ； in, Representing the The actual connections of a slice network The number of nodes; In step 3, the clustering coefficient represents the ratio of the number of actual edges connecting all nodes to their neighboring nodes to the maximum number of edges connecting those nodes. It measures the degree to which a node and its connected nodes cluster together. A larger clustering coefficient indicates that the network exhibits small-world characteristics. The formula for calculating the clustering coefficient is as follows: ； in, Represents nodes Number of adjacent nodes Indicates the first A slice network The number of edges connecting each node.

5. A medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: The centrality metrics specifically include node centrality, degree centrality, proximity centrality, betweenness centrality, and eigenvector centrality; The node degree represents the number of actual edges connecting a single node to its surrounding nodes; the number of incoming edges of a node is called its in-degree, or vertex in-degree; the number of outgoing edges of a node is called its out-degree, or vertex out-degree. In an infectious disease network, vertex out-degree measures the number of infectious disease spillovers from one city to other cities, while vertex in-degree measures the extent to which an infectious disease is spread from one city to other cities. The formula for calculating node degree is as follows: ； in, Indicates the first The first slice network The degree of each node is determined by express; The degree centrality represents the ratio of the actual number of edges a node has to the theoretical number of edges; it measures the number of connections a node has in a network. The higher the degree centrality of a city, the stronger the correlation and the closer the connection between that city and other cities. The formula for calculating degree centrality is as follows: ； The proximity centrality is the reciprocal of the average number of shortest paths between a node and all other nodes in the network. The shortest distance is the number of edges in the shortest path between two points. If we consider one point as the starting point and another point as its k-th neighbor, the shortest distance between them is k. A higher proximity centrality value indicates that the node is closer to other nodes and has the status of a "central actor" in the network. The formula for calculating proximity centrality is as follows: ； in, Indicates the first In each slice network City to The shortest path in the city; The betweenness centrality represents the number of times a node appears in all shortest paths between other nodes. A higher betweenness centrality indicates that the city corresponding to that node plays a greater intermediary role, acting as a bridge in the spread of infectious diseases. The formula for calculating betweenness centrality is as follows: ； in, express City to The number of shortest paths between cities represents City to The shortest path between cities passes through the following nodes The number of paths; The eigenvector centrality is used to measure the influence or importance of nodes in a network. In a graph, nodes influence each other; a node receives influence from its neighbors through incoming edges. Therefore, a node's eigenvector centrality score depends not only on the number of incoming edges but also on the importance of its incoming neighbors. The higher the score, the greater the influence. The formula for calculating eigenvector centrality is as follows: ； in, Indicates the first The eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the nth slice network is the eigenvector of... Each component.

6. The medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: The model assumptions of the SEIQHDR patch model in step 4 are as follows: (1) Recovered patients will not be reinfected during the study period; (2) Total population of each city Unchanged during the research period; (3) Asymptomatic carriers and their transmission to susceptible individuals are not considered; (4) Since patients do not move around at home, they are not infectious during their home stay; (5) As long as there are beds available, there will be sufficient medical staff and equipment; even during periods of medical stress, while there may be challenges in accommodating all patients, hospitalized patients will receive adequate treatment. (6) Since most people do not undergo nucleic acid testing after the control measures are lifted, people with similar symptoms are considered new cases; (7) The change in the number of hospital beds is a quantitative indicator that reflects the dynamic changes in medical resources; the increase and decrease in the allocation of hospital beds is a quantitative indicator that reflects the dynamic allocation of medical resources.

7. The medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: The population transfer relationship is as follows: susceptible individuals There is a chance of becoming a latent infected person; latent person There is a chance of developing into an symptomatic case after the incubation period; symptomatic cases There is a chance of recovery or death due to lack of timely treatment, and most of them will be placed under home quarantine; those under home quarantine... There is a chance that the condition will worsen and require hospitalization. There is a chance of recovery. There is also a certain risk of death; hospitalized patients There is a chance of death. Or recover and be discharged from the hospital; represent time The hospital bed capacity in a region reflects its medical resources. land and Susceptible, latent, infected, hospitalized, and recovered individuals can migrate between different locations.

8. The medical resource allocation method based on temporal complex networks and patch models according to claim 1, characterized in that: The dynamic allocation model for medical resources is as follows: A logistic growth model is used to simulate changes in the number of beds, expressed as: ； in, This indicates the capacity for the production and allocation of medical resources. Indicates the maximum number of beds available during an infectious disease outbreak; The daily number of beds is calculated using a piecewise function based on different stages of the disease progression, as shown below: ； in, This indicates the initial bed capacity for COVID-19 patients during an infectious disease outbreak. Representative city This is a critical moment to begin increasing medical resources; therefore, the daily potential number of empty beds is calculated as follows: =max .

9. A method for medical resource allocation based on temporal complex networks and patch models according to claim 1, characterized in that: The specific operations of step 4 are as follows: First, model parameter estimation and data fitting are performed. Latin hypercube sampling and Markov chain Monte Carlo simulation are used to estimate unknown parameters, and the SEIQHDR patch model is used to fit the daily number of new cases and hospitalizations. The ode45 function in MATLAB software is used to calculate the daily number of available hospital beds using the network density formula and clustering coefficient formula, and sensitivity analysis is performed. Then, the dynamic allocation model of medical resources is used to analyze the relationship between population flow and medical resource status in "key" areas.

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