A method for tracing the incubation period asymptomatic transmission based on the sair model

Abstract

The application relates to a tracing method for latent period asymptomatic transmission based on an SAIR model, which comprises the following steps: S1, determining a single initial transmission source node in an interpersonal relationship complex network, and transmitting according to the source node according to an SAIR model until a moment t0; S2, randomly selecting a certain number of nodes as observation nodes, and knowing the state of the observation nodes at the moment; S3, screening a candidate source node set in the complex network by using a reverse propagation algorithm according to the state of the partial observation nodes, and the hop distance and the effective distance, and the number of nodes is determined by a sampling method; S4, constructing a dynamic message transmission equation according to the SAIR model, and calculating the probability of the candidate source nodes being in different states at a moment Delta t by using the partial observation nodes; and S5, calculating the likelihood probability of the nodes in the candidate source node set becoming sources by using an average field theory, and the source is the node with the maximum probability. According to the application, the spread of infectious diseases of patients with latent period and asymptomatic can be accurately reflected, and the problem of tracing the source of infection can be solved.

Description

Technical Field

[0001] This invention relates to a method for tracing the source of infectious diseases in complex networks using maximum likelihood estimation and backpropagation, and specifically to a method for locating source nodes based on the SAIR model containing asymptomatic transmission during the incubation period and a dynamic information transfer algorithm. Background Technology

[0002] Respiratory infectious diseases such as influenza and tuberculosis have high mortality rates and are highly contagious, posing a serious threat to human health. Therefore, when an outbreak occurs, quickly and accurately determining the origin of the disease is of great practical significance for disease prevention and control. In short, the problem of disease tracing refers to inferring the source of a disease based on epidemiological disease transmission models and interpersonal contact networks, and based on the observed situation of all or some infected individuals (often referred to as a snapshot).

[0003] In recent years, the problem of source tracing in interpersonal contact networks has attracted increasing attention from scholars, resulting in a large number of influential source tracing studies based on different propagation models. Some researchers have studied the centrality of infected subgraphs. For example, Shah et al. (SHAH D, ZAMAN T. Rumor centrality: a universal sourcedetector; proceedings of the Proceedings of the 12th ACM SIGMETRICS / PERFORMANCE joint international conference on Measurement and Modeling of Computer Systems, F, 2012 [C].) proposed Rumor centrality (RC) in a tree structure based on the SI model. This algorithm posits that the probability of a node in an infected subgraph being a source of propagation is proportional to the count of all possible sequences of propagation from that node to other nodes. Zhu et al. (ZHU K, YING L. Information sourcedetection in the SIR model: A sample-path-based approach [J]. IEEE / ACMTransactions on Networking, 2014, 24(1): 408-21.) proposed Jordan centrality for the source tracing problem under the SIR model. Jordan centrality is defined as the maximum distance to all infected nodes, and based on this idea, a backpropagation algorithm was proposed for source tracing. Some researchers trace the source by calculating the likelihood probability and posterior probability of the source node. For example, Lokhov et al. (LOKHOV AY, MéZARD M, OHTA H, et al. Inferring theorigin of an epidemic with a dynamic message-passing algorithm [J]. PhysicalReview E, 2014, 90(1): 012801.) proposed a dynamic information passing algorithm based on dynamic message equations on the SIR model. This method traces the source by calculating the posterior probability of the node given snapshot information.Chang et al. (CHANGB, CHEN E, ZHU F, et al. Maximum a posteriori estimation for informationsource detection [J]. IEEE Transactions on Systems, Man, and Cybernetics:Systems, 2018, 50(6): 2242-56.) used the posterior probability of a node becoming a source by calculating the likelihood probability of the propagation path and the prior probability of centrality on a weighted network graph under SI model diffusion to trace the source. In addition, LI et al. (LIL, ZHOU J, JIANG Y, et al. Propagation source identification of infectious diseases with graph convolutional networks [J]. Journal of biomedicalinformatics, 2021, 116: 103720.) proposed a label propagation framework based on the SI model, and inspired by this, proposed a new source identification graph convolutional network framework (SIGN) for source detection tasks. YANG et al. (YANG X, ZHU Z, YU H, et al. Anaming game-based method for the location of information source in social networks [J]. Complexity, 2020, 2020.) used game theory to accurately locate the source of information. This method can be applied to various infectious disease models such as SI, SIR, and SIS.

[0004] The above work is based on traditional infectious disease models. However, many diseases currently exhibit asymptomatic transmission, and the limited node states in traditional models are insufficient to describe the transmission of infectious diseases during the incubation period. For example, the source of infection for tuberculosis is Mycobacterium tuberculosis. After a patient carries this bacterium, no symptoms of infection will appear for 4 to 8 days, but the patient is still infectious during this period. Diseases such as AIDS and measles also have similar transmission characteristics. Based on this characteristic, many works have incorporated the incubation period state into traditional transmission models. ZHOU et al. (ZHU G, FU X, CHEN G. Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks [J]. Applied Mathematical Modelling, 2012, 36(12): 5808-17.) proposed a generalized epidemic model on complex heterogeneous networks based on the SIR model with an incubation period and studied how heterogeneous connection patterns and underlying network structures affect disease transmission. Many other works have improved upon the SEIR model, establishing dynamic models of the COVID-19 epidemic with incubation periods and predicting the development trend of the epidemic, but none have studied the issue of tracing the origin of the virus in the new models. Summary of the Invention

[0005] The purpose of this invention is to address the aforementioned problems by proposing a source tracing method for asymptomatic transmission during the incubation period based on the SAIR model.

[0006] The objective of this invention is achieved as follows: a source tracing method for asymptomatic transmission during the incubation period based on the SAIR model, comprising:

[0007] S1. In a complex network of interpersonal relationships, identify a single source node for initial propagation, and propagate according to the SAIR model based on the source node until a certain moment. ;

[0008] S2. Randomly select a certain number of nodes as observation nodes, and the state of the observation nodes at that moment can be known.

[0009] S3. Based on the state of some observed nodes, combined with hop distance and effective distance, the backpropagation algorithm is used to screen the candidate source node set in the complex network. The number of nodes is determined by the sampling method.

[0010] S4. Construct a dynamic message propagation equation based on the SAIR model, and use some observation nodes to calculate the candidate source node's position. The probability of being in different states at any given time;

[0011] S5. Calculate the likelihood probability of a node in the candidate source node set becoming a source using mean field theory. The node with the highest probability is the source.

[0012] S1-1, the source node propagates up according to the SAIR model until a certain moment. This includes: abstracting real-world interpersonal interactions into complex interpersonal networks. , .in It is a set of nodes. It is an edge set. It is each edge Propagation probability The set. In the SAIR model, nodes have four states: susceptible state S, infected state I, asymptomatic latent state A, and recovered state R. At the initial time... In the network, there exists a node in the infected state I, i.e., the source of infection s*, and all other nodes are in the susceptible state A. The infection then propagates according to the SAIR model in each discrete time slice until it reaches... The moment ends. The following propagation process occurs within each unit of time:

[0013] (a) When a node in state S comes into contact with a node in state I or state A, the node will... The probability of being infected and thus carrying the pathogen and becoming a C state;

[0014] (b) A node in state C will not maintain that state, but will instead change its state based on its own characteristics with probability. Directly exhibiting symptoms of infection, thus progressing to state I, or... The probability is considered to be that the virus has entered the incubation period, and in They did not show symptoms of infection within a short period of time;

[0015] (c) The node in state A during its latency period Symptoms of infection will appear after a period of time, therefore The node that is in state A will then transition to state I;

[0016] (d) A node in state I will at each time step... The probability of a node becoming a node in state R changes, and nodes in state R no longer propagate.

[0017] S2-1, The selection of observation nodes is carried out by random selection methods, including: The number of times to randomly select is (in Nodes representing a percentage of the total number of observation nodes are considered as observation nodes. The state at any given time is known, but the states of all nodes in the graph except the observed node are unknown.

[0018] S3-1, the method for determining the number of candidate nodes includes: treating the state of a subset of observed nodes as a random sample of the propagation status of all nodes. Assume that the proportion of infected nodes among the subset of observed nodes is... Therefore, it is possible that during the entire infection process... One infected node.

[0019] S3-2, based on the states of some observed nodes combined with hop count distance and effective distance, the backpropagation algorithm is used to filter the candidate source node set in the complex network. The steps are as follows:

[0020] 1) First, create a structure of size [size missing]. min-heap set ;

[0021] 2) The A, I and R state nodes in the observation node set are placed into the set Sender_Set and initialized with their own unique information (here called ID information, including id and effective distance ed=0). The probability of this information being propagated outward is 1.

[0022] 3) Each node in the Sender_Set set propagates all the ID information it contains, and the effective distance of the edges traversed is accumulated for each propagation.

[0023] 4) After each node receives the ID information, it first determines whether it is the first time it has received the information. If it is the first time, it copies the ID information and the corresponding effective distance ed and saves them. If it is not the first time it has received the information, it updates the previously saved effective distance ed of the ID information to the smaller one.

[0024] 5) Add all nodes containing ID information to the Sender_Set collection;

[0025] 6) Repeat the above propagation process steps until... One cycle;

[0026] 7) Traverse each node, if the node If it contains all the ID information, then add it. and maintain (by adjusting) Always for (small and large min-heaps)

[0027] 8) Output the set of candidate source nodes .

[0028] S4-1, Constructing the dynamic message propagation equation based on the SAIR model includes: Deriving the dynamic message passing equation from the current time step to the next time step based on the propagation characteristics of the SAIR model and the information of known complex networks. This equation describes the probability of transitions between different states. Once the initial state of each node is determined, the probability of a node being in each state after each propagation can be derived from the dynamic message passing equation. This process is iterated until the final state is obtained. The probability value at time t.

[0029] The following dynamic message passing equations can be obtained through derivation on the SAIR model.

[0030]

[0031]

[0032]

[0033]

[0034] in Represents a node in the graph ,node exist The probability of being in state S at any given time is denoted as . The probability of being in state A is denoted as . The probability of being in state I is denoted as The probability of being in state R is From the start time to During the transmission of time, nothing happened. Node to The probability of a node spreading a pathogen can be defined as follows: From the start time to During the transmission of time, nothing happened. Node to The node spreads pathogens, but the node exist The probability of being in state A or I at any given time is defined as ;definition In order to enable nodes to perform dynamic information dissemination processes exist Never send messages to neighboring nodes The probability of disease transmission, i.e., nodes In The probability of a state.

[0035] Given the initial conditions and the source node, we can obtain the precise values ​​of the marginal probabilities of each node in various states at different times through continuous iteration. Here, the initial values ​​are set as follows:

[0036] ,

[0037] .

[0038] S5-1, using mean-field theory to calculate the likelihood probability that a node in the candidate source node set is a source includes: knowing that each observed node is in the last... Given the probability value at time t, we can use mean-field theory to calculate the likelihood probability of a candidate node as a source. The node with the highest likelihood probability is the source node. The expression is as follows:

[0039]

[0040] in The meaning is: if node If the source is an observation node in state S, then the observation node in state S is... The probability that the time is exactly in state S. The other three probability values ​​have similar meanings.

[0041] The objective function is obtained as follows:

[0042]

[0043] Where U represents the set of candidate nodes; s* is the source node obtained by this method.

[0044] The advantages and beneficial effects of this invention are as follows:

[0045] 1) This invention proposes a method for locating source nodes based on the SAIR model, which includes asymptomatic transmission during the incubation period, and a dynamic information transmission algorithm. The method simulates all node states in the network as follows: susceptible state S, infected state I, asymptomatic latent state A, and recovered state R. This model can accurately reflect the spread of infectious diseases in asymptomatic patients during the incubation period, and helps solve the problem of tracing the source of infection.

[0046] 2) A dual-distance backpropagation algorithm RPDD based on hop count distance and effective distance is proposed to filter nodes, thereby reducing the complexity of the algorithm.

[0047] 3) A source tracing method based on dynamic information transmission is proposed based on the SAIR model. This method does not require the entire infected subgraph of the underlying graph at the final time step; it only needs to know the state of some observed nodes to locate the source, which greatly reduces computational complexity and improves algorithm efficiency.

[0048] 4) This paper validates the proposed source tracing method SLADMP in three artificial networks and three real networks. The results show that, compared with other classic methods, the proposed method can more effectively identify the source of transmission, and thus play a positive role in the prevention and control of infectious diseases. Attached Figure Description

[0049] Figure 1 This is a schematic diagram of the traceability method of one or more embodiments of the present invention;

[0050] Figure 2 This is a schematic diagram of SAIR model propagation according to one or more embodiments of the present invention;

[0051] Figure 3 The accuracy graphs of different algorithms (RC, JC, DC) in one or more embodiments of the present invention at different propagation times on different networks (ER, BA, WS, Graph A, Congress Votes, Football) are shown.

[0052] Figure 4 The average error distance graphs are obtained by different algorithms (RC, JC, DC) in one or more embodiments of the present invention at different propagation times in different networks (ER, BA, WS, Graph A, Congress Votes, Football).

[0053] Figure 5 The accuracy graphs of different algorithms (RC, JC, DC) in one or more embodiments of the present invention are obtained under different networks (ER, BA, WS, Graph A, Congress Votes, Football) with different proportions of observation nodes.

[0054] Figure 6 This is an average error distance map obtained by different algorithms (RC, JC, DC) in one or more embodiments of the present invention under different networks (ER, BA, WS, Graph A, Congress Votes, Football) with different proportions of observation nodes. Detailed Implementation

[0055] To make the objectives, technical solutions, and advantages of this disclosure clearer, the present invention will be further described in detail below with reference to specific embodiments. The described embodiments are merely some examples of the present invention.

[0056] The technical solution of the present invention to solve the above-mentioned technical problems is as follows:

[0057] Figure 1 This is a flowchart of an embodiment of the present invention; as shown below. Figure 1As shown, this invention provides a method for tracing the source of asymptomatic transmission during the incubation period based on the SAIR model, comprising:

[0058] S1 identifies a single initial propagation source node in a complex interpersonal network, and propagates upwards according to the SAIR model based on the source node until a certain moment. .

[0059] For example, we can first abstract real-world interpersonal relationships into a complex network of interpersonal relationships. From the figure A single node is randomly selected as the source, and then the SAIR model is used to spread the infection to other nodes. At this point, the infection network with known sources can be obtained. Assuming the initial infection source is unknown, this can be used to verify the accuracy of the method. The propagation of the SAIR model is as follows: Figure 2 As shown.

[0060] S2 randomly selects a certain number of nodes as observation nodes, and the state of the observation nodes at that moment can be known.

[0061] For example, suppose the number of nodes in the graph is... Randomly select a quantity from the diagram. (in Nodes representing a percentage of the total number of observation nodes are considered as observation nodes. The state at a given time is known, while the states of the other nodes are unknown.

[0062] S3 uses the backpropagation algorithm to filter candidate source node sets in complex networks based on the state of some observed nodes, combined with hop distance and effective distance. The number of nodes is determined by the sampling method.

[0063] The state of a subset of observed nodes is considered as a random sample of the propagation status of all nodes. Assume that the proportion of infected nodes among the subset of observed nodes is... Therefore, it is possible that during the entire infection process... One infected node.

[0064] For example, a source node should meet the following two conditions: 1) The hop distance from the source node to all infected nodes in the graph should be less than or equal to the propagation time. 2) In backpropagation, the candidate source node should contain the ID information of all infected nodes (A, I, and R state nodes) in the observation node set, and should rank high in the Jordan centrality ranking calculated using the effective distance. .

[0065] Using the backpropagation algorithm mentioned above, nodes meeting the criteria can be selected as the candidate source node set S. S4 constructs a dynamic message propagation equation based on the SAIR model, and uses a subset of observed nodes to calculate the candidate source nodes' positions. The probability of being in different states at any given time;

[0066] Traverse all nodes in the graph, setting the visited nodes to state I and the remaining nodes to state S. Assign values ​​to the relevant parameters of all nodes and edges in the graph using the following formula.

[0067] , , , .

[0068] Calculate the value of each node i in the candidate node set S according to the dynamic message passing equation mentioned above. Time's up The probability of each node being in various states at any given time. Obtain the probability of each node being in various states at the final time step. , , , .

[0069] S5 uses mean-field theory to calculate the likelihood probability that a node in the candidate source node set becomes a source, and the node with the highest probability is the source.

[0070] Calculate the likelihood probability of the candidate node based on the calculated probability of the observed node and the actual state of the observed node. Substitute into the objective function , can be obtained This is the source node obtained by this method.

[0071] The parameters of this method are set as follows: the propagation probability of the edge. node recovery probability The probability of a node transitioning from state C to state I The latency of a node in state A In the experiment, any node in the graph was randomly selected as the source of disease transmission each time, and the SAIR model proposed in this paper was used for transmission. The transmission stopped when the set transmission time was reached. After stopping, a certain percentage of nodes were randomly selected from the transmission state at the last moment as observation nodes.

[0072] Based on the SAIR model, this method sets up two experimental scenarios: 1) using propagation time as a variable (the propagation time set in the experiment). 1) Record the source localization performance of different algorithms under different propagation times (2 to 10 units of time); 2) Record the source localization performance of different algorithms under different proportions of observation nodes (5% to 50% in this paper) using the proportion of observation nodes as a variable.

[0073] Figure 3 This chart compares the accuracy of our proposed method with other methods across various networks at different propagation times. As propagation time increases, the accuracy of all source tracing algorithms gradually decreases. This is because the increased propagation time leads to a larger number of infected nodes, thus expanding the range of candidate sources and making it more difficult to find the true source. The SLADMP algorithm proposed in this paper demonstrates the best performance across different datasets. This is partly because the SAIR model has a latency state A, where nodes can continuously infect their neighbors for a period, causing a shift in the centrality of the infection graph. Therefore, relying solely on network topology is insufficient for various centrality algorithms (RC, JC, DC), requiring them to consider the time factor. Our proposed algorithm first filters candidate nodes based on network topology and then considers the latency of state A, resulting in more accurate source location. Furthermore, with only a subset of observed nodes, centrality algorithms cannot obtain sufficient information, thus ignoring the possibility of nodes in unknown states (besides the observed nodes) becoming sources. The SLADMP algorithm can use existing observation nodes to judge every possible candidate source node in the entire underlying graph, thus obtaining more comprehensive information and more accurate results.

[0074] Figure 4 This chart compares the average error distance of our proposed method with other methods on various networks at different propagation times. As propagation time increases, the average error distance of each source tracing algorithm also gradually increases. Similarly, increased propagation time leads to an increase in the number of infected nodes, making the infection graph structure more complex and increasing the difficulty of finding the source. The average error distance of our proposed SLADMP algorithm is approximately one hop on the BA scale-free network and the WS small-world network, while the error is slightly higher on the ER network. This is because the ER network structure is relatively dispersed, resulting in multiple possible paths for infectious disease propagation. The behavior of randomly selected observation nodes may slightly differ from the various propagation scenarios in the underlying graph. In contrast, the WS small-world network and the BA scale-free network have higher clustering coefficients, making the propagation paths of infectious diseases clearer than random networks, and the infected nodes are more concentrated. Therefore, the information exhibited by some observation nodes is relatively more regular, facilitating accurate search for the source node.

[0075] from Figure 4As can be seen, the SLADMP algorithm achieves the smallest average error distance across various datasets, indicating that the source found by the algorithm in this paper is closer to the true source. In the actual spread of an epidemic, timely identification of close contacts of the source is extremely important and effective for epidemic prevention and control.

[0076] Figure 5 This chart compares the accuracy of our method with other methods across various networks with different proportions of observation nodes. As the proportion of observation nodes increases, the source tracing accuracy of the algorithm on each dataset also improves. This indicates that more observation nodes help the algorithm find the source node more effectively, which aligns with common understanding in real-world scenarios. Figure 5 It can be seen that the SLADMP algorithm has a significant advantage over other algorithms in some nodes, with an accuracy of 40% in most networks. Moreover, the increase in accuracy slows down after the proportion of observation nodes is 0.3. This shows that the number of observation nodes has a marginal effect on the accuracy of source tracing. The algorithm can detect the source well even with a low proportion of observation nodes, which is more in line with the actual situation during the epidemic.

[0077] Figure 6 This chart compares the average error distance of our proposed method with other methods across various networks with different proportions of observation nodes. As the proportion of observation nodes increases, the average error distance of the algorithms gradually decreases across different datasets. The proposed SLADMP algorithm, however, maintains an average error distance of approximately one hop across all datasets, demonstrating superior source tracing performance.

[0078] The above are merely specific embodiments of the present invention. Any omissions, modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of one or more embodiments in the specification should be included within the protection scope of the present invention.

Claims

1. A method for tracing the incubation period of asymptomatic transmission based on the SAIR model, characterized in that, The following steps are included: S1, determining a single initial spreading source node in a complex network, spreading according to the SAIR model from the source node until a certain time ; S2, a certain number of nodes are randomly selected as observation nodes, and the state of the observation nodes at the moment is known; S3, according to the state of part of the observation nodes combined with the hop distance and the effective distance, a candidate source node set is screened in the complex network by using the back propagation algorithm, and the number of nodes is determined by the sampling method; S4, constructing dynamic message propagation equation according to SAIR model, calculating the probability of candidate source node in different state at time t by using part of observed nodes ; S5, the average field theory is used to calculate the likelihood probability of the nodes in the candidate source node set to become sources, and the one with the largest probability is the source.

2. The SAIR model based contact tracing method for the incubation period of asymptomatic transmission according to claim 1, wherein, The method for determining a single initial propagation source node in a complex network, according to the source node propagates in the SAIR model until a certain time Comprising: Real-world interpersonal interactions are abstracted into a complex network of interpersonal relationships. ,in It is a set of nodes. It is an edge set. It is each edge Propagation probability The set; in the SAIR model described above, at the initial time... There exists an infected node in the network, i.e., the source of infection s*, which then propagates according to the SAIR model in each discrete time slice until it reaches... The moment has ended.

3. The SAIR model based contact tracing method of asymptomatic transmission of incubation period as claimed in claim 1, wherein, The random selection of a certain number of nodes as observation nodes, and the state of the observation nodes at the moment is known includes: After propagation according to S2, The number of times to randomly select is The nodes are used as observation nodes, where The percentage of observation nodes; the observation nodes in The state at any given time is known, but the states of all nodes except the observed node are unknown.

4. The SAIR model based contact tracing method of asymptomatic transmission of incubation period as claimed in claim 1, wherein, The candidate source node set is screened in the complex network by using the back propagation algorithm according to the state of part of the observation nodes combined with the hop distance and the effective distance, and the number of nodes is determined by the sampling method includes: The proportion of infected nodes in the partial observation nodes obtained in S3 is Therefore, there can be infected nodes in the entire infection process; and nodes are obtained as the candidate source node set by using the back propagation algorithm.

5. The SAIR model based contact tracing method of asymptomatic transmission of incubation period as claimed in claim 1, wherein, The dynamic message propagation equation is constructed according to the SAIR model, and the probability of candidate source nodes at different states at a moment is calculated by using partial observation nodes include: According to the state transition relationship of the SAIR model, the dynamic information transmission equation is derived and simplified; under the condition that the initial condition and the source node are known, the accurate value of the edge probability of each node in various states at different moments is obtained by continuous iteration.

6. The SAIR model based tracing of the incubation period asymptomatic transmission method according to claim 1, wherein, The average field theory is used to calculate the likelihood probability of the nodes in the candidate source node set to become sources, and the one with the largest probability is the source includes: According to the true state of the observation nodes obtained in S3 and the accurate value of the edge probability of each observation node in various states at different moments obtained in S5, the likelihood probability of each node in the candidate node set in S3 to become a source is calculated, and the one with the largest likelihood probability is considered to be the true source.

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