Infectious disease transmission dynamic analysis and containment control method based on double-layer weighted network

Abstract

The invention provides an infectious disease transmission dynamic analysis and containment control method based on a double-layer weighted network, and the method comprises the steps: constructing a double-layer weighted network model which comprises a disease transmission layer and an information diffusion layer, carrying out the preprocessing of the obtained data, and respectively mapping the data into weights between the nodes of the disease transmission layer and the information diffusion layer; based on a microcosmic Markov chain approximation method, performing propagation dynamics analysis on the double-layer weighted network model, establishing an infectious disease propagation system dynamics equation with periodic weight characteristics, and determining a propagation threshold value; introducing a degree threshold criterion into an information propagation layer, and forcibly converting nodes with degrees greater than or equal to a preset threshold into an informed state from an uninformed state; and constructing a comprehensive cost function composed of infection loss and intervention cost, and optimizing a control threshold and a controlled node set. According to the method, the infectious disease diffusion period can be remarkably shortened, the utilization rate of prevention and control resources is improved, and higher robustness and expandability are realized in a complex network environment.

Description

Technical Field

[0001] This invention relates to the technical field of infectious disease modeling and control, and in particular to a method for dynamic analysis and control of infectious disease transmission. Background Technology

[0002] In modern society, characterized by high population mobility and close social connections, the spread of infectious diseases has accelerated dramatically. With the advancement of globalization, international travel, trade, and urbanization are accelerating, significantly increasing population mobility between different regions. This further exacerbates the risk of cross-regional transmission of infectious diseases. Against this backdrop, outbreaks of new infectious diseases are often accompanied by rapid international spread, placing greater pressure on governments and public health institutions worldwide to control and prevent their spread.

[0003] However, how to effectively control the spread of infectious diseases remains a core issue that urgently needs to be addressed. Traditional prevention and control measures mostly focus on long-term suppression of infection rates, such as reducing the probability of transmission through vaccination or social distancing. However, these methods are often ineffective when resources are limited and the initial outbreak of infectious diseases is rapid. Especially during the critical window period of infectious disease transmission, the lack of timely and effective short-term interventions may lead to a rapid increase in the number of infections, exceeding the capacity of the medical system. Therefore, how to utilize the structural characteristics of complex networks to achieve precise and rapid containment of infectious diseases under limited resource conditions has become an important direction of current research.

[0004] To address the aforementioned issues, researchers have proposed various control strategies in recent years, including dynamic intervention methods based on optimal control theory, critical node immunization methods based on node ranking, and behavioral guidance strategies based on game theory. Among these, the restraint control method has attracted widespread attention due to its strong selectivity and high resource utilization efficiency. This method forcibly transforms some highly connected nodes from an unaware state to an informed state, thereby rapidly forming a core of public opinion at the information dissemination layer, driving the diffusion of awareness throughout the network, and indirectly reducing the risk of disease transmission. Compared to uniform or random intervention, the selective needle control strategy based on degree centrality can achieve more significant prevention and control effects at a lower cost, balancing efficiency and feasibility.

[0005] In summary, with the increasing complexity of social network structures and the diversification of infectious disease transmission patterns, traditional infectious disease models based on the homogeneity assumption can no longer meet real-world needs. A two-layer coupled model integrating complex networks, dynamic weighting mechanisms, and optimal control provides new theoretical support and application prospects for research on infectious disease transmission mechanisms and the design of precise prevention and control strategies. The introduction of needle-based control methods not only has high theoretical value but also demonstrates good practical application potential, providing an important reference for future responses to emerging infectious diseases. Existing research often only considers static or simple weighted networks, making it difficult to reflect the periodic changes in interpersonal contact intensity and the heterogeneity of information transmission in reality. Traditional control strategies mostly rely on global intervention, which is costly and has limited efficiency. Summary of the Invention

[0006] To address the shortcomings of the aforementioned background technologies, this invention proposes a dynamic analysis and control method for infectious disease transmission based on a two-layer weighted network. This method solves the problem that existing methods struggle to reflect the periodic changes in the intensity of interpersonal contact and the heterogeneity of information dissemination in reality. This invention constructs a two-layer weighted network, where the disease transmission layer is an undirected time-varying network and the information diffusion layer is a directional degree-central network. A better infectious disease prevention and control strategy is achieved using a control method.

[0007] The technical solution of this invention is implemented as follows: a method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network, the steps of which include:

[0008] Step 1: Based on complex network theory, construct a two-layer weighted network model, including a disease transmission layer and an information diffusion layer. After preprocessing the obtained population contact frequency matrix and information diffusion network, map them to the contact intensity weights between nodes in the disease transmission layer and the information diffusion weights between nodes in the information diffusion layer, respectively.

[0009] Step 2: Based on the microscopic Markov chain approximation method, conduct propagation dynamics analysis on the constructed two-layer weighted network model, establish the infectious disease propagation dynamics equation with periodic weight characteristics, and determine the propagation threshold;

[0010] Step 3: Design and optimization of restraint control: Introduce a degree threshold criterion at the information propagation layer, and forcibly change the unknown state to an known state for nodes with a degree greater than or equal to the preset threshold; construct a comprehensive cost function composed of infection loss and intervention cost, and optimize the control threshold and the set of controlled nodes.

[0011] Preferably, the disease transmission layer is an undirected weighted network, and the information diffusion layer is a directed weighted network; the undirected weighted network and the directed weighted network are structurally parallel and node-aligned.

[0012] The edge weights of the undirected weighted network are determined by the number of individual contacts and their periodic changes, truly reflecting the dynamic fluctuations in the intensity of interpersonal interaction over different time periods; the edge weights of the directed weighted network are allocated by the degree centrality of nodes, reflecting the relative influence of information sources in diffusion.

[0013] Preferably, the undirected weighted network and the directed weighted network contain the same set of nodes. Each node represents the disease state and information state of the same individual in the two layers, respectively. The interaction between the undirected weighted network and the directed weighted network is achieved through the coupling of state variables: the informed state of the nodes in the information diffusion layer affects the infection probability in the disease transmission layer, and the infection state in the disease transmission layer affects the perceived transmission probability in the information diffusion layer.

[0014] Preferably, the state of each node is modeled using a micro-Markov chain approximation method, resulting in five states: uninformed susceptibility (US), informed susceptibility (AS), informed infection (AI), uninformed recovery (UR), and informed recovery (AR). Based on the transition process between these five states, a dynamic equation for the transmission of infectious diseases with periodic weights is established as follows:

[0015] ; ;

[0016] ;

[0017] ;

[0018] ;

[0019] in, , They represent in , Time Node The probability of being in an unknown vulnerable US state. , They represent in , Time Node The probability of being in a state of informed vulnerability to AS. and They represent in , Time Node The probability of being in an informed recovery AR state. and They represent in , Time Node The probability of unknowingly reverting to the UR state. and They represent in , Time Node The probability of being in a state of informed infection with AI; This represents the information diffusion layer perception propagation probability based on node edge weights. This represents the infection probability of an unknown susceptible node based on the number of contacts and the periodic weight of the disease transmission layer; This represents the infection probability of a known susceptible node based on the number of contacts and the periodic weight of the disease transmission layer; This represents the probability that a sensing node on the information diffusion layer becomes a non-sensing node. This indicates the probability that a recovered node will become a susceptible node in the disease transmission layer; This represents the probability that an infected node becomes a recovering node on the disease transmission layer.

[0020] Preferably, the sensing propagation probability of the node ;in, The number of sensing nodes surrounding the information diffusion layer node; nodes From a single directed weight Perception nodes The probability of sensing transmission is ;

[0021] The undirected weighted network is an undirected weighted BA network, where nodes represent individuals, edges represent physical contacts, and the weight of each edge represents the number of contacts between individuals.

[0022] The infection probability of unknown susceptible nodes ;in, The total number of infected nodes surrounding a node in the disease transmission layer; the infection probability obtained at time t from an infected neighbor node. ;in, This represents the basic probability of infection at unknown susceptible nodes within the disease transmission layer. These are the initial values ​​for the basic weights of the disease transmission layer. The amplitude of the time-varying weighting formula at the disease transmission layer;

[0023] The known infection probability of susceptible nodes The infection probability obtained from an infected neighbor node at time t. ;in, This represents the basic probability of susceptible nodes being infected at the disease transmission layer.

[0024] node The out-degree is recorded as Degree centrality is ,in, This represents the maximum degree of nodes in the information diffusion layer. yes The neighboring nodes, from the node To the node weights Due to the directional nature of the information diffusion layer, the weights ,at this time, For nodes The degree center value.

[0025] Preferably, the propagation threshold is calculated as follows: at time... The probability of infection obtained from an infected neighbor node. In this case, the probability is: Assume the basic probability of infection at unknown susceptible nodes in the disease transmission layer. Small enough to have a probability Applying an exponential function and performing a first-order Taylor expansion near the probability x = 0, the infection probability of being infected by an adjacent infected node in the disease transmission layer is approximated as:

[0026] ;

[0027] At the point of maximum infection rate, the effective infection matrix The transmission threshold of infectious diseases is determined by conditions. The propagation threshold is defined as follows: ;in, The weighted adjacency matrix representing the disease transmission layer The largest eigenvalue.

[0028] Preferably, a degree-based constraint control strategy is adopted at the information propagation layer to selectively force nodes with high connectivity to transition from an unknown state to a perceived state: when the degree of a node on the information propagation layer... Exceeding the predefined threshold When the control intervention is activated, the controller with the threshold criterion is:

[0029]

[0030] in, This indicates whether node i should be forced into an informed state. 1 means the node should be forced into an informed state, and 0 means the node should not be forced into an informed state.

[0031] Preferably, by forcing unknown susceptible nodes to become known susceptible nodes and unknown recovery nodes to become known recovery nodes, the infectious disease transmission dynamics equation after adding restraint control becomes:

[0032]

[0033]

[0034] .

[0035] Preferably, the threshold By analyzing the degree distribution of the network, a comprehensive cost function is constructed, and the optimal threshold is found by iterating through the network from minimum to maximum. To minimize the overall cost function; restraint control is based on the degree threshold method to formulate an optimal control strategy, dynamically allocating sensing based on the node's connectivity and current state; at the node's degree... Exceeding the predefined threshold The system activates the restraint control mechanism, selecting nodes with a degree greater than the threshold to forcibly switch to an informed state. The intervention intensity is continuously adjustable within the operational range.

[0036] Preferably, the comprehensive cost function is:

[0037] ;

[0038] Where J1 is the cumulative value of infection loss, and J2 is the cumulative value of intervention cost. Indicates time The total number of infections This represents the social / economic cost of each infection. The unit cost reflecting control efforts, the ratio Represents the target node The proportion of uninformed individuals in the group This refers to the number of all nodes with degree k on the information diffusion layer. The information diffusion layer has a degree of The number of unsuspecting nodes, express At all times, in degree The amount received by one type of node that is forcibly switched to an informed state;

[0039] After anonymization, the population contact frequency matrix and information dissemination network are uniformly mapped to the contact intensity weights and information dissemination weights between nodes.

[0040] Compared with existing technologies, the beneficial effects of this invention are as follows: This invention innovatively constructs a two-layer network structure, where the disease transmission layer is an undirected weighted network that combines the number of contacts and a periodic weight function to characterize the dynamic fluctuations of infection risk, simulating the periodic transmission patterns caused by policy changes, seasonal effects, and behavioral adaptations; the information transmission layer is a directed weighted network that uses a degree-centrality-based weight allocation mechanism to represent the influence of individuals in information diffusion, and achieves bidirectional coupling evolution of disease and information through a five-state Markov model, thus more realistically reproducing the transmission interactions in real-world social networks. This invention proposes a degree-based constraint control strategy, thereby accelerating the global awareness diffusion through network effects and indirectly inhibiting disease transmission. This not only improves the efficiency of information transmission control but also achieves a balance between infection costs and intervention costs through an optimal control framework. It can determine the optimal control threshold under limited resources, effectively reducing infection peaks and achieving low-cost, precise prevention and control. This invention verifies the effectiveness of the theoretical results and control strategy through Monte Carlo simulation and the micro-Markov chain approximation method. The results show that this invention can significantly shorten the infectious disease spread cycle, improve the utilization rate of prevention and control resources, and achieve higher robustness and scalability in complex network environments.

[0041] This invention establishes a comprehensive and innovative method for infectious disease transmission and control, capable of characterizing disease-information co-evolution and enabling rapid intervention, through integrated innovation in modeling methods, weighted time-varying network mechanisms, and control strategies. This invention not only theoretically reveals the profound impact of complex network structures on infectious disease evolution but also provides important practical references for public health institutions to formulate scientific, efficient, and precise prevention and control measures under limited resources, demonstrating significant application value and promising prospects for wider application. Attached Figure Description

[0042] To more clearly illustrate the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the present invention.

[0044] Figure 2 This is a schematic diagram of the structure of the two-layer weighted network of the present invention.

[0045] Figure 3 This is a transmission probability tree diagram for five states.

[0046] Figure 4 This is a schematic diagram of the initial state of the two-layer weighted network of the present invention, wherein (a) is the disease transmission layer and (b) is the information diffusion layer.

[0047] Figure 5 This is a schematic diagram of the degree distribution of the two-layer weighted network of the present invention, wherein (a) is the disease transmission layer and (b) is the information diffusion layer.

[0048] Figure 6 The diagram shows the MMCA evolution of each state of the two-layer weighted network of the present invention, where (a) is the disease transmission layer and (b) is the information diffusion layer.

[0049] Figure 7 The diagram shows the MC evolution of each state of the two-layer weighted network of the present invention, where (a) is the disease transmission layer and (b) is the information diffusion layer.

[0050] Figure 8 This is a schematic diagram of the final state of the two-layer network structure of the present invention, wherein (a) is the disease transmission layer and (b) is the information diffusion layer.

[0051] Figure 9 The diagram shows the five state evolutions of the two-layer network combination of the present invention, where (a) is the disease transmission layer and (b) is the information diffusion layer.

[0052] Figure 10 For the present invention Sensitivity analysis diagrams at different levels of precision, where (a) represents the disease transmission layer and (b) represents the information diffusion layer.

[0053] Figure 11 For the present invention Sensitivity analysis plots at different levels of precision.

[0054] Figure 12 The control of the present invention begins at The state evolution diagram is shown, where (a) is the disease transmission layer and (b) is the information diffusion layer.

[0055] Figure 13 The degree of the present invention is A diagram illustrating the cost of constraint and control.

[0056] Figure 14 For the present invention The five-fold state evolution diagram.

[0057] Figure 15 For the present invention The evolution diagram of each state under optimal control is shown, where (a) is the disease transmission layer and (b) is the information diffusion layer.

[0058] Figure 16 For the present invention The diagram shows the final network state under time-optimal control, where (a) is the disease transmission layer and (b) is the information diffusion layer. Detailed Implementation

[0059] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0060] like Figure 1 This invention proposes a dynamic analysis and control method for infectious disease transmission based on a two-layer weighted network. Building upon existing research, the proposed model expands and innovates in the following aspects: First, it introduces periodically modulated dynamic weights into the disease transmission layer, enabling a more realistic simulation of the impact of seasonal factors, policy changes, or individual behavioral differences on transmission intensity, thus characterizing the periodic fluctuations of infectious diseases. Second, in the information transmission layer, it employs a degree-centrality-based weighted directed network mechanism to accurately describe the core role of high-influence nodes in risk perception and information diffusion. Third, it combines the Markov chain approximation method to construct a coupled transmission model containing five states (US, AS, AI, UR, AR), systematically describing the interaction between disease and information. Fourth, it proposes an optimal dosing control strategy based on degree thresholds, achieving a rapid decrease in the overall infection rate by controlling the state transitions of a small number of key nodes, and striking a balance between reducing infection costs and control costs. The specific implementation steps of this invention are as follows:

[0061] Step 1: Based on complex network theory, construct a two-layer weighted network model, including a disease transmission layer and an information diffusion layer. The disease layer is an undirected weighted network, and the information layer is a directed weighted network. After preprocessing the obtained population contact frequency matrix and information diffusion network, map them to the contact intensity weights between nodes in the undirected weighted network and the information diffusion weights in the directed weighted network, respectively.

[0062] like Figure 2As shown, the disease transmission layer is an undirected weighted network used to characterize the disease transmission process and simulate the physical contact and infection process between individuals. The edge weights of the undirected weighted network are determined by the number of individual contacts and their periodic changes, which can realistically reflect the dynamic fluctuations in the intensity of interpersonal interaction in different time periods (such as weekdays, weekends, and holidays). The information diffusion layer is a directed weighted network used to describe the information diffusion process and characterize the diffusion process of infectious disease-related information. The edge weights are allocated by the degree centrality of nodes, reflecting the relative influence of the information source in the diffusion. The disease transmission layer and the information diffusion layer contain the same set of individual nodes; each node represents the "physiological state" and "cognitive state" of the same individual in both layers; the interaction between the two layers is achieved through the coupling of state variables: the informed state of a node in the information diffusion layer affects its infection probability in the disease transmission layer, and the infection state in the disease transmission layer affects its perceived transmission probability in the information diffusion layer. Therefore, these two layers are parallel in structure and aligned in nodes, and dynamically achieve a two-way interaction between information and disease through state coupling.

[0063] like Figure 3 As shown, this two-layer weighted network structure, combined with the five-state Markov chain approximation method, accurately characterizes the transformation process of an individual between five states: uninformed susceptibility (US), informed susceptibility (AS), informed infection (AI), uninformed recovery (UR), and informed recovery (AR), thereby establishing a dynamic equation for the infectious disease transmission system with periodic weighting characteristics. Individuals in the model can be in five states (US, AS, AI, UR, AR), representing their health status and information awareness level, respectively. An individual refers to a person in a social network (or represents a group of individuals with similar behavioral characteristics). An information source refers to the node in the network that first obtains information about the infectious disease and begins to spread it; this can be considered as the media, government, or some early informants.

[0064] Each node represents an individual, and its data contains two layers of state variables (disease state and information state). The data processed includes node connectivity (adjacency matrix), edge weights (number of contacts or information influence), and the temporal evolution data of each node across five states. Since each node represents a person, this person has a position in both the disease transmission layer and the information diffusion layer. This individual is essentially the physical propagation process of that node in the disease transmission layer.

[0065] This two-layer network structure can simultaneously reflect the interaction between disease transmission and risk perception. Compared with traditional single-layer models, it can better reveal the transmission mechanism of "disease-information" co-evolution in real society, thereby improving the prediction accuracy and intervention feasibility of infectious disease evolution.

[0066] Table 1. Meaning of parameters in this invention

[0067]

[0068] The information diffusion layer represents the information propagation process, in which individuals receive or spread information related to infectious diseases through a directed weighted network. The weight of each directed link is determined by the out-degree of the source node and normalized over the overall size, thus capturing the relative influence of the information source. Individuals in the information diffusion layer can be unaware (U) or conscious (A), and the process of acquiring information depends on whether the individual receives information from neighbors who are already aware of it. The probability of information perception is determined by the basic probability. The input link weights from the sensing nodes, i.e., the sensing propagation probability. Control. Once informed, the risk of node infection decreases due to enhanced preventative measures. Then, the node's perceived propagation probability...

[0069] (1)

[0070] Among them, nodes From a single directed weight Perceiving neighbors The probability of sensing transmission is defined as

[0071] (2)

[0072] The disease transmission layer is an infectious disease transmission network, modeled as an undirected weighted BA (Barabási–Albert, scale-free network model). Here, nodes represent individuals, and edges represent physical contacts. The weight of each edge is a randomly assigned integer reflecting the number of contacts between individuals. Infectious diseases spread through these contacts, and the infection probability is dynamic, modulated not only by the frequency of contact but also by a periodic variation modeled as a cosine function. Importantly, two different infection rates depend on whether the susceptible individual is unaware (US) or aware (AS), respectively determined by the node infection probability. and The probability of an infected individual (AI) reaching a recovery node is indicated. And may re-enter susceptible or informed groups based on their previous status, thus allowing for reinfection cycles. Therefore, the overall effective infection probability of unknown susceptible (US) nodes is...

[0073] (3)

[0074] The probability of infection obtained from an infected neighbor at time t. for:

[0075] (4)

[0076] Similarly, for nodes in the AS state, the effective infection probability is obtained:

[0077] (5)

[0078] (6)

[0079] Considering the discrepancy between the network structure of individuals in reality and the network structure on the internet, this model better reflects the actual human condition. Node The out-degree is recorded as Its degree centrality is defined as ,in This represents the maximum node degree across the entire information diffusion layer. Assume... yes The neighboring nodes are represented by their weights. Note that due to the directional nature of the information diffusion layer, the weights... ,at this time, For nodes The degree center value is obtained, and the algorithm is described above.

[0080] Individual state transitions are modeled using MMCA (Microscopic Markov Chain Approximation), with five possible states: US (unaware of susceptibility), AS (conscious of susceptibility), AI (conscious of infection), UR (unaware of recovery), and AR (conscious of recovery). These transitions are influenced by disease and information dynamics. For example, an unknowingly susceptible individual may become infected through contact with an infected neighbor or through contact with an informed node in the information layer. Similarly, recovered individuals may lose immunity and become susceptible again, possibly retaining or losing consciousness, depending on their transition path.

[0081] This model captures the coupling between the information diffusion layer and the disease transmission layer: disease transmission dynamics depend on an individual's information state (awareness reduces the probability of infection), while information transmission is also influenced by contact with infected individuals or informed neighbors. This two-way coupling allows for a more realistic simulation of behavioral responses and epidemic evolution on complex networks.

[0082] Step 2: Based on the microscopic Markov chain approximation method, conduct propagation dynamics analysis on the constructed two-layer weighted network model, establish the dynamic equation of the infectious disease transmission system with periodic weight characteristics, and determine the propagation threshold.

[0083] Based on the Microscopic Markov Chain Approximation (MMCA) method, the propagation dynamics of the constructed two-layer weighted network model are analyzed, and the effectiveness of the model is verified using the Montto Carlo (MC) method.

[0084] Based on the transformation process between five states, the dynamic equation of the infectious disease transmission system with periodic weighting characteristics, namely the Markov chain model of the two-layer weighted network, is established as follows:

[0085] ;

[0086] ;

[0087] (7)

[0088] ;

[0089] ;

[0090] The definitions of each parameter are shown in Table 1. , They represent in , Time Node The probability of being in an unknown vulnerable (US) state. , They represent in , Time Node The probability of being in an informedly susceptible (AS) state. and They represent in , Time Node The probability of being in an informed recovery (AR) state. and They represent in , Time Node The probability of being in an unaware recovery (UR) state. and They represent in , Time Node The probability of being in a state of informed infection (AI).

[0091] In the early stages of an infectious disease outbreak, by linearizing the transmission process of the disease transmission layer, the critical condition for the transition from sparse infection to a large-scale outbreak is obtained, and the disease transmission threshold is derived from this. This threshold can directly characterize the critical point of an infectious disease outbreak and is used to determine whether the infectious disease will continue to spread in the network, thus providing a theoretical basis for the design of subsequent containment and control strategies. The method for obtaining the transmission threshold is as follows: To derive the transmission threshold of an infectious disease transmission system... The linearized form of the infection dynamics in the initial stage of the outbreak was examined. The infection threshold was obtained using equivalent infinitesimals, and its correctness was subsequently verified through simulation. The probability that a susceptible node is infected by a single infected neighbor at time t is expressed by formula (4), such that:

[0092] (8)

[0093] Assuming the basic probability of infection at unknown susceptible nodes in the disease transmission layer. Small enough to have a probability In this case, the exponential function can be applied. The nearest progressive first-order Taylor expansion. Therefore, the probability of being infected by an adjacent infected node on the disease transmission layer can be approximated as:

[0094] (9)

[0095] When probability When sufficiently small, the first-order approximation This is true. Numerical comparisons show that when the probability... hour, and The values ​​are almost indistinguishable, with a relative error of less than 0.5%. In the two-layer weighted network model of this invention, the critical infection threshold (transmission threshold) is calculated as follows: And because Therefore, under typical parameter settings, the obtained probability is The value is still far below 0.1. Therefore, the approximate probability... This is reasonable under these circumstances. The constructed model calculates the infection rate at which transmission will occur to be 0.0065 using a formula, and the subsequent threshold derived from the simulation mathematical model is 0.0068. This error is acceptable.

[0096] This first-order approximation simplifies the probability. The expression is provided for ease of subsequent analysis. At the point of maximum infection rate (i.e....) At point ), the effective infection matrix is:

[0097] (10)

[0098] The threshold for the spread of infectious diseases is determined by conditions. Definition. Therefore, a propagation threshold can be generated:

[0099] (11)

[0100] in, The maximum eigenvalue of the weighted adjacency matrix representing the disease transmission layer is obtained by calculating the eigenvalues ​​of the weighted adjacency matrix. The transmission threshold is a key parameter for assessing the epidemic potential of an infectious disease. When the actual infection rate is below this threshold, the disease will gradually subside; conversely, if it reaches or exceeds the threshold, the epidemic may gradually spread over time. Determining this parameter provides an important theoretical basis for the formulation of subsequent prevention and control strategies.

[0101] This invention introduces a time-varying weighting mechanism that integrates the number of contacts and a periodic modulation factor into the disease transmission layer, overcoming the limitations of the uniform contact assumption in traditional models. By embedding periodic modulation in the form of a cosine function into the edge weights, the model can capture the dynamic fluctuations in transmission rates caused by seasonal epidemics, policy adjustments, or changes in individual behavior. Simultaneously, the weights of the information transmission layer are normalized based on the degree centrality distribution of nodes, highlighting the crucial role of highly specialized nodes in the diffusion of information within a group. This mechanism not only characterizes the spatial heterogeneity of the network but also its temporal evolutionary characteristics, enabling the model to identify the key roles of super-spreaders and core nodes of public opinion in complex transmission processes, thereby providing theoretical support for developing targeted intervention measures.

[0102] Step 3: Design and optimization of restraint control: Introduce a degree threshold criterion at the information propagation layer, apply control to nodes with a degree greater than or equal to the preset threshold, and force them to change from an unaware state to an aware state; construct a comprehensive cost function composed of infection loss and intervention cost.

[0103] Degree-based constraint control introduces a degree threshold criterion at the information dissemination layer. Nodes with a degree greater than or equal to a preset threshold are controlled, forcibly transitioning from an unaware state to an aware state, thereby accelerating awareness dissemination and indirectly reducing the risk of disease transmission. A comprehensive cost function, composed of infection loss and intervention costs, is further constructed to optimize the control threshold and the set of controlled nodes, achieving a balance between resource utilization and suppression effects. Individual awareness is a key line of defense against the spread of epidemics; informed individuals spontaneously take protective actions, collectively changing the transmission pattern of infectious diseases. This self-organized risk mitigation effect is amplified through social networks, creating cascading protection beyond directly controlled nodes. To mitigate infectious disease outbreaks through behavioral intervention, a degree-based constraint control strategy is adopted at the information dissemination layer. The control mechanism selectively forces nodes with high connectivity to transition from an unaware state to an aware state. This targeted approach optimizes network resource allocation while reducing the transmissibility of influential communicators.

[0104] When the degree of the node on the information propagation layer Exceeding the predefined threshold When the control intervention is activated, the threshold criterion is implemented through the following controller:

[0105] (11)

[0106] in, This indicates whether node i should be forced into an informed state. 1 means the node should be forced into an informed state, and 0 means the node should not be forced into an informed state.

[0107] The dynamics of controlled infectious disease transmission systems include two main interventions: forcing unknown susceptible nodes to become known susceptible nodes, and forcing unknown recovered nodes to become known recovered nodes. In this case, equation (7) becomes:

[0108] ,

[0109] ,

[0110] (12)

[0111] ,

[0112] ,

[0113] Restraint and Control By strategically altering state transitions by transforming selected high-level nodes from an unaware state to a conscious state, the intrinsic coupling between information dissemination and disease transmission dynamics is maintained. This targeted intervention alters the system's natural evolution without disrupting its fundamental interaction mechanisms.

[0114] threshold The threshold should be determined by analyzing the degree distribution of the network, typically choosing a value to identify the most influential propagators. Each network node has its own degree, and the optimal threshold value is determined later by constructing a cost function to adjust the threshold. The optimal threshold is found by iterating through the thresholds from minimum to maximum. Practical considerations include available intervention resources and the required speed of containment. A lower threshold. It expands the scope of control but increases implementation costs, while the higher threshold... Focusing on the most critical nodes may produce localized impacts more quickly. The optimal method for constraint control is to develop an optimal control strategy based on the degree threshold method, dynamically allocating perception enhancement resources according to the node's connectivity and current state. The control mechanism is based on the node's degree... Exceeding the predefined threshold It is activated at any time, and the intervention intensity is continuously adjustable within the operational range.

[0115] The first component of the cost function is the disease prevalence within the penalty control range:

[0116] (13)

[0117] in, Indicates time The total number of infections This represents the social / economic cost of each infection. This factor drives the controller to reduce overall transmission. This refers to the cumulative value of infection losses, i.e., the total cost of infection over a period of time; This refers to the simulation time steps or the upper limit of the control cycle (integral interval).

[0118] The second cost component quantifies the resource expenditure for improving awareness of movement, with the objective function as follows:

[0119] (14)

[0120] in, The unit cost reflecting the control effort, while The ratio represents the target node The proportion of uninformed individuals in the group Refers to all on the information diffusion layer The number of degree nodes. Refers to the information diffusion layer The number of nodes that are unaware of the situation. Indicates that at time t, the degree is The amount of data received by nodes in this category that are forced to become aware of the situation. This normalization ensures effective resource allocation by adjusting the scale of work based on the actual controllable population. An optimal target node degree is obtained through simulation using a cost function.

[0121] The combined objective function minimizes the impact of the epidemic and the cost of control; the combined cost function, i.e., the comprehensive cost function, is as follows:

[0122] (15)

[0123] This method provides the greatest intervention effect when two conditions are met. First, the node must exceed the degree threshold. Secondly, the predicted cost savings from infectious disease control must exceed the control expenditures. After defining and setting the threshold, nodes with a degree greater than the threshold are forcibly switched to an informed state. The overall cost function is then adjusted after the control parameters are changed. When the minimum value is reached (i.e., the reduction in infection loss > the increase in control costs), the cost savings are considered to exceed the control expenditure. Threshold It serves as both a topology filter for node selection and a strategy parameter for controlling the strength of containment measures.

[0124] Data Sources and Preprocessing: Real-world social contact data and information dissemination data can be obtained from the following channels: Disease transmission layer: Contact frequency matrices are obtained through epidemiological surveys, mobile communication trajectories, public transportation travel records, or medical institution reports; Information dissemination layer: Information dissemination networks are obtained through social media interaction data, news dissemination records, or forwarding relationships on public opinion platforms. After anonymization, this raw data is uniformly mapped to contact intensity weights and information dissemination weights between nodes.

[0125] Network mapping and parameter estimation: A two-layer network is established with individuals (or groups) as nodes, maintaining a one-to-one correspondence between nodes; the topological characteristics of the disease transmission layer and the information diffusion layer are determined through statistical analysis (such as weighted degree distribution, community detection, etc.); and transmission parameters (infection rate, recovery rate, probability of forgetting, etc.) are estimated by recording historical infection data and then fitting the parameters.

[0126] Model embedding and dynamic updating: Actual network data is input into the two-layer weighted propagation model of this invention; the model updates node state probabilities in real time to predict the evolution trend of infectious diseases; the latest monitoring data can be accessed through a short-term sliding window to form a dynamic updating system.

[0127] The implementation of the control strategy in reality: In the information diffusion layer, nodes with high centrality of identification (such as key media accounts, community leaders or high-traffic platforms) are identified; "cognitive intervention" or "key publicity" is implemented for these nodes, which is equivalent to mandatory informed consent in the model; the disease transmission layer can correspond to enhanced protection and early vaccination for key areas or high-contact groups (such as medical staff and service industry personnel); by optimizing the control threshold, the optimal allocation of prevention and control resources can be achieved.

[0128] Results Interpretation and Decision Support: The model outputs infection curves, awareness spread curves, and a comprehensive cost function to evaluate the cost-effectiveness of different intervention programs; public health departments can use this to determine the "minimum input, maximum suppression" strategy for practical prevention and control decisions.

[0129] Step 4: Numerical simulation experiments verify the effectiveness of the two-layer weighted network model.

[0130] Numerical simulations of the system, i.e., equation (7), are performed on a typical complex network topology - BA scale-free network. The initial network state is as follows: Figure 4 As shown, these two networks are more consistent with human communication patterns, validating the effectiveness of the model and control strategy. Figure 4 The initial state of the two-layer network is shown. In the disease transmission layer, a few nodes are infected, forming a local concentration; in the information diffusion layer, only a few nodes are aware of the infection, and information has not yet spread. This illustrates that the initial conditions of the model correspond to the early stage of infectious disease transmission, and subsequent evolution verifies the interaction between information diffusion and disease spread. Initial parameters are set, and the initial values ​​of each state are... , , , , , Represents amplitude, Represents the basic weight. Represents phase, the number of times two nodes come into contact. Randomly select a number between 1 and 20, and the remaining parameters are shown in Table 2.

[0131] Table 2 Initial Parameter Settings

[0132]

[0133] Once the network is determined, all information about the network, including its degree, nodes, and degree distribution, can be obtained. The degree distribution is as follows: Figure 5 As shown, this initial configuration reflects a reality where early awareness is sparse and unevenly distributed, while infection has already begun to spread within the network. Since both layers are generated via a BA network, they exhibit a power-law distribution. The disease propagation layer shows a higher degree of concentration in a few hub nodes, while the information diffusion layer exhibits a more dispersed structure due to the directional nature of its edges.

[0134] To evaluate the performance and behavior of the model under the MMCA framework and Monte Carlo simulations, multiple experiments were conducted on the constructed two-layer BA network. The corresponding results are as follows: Figures 6 to 9 As shown.

[0135] MMCA and MC simulations show a consistent co-evolutionary pattern, but differ in dynamic details. Figure 6 , Figure 7 Due to the deterministic probability updates, the MMCA results show more pronounced fluctuations, capturing transient local dynamics through oscillating behavior of infection peaks and consciousness growth.

[0136] In contrast, MC simulations, through ensemble averaging, produce smoother curves that better represent the expected system-level trends. Both approaches are consistent on key phenomena: initial infections grow rapidly, then stabilize, and increased awareness, while slightly lagging but stable, ultimately contributes to epidemic control. While MMCA offers computational efficiency for observing localized changes, MC provides robust validation of the mean dynamics, together validating the model's core mechanism.

[0137] Figure 8The uninterrupted steady-state behavior revealed a local equilibrium within the disease transmission layer, where some nodes continue to recover, but the infection is not completely eradicated. Simultaneously, the perceptual coverage of the information diffusion layer is incomplete; most nodes have acquired protective awareness, while some remain unaware. This steady state illustrates the complex interaction between disease transmission and information diffusion, demonstrating that the system naturally maintains a stable coexistence of infection and awareness without targeted control, highlighting the necessity of precise intervention strategies.

[0138] Figure 9 Comparative analysis of MMCA and MC simulations shows a high degree of consistency in dynamic patterns across the temporal evolution of all five node states. Both methods capture the characteristic popularity curves that gradually decline after the initial rapid growth of artificial intelligence, while cognitively relevant states exhibit a continuous growth characteristic, confirming the synergistic effect between information dissemination and behavioral adaptation. Despite small amplitude and phase differences, this overall consistency strongly validates the reliability of the coupled model.

[0139] Finally, the basic probability was studied through a series of simulations. and The impact on system behavior. For example... Figure 10 As shown in (a), a step size of 0.01 is used for the basic probability. Sensitivity analysis was performed. The results show that when the basic probability... When increased in finer increments, the system exhibits a more pronounced change, especially near the critical point. Figure 10 (b)). This finer granularity helps to determine the precise range of the transition from subcritical to supercritical behavior in infectious diseases. It also reveals the impact of the size of the infected population on the basic probability. The varying nonlinear response underscores the importance of accurately estimating this parameter in real-world implementations. It is worth noting that... Figure 10 The basic probability shown in (b) The critical threshold corresponds to The theoretical calculation result derived in the previous section is the propagation threshold. The close correlation between these empirical and analytical values ​​confirms the robustness of the theoretical framework of this invention and highlights the consistency between numerical simulations and mathematical predictions. This finding further emphasizes that even with basic probabilities... Even small deviations from the threshold can drastically alter epidemic dynamics, thus validating the practical relevance of the derived threshold condition. Beyond this critical value, the basic probability... Even a small increase can significantly amplify disease transmission, thus reinforcing the need for accurate parameter estimation in pandemic preparedness strategies.

[0140] Further analysis of basic probabilities Sensitivity analysis was performed using a very fine step size of 0.0001, such as... Figure 11 As shown. This parameter governs the probability of infection in conscious susceptible individuals, and its influence is relatively more subtle due to the protective effect of consciousness. However, the results indicate that even the base probability... Even a small increase in the number of infected individuals can lead to a significant increase in the infected population, especially when awareness mechanisms are not strong enough. These findings highlight the need for strict control over awareness, effectiveness, and disease transmissibility in coupled network interventions.

[0141] Figure 10 and Figure 11 The oscillations visible in the fitted curves highlight the impact of periodic weight changes on transmission dynamics. The resulting periodic infection rates cause the evolution of the infected population to deviate slightly from a smooth trend, reflecting real-world epidemic behavior under time-dependent influences. Despite these fluctuations, the overall trajectories of the MMCA and MC simulations remain consistent, indicating that the model reliably and realistically captures the underlying dynamics.

[0142] In many existing studies, the fit between MMCA and MC simulations tends to be smooth and visually ideal. However, in our simulations, the MMCA and MC curves exhibit slight oscillations during the fitting process. This phenomenon is caused by the periodic changes in the weights over time, which in turn leads to periodic fluctuations in the infection rate. This behavior is not a drawback, but rather a realistic reflection of the model design. It highlights the influence of periodic effects in the disease transmission process and increases the credibility of the simulation results. Despite these oscillations, the overall trend of the MMCA and MC simulations remains consistent, validating the consistency and accuracy of our modeling approach.

[0143] Step 5: Simulation experiment to verify the effectiveness of the restraint control.

[0144] The effectiveness of the proposed pinning control is initially as follows: Figure 12 As shown, Figure 12 Depicting in The dynamic evolution of the disease transmission layer and system, i.e., equation (12), at the start of control. A clear inflection point was observed, where the infection prevalence began to decline sharply after control interventions were implemented, indicating that timely targeting of specific lymph nodes can significantly suppress the spread of the disease. Figure 12 By capturing the evolution of all five node states (including the perception dimension), a more detailed view of the system is provided. Upon activation of control, the proportion of conscious individuals (AS and AR) steadily increases, while the proportion of unaware and infected individuals (US and AI) decreases accordingly. This result validates the strategic advantage of applying consciousness-based control to a high-level subset of nodes, where the threshold is customized. This allows for early and effective disruption of the infection pathway. The purpose of this threshold is to demonstrate the feasibility of the control method.

[0145] It is important to note that the thresholds used in the above control strategies This is an empirical choice to demonstrate the feasibility and preliminary effectiveness of the proposed method. The choice has not yet been optimized, and a systematic exploration of the optimal threshold will be conducted.

[0146] Simulation experiments verify the optimality of the restraint control:

[0147] Building on these initial findings, the next step will be to explore how changes in the control range affect the impact of adjusting the threshold. Adjusting overall performance presents a crucial trade-off between control effectiveness and implementation costs. Setting thresholds too low can lead to widespread information dissemination but also excessive control costs, while setting them too high restricts information dissemination, resulting in compromised infectious disease control effectiveness. This fundamental balance requires careful optimization of threshold parameters to achieve effective disease control with minimal resource expenditure. To evaluate the effectiveness of different control strategies, the control thresholds were adjusted. A series of simulations were conducted, and this parameter determined the minimum level of direct sensing intervention at the target node.

[0148] use Figure 13 Two key metrics are shown to evaluate the cost-effectiveness of control strategies. The first metric quantifies the direct cost of infection, while the second measures the intervention cost based on the number of control points. With control thresholds... With the increase in [the number of nodes requiring intervention], the number of nodes needing intervention decreases, reducing control costs. However, this limited scope of intervention restricts the spread of information, potentially increasing the risk of infection. Simulation results show that the total cost function [is affected by this]. Follow First decrease, then increase, at the threshold The minimum value is reached at this point (derived through cost function simulation). This optimal threshold represents the best balance between intervention costs and infection control.

[0149] To further understand the dynamic response of the system under this optimal control scheme, i.e., equation (12), as follows: Figure 14 As shown, the temporal evolution of these five node states clearly demonstrates the rapid suppression of the infectious disease while related awareness steadily increased across states. These results confirm the effectiveness of targeting high nodes in the information layer as a control strategy, effectively utilizing network heterogeneity to maximize public health benefits while controlling resource usage. Overall, the findings demonstrate that degree-based awareness enhancement is a feasible and effective method for managing coupled information epidemic propagation processes, providing practical guidance for designing targeted intervention policies in real-world situations.

[0150] The time evolution of the state of each node under the optimal control strategy is as follows: Figure 15 As shown, the infected population (AI) declined significantly faster than in the uncontrolled scenario, reflecting the effectiveness of the intervention in containing disease transmission. Simultaneously, the proportions of aware susceptible (AS) and aware recovered (AR) individuals steadily increased over time, indicating successful dissemination of awareness throughout the network. Importantly, the proportion of uninformed susceptible nodes (US) remained low throughout the simulation. This sustained suppression of the uninformed susceptible population highlights the control scheme's ability to rapidly transition individuals to an informed behavioral state, thereby reducing their susceptibility to infection. Overall, these dynamic models validate that the proposed optimal control framework is a robust and effective approach that can simultaneously suppress epidemic transmission and promote awareness diffusion in coupled information-epidemic systems.

[0151] At the determined optimal control threshold Based on this, subsequent simulations are conducted to evaluate the long-term impact of the target intervention strategy. Under this optimal control, the final state of the network is as follows: Figure 16 As shown, at the disease transmission layer, the infection prevalence rate dropped significantly, indicating successful control of disease transmission. Simultaneously, the information layer almost completely permeated the awareness of the entire population, reflecting the efficiency of awareness-boosting strategies in reaching a broad audience. This result clearly demonstrates that selectively raising awareness at nodes with high density is an effective mechanism for suppressing the spread of epidemics while maximizing information dissemination. The findings highlight the potential of leveraging network heterogeneity to design cost-effective and impactful public health interventions within complex, coupled information epidemic systems.

[0152] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network, characterized in that: The steps include: Step 1: Based on complex network theory, construct a two-layer weighted network model, including a disease transmission layer and an information diffusion layer. After preprocessing the obtained population contact frequency matrix and information diffusion network, map them to the contact intensity weights between nodes in the disease transmission layer and the information diffusion weights between nodes in the information diffusion layer, respectively. Step 2: Based on the microscopic Markov chain approximation method, conduct propagation dynamics analysis on the constructed two-layer weighted network model, establish the infectious disease propagation dynamics equation with periodic weight characteristics, and determine the propagation threshold; Step 3: Design and optimization of restraint control: Introduce a degree threshold criterion at the information propagation layer, and forcibly change the unknown state to an known state for nodes with a degree greater than or equal to the preset threshold; construct a comprehensive cost function composed of infection loss and intervention cost, and optimize the control threshold and the set of controlled nodes.

2. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 1, characterized in that: The disease transmission layer is an undirected weighted network, and the information diffusion layer is a directed weighted network; the undirected weighted network and the directed weighted network are structurally parallel and node-aligned. The edge weights of the undirected weighted network are determined by the number of individual contacts and their periodic changes, truly reflecting the dynamic fluctuations in the intensity of interpersonal interaction over different time periods; the edge weights of the directed weighted network are allocated by the degree centrality of nodes, reflecting the relative influence of information sources in diffusion.

3. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 2, characterized in that: The undirected weighted network and the directed weighted network contain the same set of nodes. Each node represents the disease state and information state of the same individual in the two layers, respectively. The interaction between the undirected weighted network and the directed weighted network is achieved through the coupling of state variables: the informed state of the nodes in the information diffusion layer affects the infection probability in the disease transmission layer, and the infection state in the disease transmission layer affects the perceived transmission probability in the information diffusion layer.

4. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 2 or 3, characterized in that: The state of each node is modeled using a micro-Markov chain approximation method, resulting in five states: Unaware Susceptibility (US), Aware Susceptibility (AS), Aware Infection (AI), Unaware Recovery (UR), and Aware Recovery (AR). Based on the transition process between these five states, a dynamic equation for the spread of infectious diseases with periodic weights is established: ； ； ； ； ； in, , They represent in , Time Node The probability of being in an unknown vulnerable US state. , They represent in , Time Node The probability of being in a state of informed vulnerability to AS. and They represent in , Time Node The probability of being in an informed recovery AR state. and They represent in , Time Node The probability of unknowingly reverting to the UR state. and They represent in , Time Node The probability of being in a state of informed infection with AI; This represents the information diffusion layer perception propagation probability based on node edge weights. This represents the infection probability of an unknown susceptible node based on the number of contacts and the periodic weight of the disease transmission layer; This represents the infection probability of a known susceptible node based on the number of contacts and the periodic weight of the disease transmission layer; This represents the probability that a sensing node on the information diffusion layer becomes a non-sensing node. This indicates the probability that a recovered node will become a susceptible node in the disease transmission layer; This represents the probability that an infected node becomes a recovering node on the disease transmission layer.

5. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 4, characterized in that: The node's perception propagation probability ;in, The number of sensing nodes surrounding the information diffusion layer node; nodes From a single directed weight Perception nodes The probability of sensing transmission is ; The undirected weighted network is an undirected weighted BA network, where nodes represent individuals, edges represent physical contacts, and the weight of each edge represents the number of contacts between individuals. The infection probability of unknown susceptible nodes ;in, The total number of infected nodes surrounding a node in the disease transmission layer; the infection probability obtained at time t from an infected neighbor node. ;in, This represents the basic probability of infection at unknown susceptible nodes within the disease transmission layer. These are the initial values ​​for the basic weights of the disease transmission layer. The amplitude of the time-varying weighting formula at the disease transmission layer; The known infection probability of susceptible nodes The infection probability obtained from an infected neighbor node at time t. ;in, This represents the basic probability of susceptible nodes being infected at the disease transmission layer. node The out-degree is recorded as Degree centrality is ,in, This represents the maximum degree of nodes in the information diffusion layer. yes The neighboring nodes, from the node To the node weights Due to the directional nature of the information diffusion layer, the weights ,at this time, For nodes The degree center value.

6. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 5, characterized in that: The propagation threshold is calculated as follows: at time... The probability of infection obtained from an infected neighbor node. In this case, the probability is: Assume the basic probability of infection at unknown susceptible nodes in the disease transmission layer. Small enough to have a probability Applying an exponential function and performing a first-order Taylor expansion near the probability x = 0, the infection probability of being infected by an adjacent infected node in the disease transmission layer is approximated as: ； At the point of maximum infection rate, the effective infection matrix The transmission threshold of infectious diseases is determined by conditions. The propagation threshold is defined as follows: ;in, The weighted adjacency matrix representing the disease transmission layer The largest eigenvalue.

7. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 5 or 6, characterized in that: A degree-based constraint control strategy is adopted at the information propagation layer, selectively forcing nodes with high connectivity to transition from an unknown state to a perceived state: when the degree of a node on the information propagation layer... Exceeding the predefined threshold When the control intervention is activated, the controller with the threshold criterion is: in, This indicates whether node i should be forced into an informed state. 1 means the node should be forced into an informed state, and 0 means the node should not be forced into an informed state.

8. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 7, characterized in that: By forcing unknown susceptible nodes to become known susceptible nodes and unknown recovery nodes to become known recovery nodes, the infectious disease transmission dynamics equation becomes: 。 9. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 7, characterized in that: The threshold By analyzing the degree distribution of the network, a comprehensive cost function is constructed, and the optimal threshold is found by iterating through the network from minimum to maximum. To minimize the overall cost function; restraint control is based on the degree threshold method to formulate an optimal control strategy, dynamically allocating sensing based on the node's connectivity and current state; at the node's degree... Exceeding the predefined threshold The system activates the restraint control mechanism, selecting nodes with a degree greater than the threshold to forcibly switch to an informed state. The intervention intensity is continuously adjustable within the operational range.

10. The method for dynamic analysis and control of infectious disease transmission based on a two-layer weighted network according to claim 9, characterized in that: The comprehensive cost function is: ； Where J1 is the cumulative value of infection loss, and J2 is the cumulative value of intervention cost. Indicates time The total number of infections This represents the social / economic cost of each infection. The unit cost reflecting control efforts, the ratio Represents the target node The proportion of uninformed individuals in the group This refers to the number of all nodes with degree k on the information diffusion layer. The information diffusion layer has a degree of The number of unsuspecting nodes, express At all times, in degree The amount received by one type of node that is forcibly switched to an informed state; After anonymization, the population contact frequency matrix and information dissemination network are uniformly mapped to the contact intensity weights and information dissemination weights between nodes.

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