A propagation risk continuous prediction method based on graph neural delay differential equation

Abstract

The application discloses a kind of based on graph neural delay differential equation's propagation risk continuous prediction method, belong to the popular disease prediction technical field.This method first inputs multi-region popular disease monitoring time series data, obtains logarithmic growth sequence by sequence transformation, uses lead-lag correlation estimation interregional and interregional propagation time lag, constructs propagation time lag priori.Then, in combination with regional historical representation, learnable region and delay embedding, the score of delay perception relationship is calculated and normalized, and the delay perception propagation graph is built.Further, the graph is coupled with the graph to build a neural delay differential equation, and the delay state inside and outside the region is aggregated to realize the continuous time evolution of the hidden state of the popular disease.Finally, the evolution hidden state is mapped and inverse transformed, and the prediction results of the number of cases and the propagation risk of multiple time steps in multiple regions in the future are output.The application effectively captures the time lag characteristics and spatiotemporal correlation of the popular disease propagation, and can realize high-precision, continuous multi-region popular disease propagation risk prediction.

Description

Technical Field

[0001] This invention relates to the field of epidemic prediction technology, specifically to a method for continuous prediction of transmission risk based on graph neural delay differential equations. Background Technology

[0002] With the acceleration of globalization and the increased cross-regional movement of people, the spread and speed of infectious diseases such as influenza and COVID-19 have significantly increased, posing continuous challenges to people's health, public safety, and social stability. In the process of epidemic risk prevention and control, a scientifically effective epidemic prediction system is a crucial foundation for early warning, medical resource allocation, and the formulation of public health intervention strategies. Accurate risk prediction can buy valuable response time for public health departments and provide a basis for epidemic hazard classification, key area identification, and adjustment of intervention measures. However, the spread of epidemics in the real world is nonlinear and spatially heterogeneous, and is affected by factors such as incubation period, reporting lag, non-equidistant sampling, and seasonal peak shifts. Epidemic curves in different regions often exhibit a significant lead-lag relationship, making it difficult to accurately characterize transmission lags, thus affecting the stability of long-term predictions and the accuracy of regional transmission relationship modeling. Therefore, how to fully explore the potential asynchronous transmission relationships between regions based on limited and irregular multi-regional epidemic monitoring sequences during an outbreak, and to make stable predictions of future transmission trends, has become a key technical issue in the construction of intelligent public health early warning systems.

[0003] To accurately quantify and assess the spread of epidemics, various epidemiological prediction methods based on different technical approaches have emerged in recent years. Traditional epidemiological prediction methods typically employ SIR models based on compartmental structures and their metapopulation variants, using differential equations to characterize the transition processes between susceptible, infected, and recovered population states. These methods offer good epidemiological interpretability and can reflect the basic mechanisms of disease transmission to some extent. However, in practical applications, these methods often rely on the precise calibration of parameters such as transmission rate, contact intensity, and inter-regional flow intensity, and are usually based on strong assumptions about population mixing or fixed transmission structures. When facing emerging infectious diseases, rapidly changing transmission environments, or heterogeneous multi-regional transmission processes, these parameters are difficult to obtain accurately and in a timely manner, and the models struggle to adapt to complex nonlinear transmission patterns. Especially when monitoring data is incomplete, external flow data is unavailable, or the epidemic's transmission stage changes rapidly, the ability of traditional mathematical models to meet real-time requirements and provide high-precision multi-regional predictions is significantly limited.

[0004] Data-driven deep learning methods offer a new technological path for epidemic prediction. Models such as recurrent neural networks (RNNs), convolutional neural networks (CNNs), attention mechanisms, and spatiotemporal graph neural networks (STGNNs) can directly learn nonlinear temporal patterns and inter-regional dependencies from historical monitoring data. In particular, STGNNs can model different regions as nodes and inter-regional interactions as edges, thereby capturing spatial dependencies in disease transmission through graph message passing, showing great application potential in multi-regional epidemic prediction tasks. Nevertheless, most existing STGNN prediction models are still based on fixed discrete time steps and regular sampling data, usually requiring monitoring sequences from different regions to be aligned at a uniform time interval, and assuming that the effects of inter-regional transmission occur instantly within the same time step, thus making it difficult to model dynamic and continuous systems.

[0005] To characterize dynamic systems with continuous evolutionary features, such as epidemic transmission, neural differential equations (NDEs) offer a novel technical approach for modeling continuous-time dynamic systems. Compared to traditional discrete-time prediction models, NDEs can describe the evolution of hidden states in continuous time. Specifically, neural ordinary differential equations (NODEs) provide a flexible data-driven framework for disease trend modeling by learning the continuous changes in hidden states; neural controlled differential equations (NCDEs) further construct continuous control paths from observation sequences, adapting to scenarios such as non-equidistant sampling, partially missing observations, and irregular monitoring records. However, existing NDE methods still have significant limitations in epidemic prediction tasks. On the one hand, standard NDEs typically employ the Markov state evolution assumption, meaning that changes in the current state are primarily determined by the current hidden state, making it difficult to explicitly utilize the delayed impact of historical states on the current transmission process. Actual epidemic transmission is often influenced by factors such as incubation period, monitoring and reporting lags, cross-regional transmission processes, and seasonal peak shifts. Case changes in different regions do not occur synchronously but exhibit a clear lead-lag relationship. If models rely solely on the current state for continuous evolution, they cannot accurately characterize such asynchronous transmission processes. On the other hand, while Neural Delay Differential Equations (NDDEs) can incorporate historical states into continuous-time evolution, existing methods typically rely on predefined delays, uniform delays, or fixed delay parameters, making it difficult to adapt to the heterogeneous transmission lags in real epidemic transmission, which are characterized by regional specificity, directional specificity, and stage changes. For multi-regional epidemic prediction tasks, the transmission impact between different regions may appear at different time intervals, and the same region may also experience time delays due to local epidemic inertia and peak misalignment. Therefore, how to adaptively identify intra-regional time delays and inter-regional transmission lags from the limited epidemic monitoring sequences within a continuous-time modeling framework, and use them to guide the evolution of multi-regional states, remains a key problem that has not yet been fully solved in the interdisciplinary research of artificial intelligence and epidemic prediction.

[0006] Based on the above analysis, the problems and shortcomings of the existing technology are as follows: (1) Insufficient characterization of asynchronous propagation time delay: Most existing spatiotemporal prediction methods assume that the impact of inter-regional propagation occurs immediately, or only use a unified, predefined or fixed propagation time delay, which is difficult to adapt to the heterogeneous propagation time delay relationship under different regions, different propagation directions and different epidemic stages. Therefore, they cannot fully reflect the incubation period, reporting lag and cross-regional propagation process in the real epidemic.

[0007] (2) Insufficient ability to continuously model irregular monitoring: Most discrete time prediction models require multi-region monitoring sequences to be aligned at fixed time steps, making it difficult to directly handle missing, delayed reporting, and non-equal interval sampling of epidemic monitoring data; even if interpolation or completion methods are used, additional estimation errors may be introduced, affecting the reliability of subsequent prediction results.

[0008] (3) Strong dependence on external graph structure or auxiliary data: Some spatiotemporal graph neural network methods require predefined external information such as geographic adjacency graphs, population flow graphs or traffic flow to construct inter-regional transmission relationships. In the early stages of a new epidemic or in scenarios with limited data, such external information is often difficult to obtain in a timely manner and may not be consistent with the actual transmission path, which limits the deployment applicability and generalization ability of the model.

[0009] Therefore, in order to address the problems of asynchronous transmission processes, irregular monitoring records, and difficulty in directly observing potential inter-regional dependencies in multi-regional epidemic prediction, it is urgent to construct an epidemic prediction method oriented towards continuous time evolution. This method should extract the regional evolutionary inertia and cross-regional transmission lag information from historical monitoring sequences, and characterize the heterogeneous delay dependencies in different regions and different transmission directions without relying on fixed transmission time lags or artificially set transmission structures. This would support efficient, accurate, and robust prediction of future case trends in multiple regions.

[0010] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of the present invention, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0011] The purpose of this invention is to provide a continuous prediction method for transmission risk based on graph neural delay differential equations, in order to solve the problem that existing epidemic prediction methods are usually based on fixed discrete time modeling, which makes it difficult to effectively characterize the asynchronous delay dependence and irregular monitoring data in the multi-regional epidemic transmission.

[0012] Other features and advantages of the invention will become apparent from the following detailed description, or may be learned in part by practice of the invention.

[0013] According to a first aspect of the present invention, a method for continuous prediction of propagation risk based on graph neural delay differential equations is provided, the method comprising: S1, Input Multi-Regional Epidemic Surveillance Sequence: Obtain and input time-series data containing historical epidemic surveillance records for N regions; S2, Epidemic Transmission Lag Estimation: Transform the original monitoring values ​​in step S1, calculate the logarithmic growth sequence, calculate the lead-lag correlation between the source region and the target region under different candidate lags within the candidate lag set, and estimate the inter-regional transmission lag accordingly; at the same time, estimate the intra-regional time lag, and construct the epidemic transmission lag prior based on the inter-regional transmission lag and the intra-regional time lag. S3, Delay-aware propagation graph construction: Based on the delay prior obtained in step S2, combined with the continuous historical representation of each region, the learnable region embedding, and the learnable delay embedding, calculate the delay-aware relationship scores corresponding to the target region, source region, and candidate delay; normalize the delay-aware relationship scores, and construct the delay-aware propagation graph based on the normalized delay-aware relationship scores. S4, Graph-coupled neural delay differential equation evolution: Based on the delay-aware propagation graph obtained in step S3, a continuous history function is constructed and initialized with the hidden state of the predicted starting point; in continuous time, the delayed state within the region and the delayed propagation state between regions are aggregated according to the delay-aware propagation graph to obtain the delayed message of the target region, and a graph-coupled neural delay differential equation is constructed to realize the continuous time evolution of the hidden state of the multi-region epidemic. S5, Continuous prediction of multi-regional transmission risk: Read out the future hidden state obtained from step S4 to obtain the predicted logarithmic number of cases, and obtain the predicted number of cases through inverse transformation; finally, output the multi-regional epidemic prediction results for multiple future time steps.

[0014] In some exemplary embodiments, in step S1, the time series data is represented as ,in, , indicating the first A vector of monitoring values ​​for all regions at each observation time point. Represents the monitoring value vector The Each component, i.e., the region In the Epidemiological monitoring records at each observation time point.

[0015] In some exemplary embodiments, the specific method for estimating the epidemic transmission time lag in step S2 is as follows: Step 2.1, Calculation of Lag Correlation: First, calculate the lag correlation of the input in step S1. The region in the first Raw monitoring values ​​at each observation time step Perform a logarithmic transformation to obtain the logarithmic case sequence. Log-growth sequence is calculated based on the logarithmic difference between adjacent observation time steps. ; Let the candidate delay set be: , in, Indicates the maximum candidate propagation delay; for the target area Heyuan area In candidate delay The following calculates the lead-lag correlation: , in, Indicates the region From the 2nd to the 1st A historical logarithmically growing subsequence at each time step Indicates the target area From the The first to the second A time-step lag log-grown subsequence; This represents the Pearson correlation coefficient, used to measure the source region. Changes and target area Postponed The degree of linear correlation between the changes after each time step; Step 2.2, Inter-regional propagation time delay estimation: For the propagation time delay estimation between different regions, when At that time, in the candidate delay set Within the source region, the candidate delay that maximizes the lead-lag correlation is selected as the source delay. To the target area Inter-regional propagation time lag: , The corresponding correlation is then nonnegated and used as the reliability strength of the propagation delay:

[0016] in, Indicates nonnegativity; Step 2.3, Time Delay Estimation within the Region: For time self-delay estimation within the same region, when At that time, a candidate delay set that excludes zero delay is used. and in Within the region, the candidate delay that maximizes the correlation between the current state of change and its historical state of change is selected as the region. Time delay within the region: , Meanwhile, the corresponding correlation is nonnegated and used as the reliability strength of the time delay within this region:

[0017] Step 2.4, Unified Time-Lag Prior Generation: Based on the inter-regional propagation time lag obtained in Step 2.2 and the intra-regional time lag obtained in Step 2.3, a unified epidemic propagation time lag prior is constructed. .

[0018] In some exemplary embodiments, the specific method for constructing the delayed sensing propagation map in step S2 includes the following steps: Step 3.1, Continuous Control Path Construction: Construct a continuous control path for each region's logarithmic case observation sequence input in Step S1; for the... Each region, based on its available observation points Constructing continuous control paths using natural cubic spline interpolation ,in, Indicates the region The actual observation timestamp set; Step 3.2, Continuous history representation encoding: based on the continuous control path The region is obtained by encoding using neural control differential equations. Continuous historical representation Its expression is:

[0019] in, Represents the time variable of integration. Indicates the region In time The continuous history represents, Indicates based on region The logarithmic case observations were used to obtain the continuous control path in time using natural cubic spline interpolation. The value of , Represents a learnable encoding function

[0020] Step 3.3, Calculation of delayed perception relationship score: The delayed perception relationship score It is jointly determined by the learnable relation scoring term and the propagation delay prior soft constraint term, and its expression is:

[0021] in, The parameter is Learnable relational scoring function; Indicates the target area At the starting point of the prediction A continuous historical representation; Indicates the source region At the delayed time point A continuous historical representation; and Representing the target area Heyuan area Learnable region embedding; Indicates candidate delay Learnable delayed embeddings; The weighting coefficients represent the propagation delay prior soft constraints; This represents the Gaussian kernel type time-delay prior bias term; Step 3.4, Delayed-aware propagation graph generation: Based on the delayed-aware relationship scores obtained in Step 3.3, construct the delayed-aware propagation graph: , in, Indicates candidate delay The corresponding propagation relationship matrix, matrix elements Indicates the source region The state after candidate delay Then target area Influence weight; For intra-regional delayed dependencies, i.e. At that time, in the region Normalize across the entire range of candidate delays:

[0022] For inter-regional delayed propagation dependency, i.e. At that time, in the target area Normalization is performed across the entire source region and the candidate delay combination range:

[0023] in, Indicates the region In candidate delay The regional delay relationship score is below. Indicates the source region In candidate delay Below the target area Inter-regional delayed propagation relationship score; and These represent the summation indexes for the candidate delay and the source region, respectively.

[0024] In some exemplary embodiments, the specific method for the evolution of the graph-coupled neural delayed differential equation in step S4 includes the following steps: Step 4.1, Construction of continuous history function: Based on the region continuous history representation obtained in step S3 Mapping it to the hidden state space of the neural delay differential equation, constructing the region Continuous history function: , in, This represents a learnable linear mapping matrix. Indicates the region The historical hidden state trajectory prior to the prediction starting point is determined; and the region is initialized with the historical function value at the prediction starting point. Future hidden state ; Step 4.2, Delayed State Query: During continuous-time evolution, when a query region is needed... exist When there is a time delay, it depends on whether the query time is earlier than the prediction start time. Obtain the delayed state using different methods:

[0025] in, This represents the region maintained by the numerical solver. The future hidden state trajectory This indicates an interpolation query operation; Step 4.3, Delayed Message Aggregation: Based on the delay-aware propagation graph obtained in step S3 The target region is obtained by weighted aggregation of the delay propagation states within and between regions. In time Delayed message : , in, and Represents the learnable state transition matrix. This represents the balance coefficient of the inter-regional propagation contribution; the first term represents the intra-regional delayed state aggregation term, and the second term represents the inter-regional delayed propagation state aggregation term. Step 4.4, Construction of the graph-coupled neural delayed differential equation: The current hidden state... Delayed messages and continuous time Input a vector field parameterized by a neural network to construct a region The graph-coupled neural delay differential equation:

[0026] in, The parameter is The neural network vector field is used to describe the evolution of the hidden state of regional epidemics over continuous time; Step 4.5, Gated Continuous State Update: The neural network vector field A gated update structure is adopted to map the current hidden state and delayed messages to the same hidden space, and adaptively control the speed at which the current hidden state evolves into a delay-aware candidate state. Its expression is:

[0027]

[0028] in, Indicates delayed perception candidate state, Indicates the gating coefficient. This represents element-wise multiplication. All are learnable parameters; Step 4.6, Solving for the future hidden state: A numerical solver is used to solve the graph-coupled neural delay differential equation to obtain the future hidden state. A multi-region hidden state sequence at each time step: .

[0029] In some exemplary embodiments, the specific method for continuous prediction of multi-regional propagation risk in step S5 includes the following steps: Step 5.1, Predicting the logarithmic number of cases: The future hidden state sequence obtained from step S4. In the middle, read the area In the future Hidden states at each prediction time step ,in, Then, the future hidden state is input into the prediction readout network to obtain the region. In the future The predicted logarithm of cases at each predicted time step :

[0030] in, The parameter is Multilayer perceptron readout network To ensure the non-negativity of the predicted logarithmic number of cases; an inverse exponential transform is performed on the predicted logarithmic number of cases to obtain the region. In the future Predicted number of infected cases at each prediction time step:

[0031] Step 5.2, Multi-region Prediction Result Output: Execute Step 5.1 for all regions and all future prediction time steps to obtain the future... Multi-regional epidemiological prediction results at each time step: Indicates the first The predicted number of infected cases in all regions at each predicted time step.

[0032] In some exemplary embodiments, step S5 further includes step 5.3, prediction loss optimization: during the training phase, the actual number of infected cases is... Transformed into the true logarithm of the number of cases:

[0033] The model is then optimized using the Huber loss function in the logarithmic case space.

[0034] in, This represents the threshold parameter of the Huber loss function.

[0035] According to a second aspect of the present invention, a storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the method for continuous prediction of propagation risk based on graph neural delay differential equations as described in the first aspect.

[0036] According to a third aspect of the present invention, a computer program product is provided, on which a computer program is stored, wherein when the computer program is executed by a processor, it implements the method for continuous prediction of propagation risk based on graph neural delay differential equations as described in the first aspect.

[0037] According to a fourth aspect of the present invention, an electronic device is provided, comprising: Processor; and Memory for storing the executable instructions of the processor; The processor is configured to implement the graph neural delay differential equation-based continuous prediction method for propagation risk described in the first aspect above by executing the executable instructions.

[0038] The continuous propagation risk prediction method based on graph neural delay differential equations provided in the embodiments of the present invention has the following advantages compared with the prior art: (1) The present invention can adaptively learn the time delay of epidemic transmission: Based on the historical epidemic monitoring sequence of multiple regions, this method estimates the time delay within the region and the transmission time delay between regions, which can characterize the asynchronous transmission relationship caused by local epidemic inertia, incubation period, reporting delay and cross-regional transmission process, and overcome the limitations of the instant transmission assumption or unified fixed delay assumption in the existing methods.

[0039] (2) The present invention can construct a delayed-perception propagation map to characterize heterogeneous propagation dependence: This method combines the epidemic propagation time lag prior with the learnable region representation to generate a directional, time-lag and heterogeneous delayed-perception propagation map. It does not rely on predefined population flow maps or geographical adjacency maps, and can learn potential propagation sources, propagation intensity and propagation time lag from monitoring sequences, thereby improving the flexibility and interpretability of multi-regional propagation relationship modeling.

[0040] (3) The present invention can improve the stability under irregular observation and long time domain prediction conditions: The method uses continuous time encoding to construct historical state functions and performs continuous state evolution through graph-coupled neural delay differential equations under the constraint of delay perception propagation graph. This allows the prediction process to consider both the regional delay state and the cross-regional delay propagation state, thereby enhancing the model's adaptability to missing, delayed reporting and non-equal interval monitoring data, and improving the robustness of multi-step epidemic prediction.

[0041] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit the invention. Attached Figure Description

[0042] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0043] Figure 1 This is a flowchart of the method involved in the present invention; Figure 2 This is a modular detailed diagram illustrating the method involved in this invention; Figure 3 A schematic diagram of long-term and short-term prediction results on a specific dataset; Figure 4 This is a comparison chart showing the prediction performance of key hyperparameters at different values. Detailed Implementation

[0044] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided so that the invention will be more comprehensive and complete, and will fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0045] Furthermore, the accompanying drawings are merely illustrative of the invention and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0046] To address the shortcomings and deficiencies of existing technologies, this example implementation provides a method for continuous prediction of transmission risk based on graph neural delay differential equations. The method includes: inputting multi-regional epidemic monitoring sequences; estimating epidemic transmission lag; constructing a delayed-sensing transmission graph; evolving the graph-coupled neural delay differential equation; and continuously predicting multi-regional transmission risk. This invention can adaptively learn heterogeneous delayed transmission relationships within and outside the region from irregular, missing, or delayed-reported epidemic monitoring records. It does not rely on predefined population flow maps or fixed transmission lags and is applicable to short-term, medium-term, and long-term trend prediction of infectious diseases such as influenza and COVID-19. Compared with existing methods, this invention effectively solves the problems of epidemic prediction models' difficulty in characterizing transmission lag effects, insufficient continuous modeling capability for irregular monitoring data, and weak long-term prediction stability. It has advantages such as low external data dependence, strong asynchronous transmission characterization capability, good continuous-time modeling capability, and strong robustness to irregular observations.

[0047] refer to Figure 1 As shown, the specific steps may include: S1, Input Multi-Regional Epidemic Surveillance Sequence: Obtain and input time-series data containing historical epidemic surveillance records for N regions; S2, Epidemic Transmission Lag Estimation: Transform the original monitoring values ​​in step S1, calculate the logarithmic growth sequence, calculate the lead-lag correlation between the source region and the target region under different candidate lags within the candidate lag set, and estimate the inter-regional transmission lag accordingly; at the same time, estimate the intra-regional time lag, and construct the epidemic transmission lag prior based on the inter-regional transmission lag and the intra-regional time lag. S3, Delay-aware propagation graph construction: Based on the delay prior obtained in step S2, combined with the continuous historical representation of each region, the learnable region embedding, and the learnable delay embedding, calculate the delay-aware relationship scores corresponding to the target region, source region, and candidate delay; normalize the delay-aware relationship scores, and construct the delay-aware propagation graph based on the normalized delay-aware relationship scores. S4, Graph-coupled neural delay differential equation evolution: Based on the delay-aware propagation graph obtained in step S3, a continuous history function is constructed and initialized with the hidden state of the predicted starting point; in continuous time, the delayed state within the region and the delayed propagation state between regions are aggregated according to the delay-aware propagation graph to obtain the delayed message of the target region, and a graph-coupled neural delay differential equation is constructed to realize the continuous time evolution of the hidden state of the multi-region epidemic. S5, Continuous prediction of multi-regional transmission risk: Read out the future hidden state obtained from step S4 to obtain the predicted logarithmic number of cases, and obtain the predicted number of cases through inverse transformation; finally, output the multi-regional epidemic prediction results for multiple future time steps.

[0048] The steps in this exemplary embodiment will now be described in more detail with reference to the accompanying drawings and embodiments.

[0049] Example 1 S1, Input multi-regional epidemiological surveillance sequences: Obtain and input sequences containing... Time-series data of historical epidemiological surveillance records in each region ,in, , indicating the first A vector of monitoring values ​​for all regions at each observation time point. Represents the monitoring value vector The Each component, i.e., the region In the Epidemiological surveillance records at each observation time point. These records include data from regular and irregular sampling. Regularly sampled data refers to monitoring data collected at the same fixed time intervals across all areas. Irregularly sampled data refers to data where adjacent valid observation time intervals in at least one area are not entirely identical, or where valid monitoring values ​​are only available at certain time points due to missing records, delayed reporting, or inconsistent collection frequencies.

[0050] This embodiment uses four publicly available real-world datasets widely used in epidemiological prediction research for validation. All datasets contain only historically reported infection cases, and for each dataset, it is divided chronologically into a 50% training set, a 20% validation set, and a 30% test set.

[0051] (1) US-Regions: contains weekly influenza activity data from 2002 to 2017 for 10 regions under the US Department of Health and Human Services (HHS). The sequence length is 785, which can be used to evaluate the model’s continuous prediction ability and long-term prediction stability under multi-seasonal influenza transmission patterns. (2) US-States Influenza Dataset: This dataset contains weekly influenza-like illness patient visit counts from 2010 to 2017 for 49 US states, with a sequence length of 360. The dataset has a large number of regional nodes, making it suitable for testing the model's ability to model propagation dependencies in large-scale, multi-regional epidemiological surveillance networks. (3) Australian COVID-19 Dataset (AU-COVID): This dataset contains the daily number of new COVID-19 cases in eight Australian states or territories from January 2020 to August 2021, with a sequence length of 556. This dataset has a daily monitoring granularity and can reflect the characteristics of case reporting, regional transmission, and short-term fluctuations in public health emergencies. It is suitable for evaluating the model's ability to predict the spread of emerging infectious diseases. (4) Japan COVID-19 Dataset (JP-COVID): This dataset contains the daily number of new COVID-19 cases in 47 prefectures of Japan from April 1, 2020 to September 21, 2021, with a sequence length of 349. This dataset has a large number of regions and obvious differences in cross-regional transmission, making it suitable for evaluating the model's ability to model the time lag of transmission in heterogeneous regions, daily case fluctuations, and the evolution trend of the epidemic in multiple regions.

[0052] The dataset and its attributes involved in this embodiment are shown in Table 1: Table 1

[0053] S2, Epidemic Transmission Lag Estimation: Estimation of the original monitoring values ​​from step S1 Perform a logarithmic transformation to obtain the logarithmic case sequence. And further calculate the logarithmic growth sequence In the candidate delay set Within, computation source region To the target area Different candidate delays Lead-lag correlation Based on this, the propagation time delay between regions is estimated; for the time delay estimation within a region, the following method is used: As a candidate delay set to exclude zero-delay, a delay prior that includes both delay value and reliability strength is ultimately obtained. The specific process includes: S2.1, Calculation of Lag Correlation: First, for the input in step S1... The region in the first Raw monitoring values ​​at each observation time step Perform a logarithmic transformation to obtain the logarithmic case sequence. Log-growth sequence is calculated based on the logarithmic difference between adjacent observation time steps. Since the logarithmic growth value needs to be calculated from data at two adjacent time points, the sequence starts from the second time step. Let the candidate delay set be: , in, Indicates the maximum candidate propagation delay; for the target area Heyuan area In candidate delay Lower calculation source region Historical logarithmic growth sequence and target region Correlation between lagged logarithmic growth sequences: , in, Indicates the region From the 2nd to the 1st A historical logarithmically growing subsequence at each time step Indicates the target area From the The first to the second A time-step lag log-grown subsequence; This represents the Pearson correlation coefficient, used to measure the source region. Changes and target area Postponed The degree of linear correlation between the changes after each time step; S2.2, Inter-regional propagation time delay estimation: For propagation time delay estimation between different regions, when At that time, in the candidate delay set Within the source region, the candidate delay that maximizes the lead-lag correlation is selected as the source delay. To the target area Inter-regional propagation time lag: , The corresponding correlation is then nonnegated and used as the reliability strength of the propagation delay:

[0054] S2.3, Time Delay Estimation within the Region: For time self-delay estimation within the same region, when At that time, a candidate delay set that excludes zero delay is used. and in Within the region, the candidate delay that maximizes the correlation between the current state of change and its historical state of change is selected as the region. Time delay within the region: Meanwhile, the corresponding correlation is nonnegated and used as the reliability strength of the time delay within that region:

[0055] S2.4, Unified Time-Lag Prior Generation: Based on the inter-regional propagation time delay obtained in step S2.2 and the intra-regional time delay obtained in step S2.3, a unified epidemic propagation time-lag prior is constructed. ,in, Indicates from the source region To the target area The estimated propagation time delay, This indicates the reliability strength corresponding to the estimated propagation delay; when hour, Indicates the region Time delay within the region; when hour ， Indicates the source region To the target area Inter-regional propagation time lag.

[0056] Finally, the epidemic transmission lag prior is used as a soft constraint in the subsequent delayed sensing transmission graph construction process to guide the model to focus on candidate transmission lags that are consistent with the lead-lag relationship in historical monitoring sequences; the epidemic transmission lag prior is not used as a fixed transmission graph constraint, so that the model can adaptively correct unreliable lag estimates during end-to-end training.

[0057] S3, Delay-Aware Propagation Graph Construction: Based on the delay prior obtained in step S2 Combining continuous historical data of each region Learnable region embedding and learnable delayed embeddings Calculate the target region Source region and candidate delay Corresponding delayed perception relationship score The relationship scores are normalized to obtain the delayed-perception propagation graph. ,in, Characterizes the strength of the delay dependency of the region itself. Characterizing the source region In delay Then target area The intensity of its propagation impact. The specific calculation steps can be expressed as follows: S3.1, Construction of Continuous Control Path: Construct a continuous control path for each region's logarithmic case observation sequence input in step S1; for the... Each region, based on its available observation points Constructing continuous control paths using natural cubic spline interpolation ,in, Indicates the region The actual observation timestamp set; S3.2, Continuous history representation encoding: based on the continuous control path The region is obtained by encoding using neural control differential equations. Continuous historical representation Its expression is:

[0058] in, Represents the time variable of integration. Indicates the region In time The continuous history represents, Indicates based on region The logarithmic case observations were used to obtain the continuous control path in time using natural cubic spline interpolation. The value of , This represents a learnable encoding function. The neural control differential equation is obtained through a continuous control path. Driver hidden state Evolution, thereby generating a continuous history representation without imputing missing observations with fixed values.

[0059] S3.3, Calculation of delayed perception relationship score: The delayed perception relationship score It is jointly determined by the learnable relation scoring term and the propagation delay prior soft constraint term, and its expression is:

[0060] in, The parameter is Learnable relational scoring function; Indicates the target area At the starting point of the prediction A continuous historical representation; Indicates the source region At the delayed time point A continuous historical representation; and Representing the target area Heyuan area Learnable region embedding; Indicates candidate delay Learnable delayed embeddings; This represents the source region estimated by step S2. To the target area Propagation time lag; This indicates the reliability strength corresponding to the propagation delay; The weighting coefficients represent the propagation delay prior soft constraints; This represents the Gaussian kernel type time-delay prior bias term; S3.4, Generation of Delayed-Aware Propagation Graph: Based on the delayed-aware relationship score obtained in step S3.3, construct the delayed-aware propagation graph: , in, Indicates candidate delay The corresponding propagation relationship matrix, matrix elements Indicates the source region The state after candidate delay Then target area The influence weight.

[0061] For both intra-regional and inter-regional delayed propagation dependencies, the scores of delayed-perceived relationships are normalized separately. For intra-regional delayed dependencies, i.e. At that time, in the region Normalize across the entire range of candidate delays: . For inter-regional delayed propagation dependency, i.e. At that time, in the target area Normalization is performed across the entire source region and the candidate delay combination range:

[0062] in, Indicates the region In candidate delay The regional delay relationship score is below. Indicates the source region In candidate delay Below the target area Inter-regional delayed propagation relationship score; and These represent the summation indexes for the candidate delay and the source region, respectively.

[0063] The diagonal elements of the delayed-aware propagation map are used to characterize the time delay dependency within a region, while the off-diagonal elements are used to characterize the directed delayed propagation relationship between different regions, thereby simultaneously providing information on the propagation source, propagation intensity, and propagation time delay.

[0064] S4, Graph-coupled neural delay differential equation evolution: Based on the delay perception propagation graph obtained in step S3 Construct a continuous history function and with The hidden state serves as the starting point for prediction; in continuous time... Above, the delayed states within the region are aggregated according to the delayed perception propagation map. and inter-regional delayed propagation status , obtain the target area Delayed message Furthermore, a graph-coupled neural delayed differential equation is constructed to realize the continuous-time evolution of hidden states in multi-regional epidemics. The specific process includes: S4.1 Construction of continuous history function: Based on the region continuous history representation obtained in step S3 Mapping it to the hidden state space of the neural delay differential equation, constructing the region Continuous history function: , in, This represents a learnable linear mapping matrix. Indicates the region The historical hidden state trajectory prior to the prediction starting point is determined; and the region is initialized with the historical function value at the prediction starting point. Future hidden state .

[0065] S4.2, Delayed State Query: During continuous-time evolution, when a query region is needed... exist When there is a time delay, it depends on whether the query time is earlier than the prediction start time. Obtain the delayed state using different methods:

[0066] in, This represents the region maintained by the numerical solver. The future hidden state trajectory cache This indicates an interpolation query operation; S4.3 Delayed Message Aggregation: Based on the delay-aware propagation graph obtained in step S3 The target region is obtained by weighted aggregation of the delay propagation states within and between regions. In time Delayed message : , in, and Represents the learnable state transition matrix. This represents the balance coefficient of the inter-regional propagation contribution; the first term represents the intra-regional delayed state aggregation term, and the second term represents the inter-regional delayed propagation state aggregation term. S4.4, Construction of graph-coupled neural delayed differential equations: The current hidden state... Delayed messages and continuous time Input a vector field parameterized by a neural network to construct a region The graph-coupled neural delay differential equation:

[0067] in, The parameter is The neural network vector field is used to describe the evolution of the hidden state of regional epidemics over continuous time; S4.5, Gated Continuous State Update: The Neural Network Vector Field A gated update structure is adopted to map the current hidden state and delayed messages to the same hidden space, and adaptively control the speed at which the current hidden state evolves into a delay-aware candidate state. Its expression is:

[0068]

[0069] in, Indicates delayed perception candidate state, Indicates the gating coefficient. This represents element-wise multiplication. All are learnable parameters; S4.6, Solving for the Future Hidden State: A numerical solver is used to solve the graph-coupled neural delay differential equation to obtain the future hidden state. A multi-region hidden state sequence at each time step:

[0070] in, Indicates the entire region in time The hidden state matrix, and the future hidden state sequence are used for subsequent continuous prediction of multi-regional propagation risk.

[0071] S5, Continuous prediction of multi-regional transmission risk: This involves analyzing the future hidden states obtained from step S4. Perform readout mapping to obtain the future... The predicted logarithm of cases at each predicted time step The predicted number of cases is obtained through inverse exponential transformation; the final output is the future number of cases. The multi-regional epidemiological prediction results at each time step are used to characterize the future trend of infection cases in each region. The specific calculation process is as follows: S5.1, Predicting the logarithmic number of cases: the future hidden state sequence obtained from step S4. In the middle, read the area In the future Hidden states at each prediction time step ,in, Then, the future hidden state prediction readout network is used to obtain the region. In the future The predicted logarithmic number of cases at each time step :

[0072] in, The parameter is Multilayer perceptron readout network This is used to ensure the non-negativity of the predicted logarithmic number of cases. An inverse exponential transform is performed on the predicted logarithmic number of cases to obtain the region. In the future Predicted number of infected cases at each prediction time step:

[0073] S5.2, Multi-region Prediction Result Output: Execute step S5.1 for all regions and all future prediction time steps to obtain the future... Multi-regional epidemiological prediction results at each time step: in, Indicates the first The predicted number of infected cases in all regions at each prediction time step; the multi-regional epidemic prediction results are used to characterize the future trend of infection cases in different regions and further reflect the level of multi-regional epidemic transmission risk. S5.3, Prediction Loss Optimization: During the training phase, the actual number of infected cases is... Transformed into logarithmic case count:

[0074] The model is then optimized using the Huber loss function in the logarithmic case space.

[0075] in, This represents the threshold parameter of the Huber loss function. Robust optimization in the logarithmic case space reduces the impact of monitoring noise, sudden spikes, and backtracking corrections on the continuous prediction results of multi-regional transmission risk.

[0076] This embodiment selects several existing epidemic prediction and spatiotemporal prediction technologies as comparative models to verify the advancement of the present invention. Specifically, it includes: 1) Traditional statistical forecasting models: ARIMA, VAR; Among them, ARIMA is the differential autoregressive moving average model, and VAR is the vector autoregressive model. Both are used to evaluate the benchmark performance of traditional time series statistical methods in multi-regional epidemic prediction tasks.

[0077] ARIMA: D. Benvenuto, M. Giovanetti, L. Vassallo, S. Angeletti, M.Ciccozzi, “Application of the ARIMA model on the COVID-2019 epidemic dataset,” Data in brief, 2020, 29: 105340. VAR: F. Khan, A. Saeed, and S. Ali, “Modelling and forecasting of new cases, deaths and recover cases of COVID-19 by using Vector Autoregressivemodel in Pakistan,” Chaos, solitons&fractals, 2020, 140: 110189. 2) Temporal prediction models based on deep learning: CNN, RNN-Res, LSTNet; CNNRNN-Res: Y. Wu, Y. Yang, H. Nishiura, and M. Saitoh, “Deeplearning for epidemiological predictions,” in The 41st International ACMSIGIR Conference on Research&Development in Information Retrieval, 2018, pp.1085–1088. LSTNet: G. Lai, W. Chang, Y. Yang, and H. Liu, “Modeling long- and short-term temporal patterns with deep neural networks,” in The 41st International ACM SIGIR Conference on Research & Development in Information Retrieval, 2018, pp. 95–104. 3) General spatio-temporal prediction models: STGCN, STGODE, STG-NCDE, STDDE; STGCN: B. Yu, H. Yin, and Z. Zhu, “Spatio-temporal graph convolutional networks: a deep learning framework for traffic forecasting,” in Proceedings of the 27th International Joint Conference on Artificial Intelligence, 2018, pp. 3634–3640. STGODE: Z. Fang, Q. Long, G. Song, and K. Xie, “Spatial-temporal graph ODE networks for traffic flow forecasting,” in Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, 2021, pp. 364–373. STG-NCDE: J. Choi, H. Choi, J. Hwang, and N. Park, “Graph neural controlled differential equations for traffic forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 36, 2022, pp. 6367–6374. STDDE: Q. Long, Z. Fang, C. Fang, C. Chen, P. Wang, and Y. Zhou,“Unveiling delay effects in traffic forecasting: a perspective from spatial-temporal delay differential equations,” in Proceedings of the ACM WebConference 2024, 2024, pp. 1035–1044. 4) Specialized models for epidemic prediction: EpiGNN, EpiColaGNN, EINN, EARTH; EpiGNN: F. Xie, Z. Zhang, L. Li, B. Zhou, and Y. Tan, “EpiGNN:Exploring spatial transmission with graph neural network for regionalepidemic forecasting,” in Joint European Conference on Machine Learning andKnowledge Discovery in Databases, Springer, 2022, pp. 469–485. EpiColaGNN: M. Liu, Y. Liu, and J. Liu, “Epidemiology-aware deeplearning for infectious disease dynamics prediction,” in Proceedings of the32nd ACM International Conference on Information and Knowledge Management,2023, pp. 4084–4088. EINN: A. Rodríguez, J. Cui, N. Ramakrishnan, B. Adhikari, and BAPrakash, “EINNs: Epidemiologically-informed neural networks,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, 2023, pp. 14453–14460. EARTH: G. Wan, Z. Liu, X. Shan, MSY Lau, BA Prakash, and W.Jin, “EARTH: Epidemiology-Aware Neural ODE with Continuous DiseaseTransmission Graph,” in Forty-second International Conference on MachineLearning, 2025. This embodiment involves short-term (5 steps), medium-term (10 steps), and long-term (15 steps) prediction of epidemics, using 20 observed time steps (days / weeks) as input, and simultaneously predicting the number of future epidemic infections in multiple regions.

[0078] This embodiment uses Root Mean Square Error (RMSE) and Concordance Correlation Coefficient (CCC) as the main evaluation metrics. RMSE measures the deviation between predicted and actual values; a smaller RMSE value indicates lower prediction error and higher accuracy. CCC measures the linear correlation between the trends of predicted and actual values; a value closer to 1 indicates a stronger ability of the model to capture the evolution of the epidemic.

[0079] This embodiment compares the prediction results with other state-of-the-art models on four public datasets. The US-Regions and US-States datasets show the performance comparison on the influenza dataset, while the AU-COVID and JP-COVID datasets show the performance comparison on the COVID-19 dataset. The results are shown in Tables 2 and 3, respectively. The best and second-best values ​​are shown in bold and underlined, respectively. Table 2

[0080] Table 3

[0081] As shown in Tables 2 and 3, the method of this invention exhibits strong predictive performance and stability in both seasonal influenza and COVID-19 real-world epidemic prediction tasks. Overall, the method achieves optimal or near-optimal prediction results across multiple datasets and prediction time domains, demonstrating that the proposed epidemic transmission delay estimation, delayed-sensing propagation graph construction, and graph-coupled neural delayed differential equation evolution mechanism can effectively improve the continuous prediction capability of multi-regional epidemics.

[0082] On the Influenza-like Illness (ILI) dataset, the US-Regions dataset exhibits a long time span and clear periodic influenza transmission characteristics. The method of this invention achieves the lowest RMSE in short-term (H=5), medium-term (H=10), and long-term (H=15) predictions, and the highest CCC under H=5 and H=15 conditions, while maintaining near-optimal prediction consistency under H=10. This indicates that the method of this invention can effectively characterize the intra-regional time dependence and inter-regional delayed transmission relationships in long-term seasonal influenza sequences, thereby reducing prediction errors and improving multi-step prediction stability. On the US-States dataset, the number of regional nodes is greater, and the inter-regional transmission structure is more complex. The method of this invention achieves the lowest RMSE in the short-term prediction task and the highest CCC in the medium- and long-term prediction tasks. This shows that although the method is not always the lowest in some medium- and long-term RMSE indicators, it can more stably maintain the consistency between the predicted trajectory and the actual epidemic trend, demonstrating its adaptability to large-scale multi-regional epidemic prediction tasks.

[0083] The advantages of the proposed method are even more pronounced on the COVID-19 dataset. Both AU-COVID and JP-COVID are daily case surveillance data, which are more susceptible to reporting delays, asynchronous regional outbreaks, and short-term case fluctuations compared to weekly ILI data. On the AU-COVID dataset, the proposed method achieves the lowest RMSE in all three prediction time domains (H=5, H=10, and H=15) and the best CCC in medium- to long-term predictions, demonstrating its ability to effectively mitigate prediction instability caused by short-term fluctuations and delayed transmission in daily epidemic data. On the JP-COVID dataset, the proposed method achieves both the best RMSE and the best CCC across all prediction time domains, indicating its strong modeling capability for COVID-19 prediction tasks involving a large number of regions and significant differences in transmission time lags.

[0084] In summary, the method of this invention exhibits superior overall performance compared to traditional statistical models, deep time-series models, general spatiotemporal prediction models, and epidemic-specific prediction models. Particularly under long-term prediction conditions, this method can more fully utilize the historical state of a region and the delayed cross-regional transmission status, reducing the risk of error accumulation in long-term predictions. These results demonstrate that the method of this invention can learn heterogeneous delayed transmission relationships using only multi-regional historical epidemic monitoring sequences, achieving stable predictions of data with different transmission characteristics, such as seasonal influenza and COVID-19, providing effective technical support for early warning of multi-regional epidemics and public health risk assessment.

[0085] Example 2 This embodiment provides a method for continuous prediction of epidemics based on delayed-sensing graph neural delayed differential equations. This embodiment aims to further verify the necessity of the core technical features of this invention, such as epidemic propagation delay estimation, delayed-sensing propagation graph construction, continuous history encoding, and graph-coupled neural delayed differential equation evolution, through ablation experiments, robustness testing, case analysis, and parameter sensitivity analysis. It also demonstrates the robustness of the method under different epidemic datasets, different prediction time domains, and different regional propagation patterns.

[0086] To verify the effectiveness of each core module of this invention, this embodiment designs ablation experiments on US state-level influenza datasets and Japanese COVID-19 datasets, and compares the complete model with the following variants: 1) Variant 1 (Fixed Delay): The delay index propagation graph in this invention is degenerated into a single fixed delay shared by dependencies within all regions and propagation relationships between regions, used to verify the necessity of heterogeneous propagation delay modeling; 2) Variant 2 (w / o Delay): Set the candidate delay set to D={0} and remove the propagation delay prior, so that both intra-regional state aggregation and inter-regional propagation state aggregation degenerate into instantaneous propagation, which is used to verify the contribution of explicit delay modeling to epidemic prediction. 3) Variant 3 (w / o DATG): Remove the delayed-aware propagation graph learning module, replace the learned delayed-aware propagation graph with a fixed graph constructed from predefined geographical adjacency relationships, and spread it evenly on the candidate delayed set to verify the importance of the adaptive delayed-aware propagation graph for learning the propagation dependency of potential regions. 4) Variant 4 (w / o DDE): The graph-coupled neural delayed differential equation evolution module is removed, and the discrete-time GRU propagation module is used to replace the continuous-time delayed state evolution, which is used to verify the role of continuous-time neural delayed differential equations in multi-step prediction. 5) Variant 5 (w / o NCDE): The continuous history encoding and history function construction based on neural control differential equations are removed and replaced with a discrete-time encoder to verify the contribution of continuous history trajectories to delayed perception propagation graph learning and subsequent delayed differential equation evolution.

[0087] Experimental results show that the complete model of this invention achieves the lowest RMSE and the highest CCC in different prediction time domains on both the US state-level influenza dataset and the Japanese COVID-19 dataset. This indicates that there is a complementary relationship between epidemic transmission delay estimation, delayed-aware transmission graph construction, continuous history encoding, and graph-coupled neural delay differential equation evolution, which can jointly improve the continuous prediction performance of multi-regional epidemics. Among them, the removal of the delayed-aware transmission graph module and the graph-coupled neural delay differential equation module resulted in the most significant performance degradation, indicating that adaptive delayed transmission relationship learning and continuous time delay state evolution are the key technical aspects of this invention. The performance of the Fixed Delay and w / o Delay variants is lower than that of the complete model, indicating that the real epidemic transmission process cannot be simply regarded as instantaneous transmission or global fixed-delay transmission, but requires learning heterogeneous transmission delays for different regions and different transmission directions. The performance degradation of the w / o NCDE variant further illustrates that continuous history encoding can provide more sufficient historical trajectory information for delayed-aware transmission graph construction and neural delay differential equation evolution, especially suitable for monitoring scenarios with missing, delayed reporting, or non-equal interval sampling. The ablation experiment results are shown in Table 4.

[0088] Table 4

[0089] Furthermore, to verify the robustness of this invention in irregular epidemic monitoring scenarios, this embodiment employs a random downsampling strategy to simulate missing observations and delayed reporting, setting missing rates from 10% to 50%, and evaluating the predictive performance on two real-world epidemic datasets. The experimental results are shown in Table 5. The data missing rate experiment results show that as the missing rate increases, the RMSE of the method of this invention gradually increases, while the CCC slightly decreases, but the overall degradation remains within an acceptable range. This indicates that continuous-time history encoding can construct continuous control paths from irregular observations, and the graph-coupled neural delayed differential equation can utilize delayed historical states for propagation and evolution without strict discrete alignment, thereby maintaining a relatively stable multi-regional epidemic prediction capability.

[0090] Table 5

[0091] To more intuitively demonstrate the predictive effect of the method of this invention, this embodiment selects four representative regions for case analysis, see... Figure 3 As shown. Figure 3The paper presents case prediction curves for Region 4 in the US regional influenza dataset, Region 45 in the US state-level influenza dataset, Region 7 in the Australian COVID-19 dataset, and Region 12 in the Japanese COVID-19 dataset at a prediction time domain H=10. The method of this invention is compared with two competing baseline models, EARTH and STDDE. Figure 3 As can be seen, the predicted curve generated by the method of this invention is generally closer to the actual observed curve, with smaller deviations near the peak of the major epidemic, and can more accurately capture fine-grained changes in the number of cases within a local time window. In contrast, some baseline models are prone to large deviations or unstable fluctuations during rapid rises, rapid falls, or peak transitions. Because the method of this invention simultaneously utilizes the regional delay state and the inter-regional delay transmission state, it can better adapt to the epidemic change trends under different data scales, different disease types, and different regional transmission patterns.

[0092] Furthermore, this embodiment performs sensitivity analysis on the key hyperparameters of the method of the present invention, and the results are shown in [the table below]. Figure 4 As shown. The hyperparameters analyzed include the history window length. Maximum candidate delay and cross-regional balance coefficient Among them, the historical window length is used to control the range of historical monitoring sequences that the model can utilize; the maximum candidate delay is used to limit the search range of time delay within the region and propagation delay between regions; and the cross-regional balance coefficient is used to adjust the contribution intensity of inter-regional delayed propagation messages in continuous state evolution. Figure 4 The results show that the method of this invention maintains relatively stable predictive performance under different parameter settings, indicating that the method has good robustness to key hyperparameters. In the analysis of historical window length, a moderate historical window can provide the model with sufficient historical propagation information while avoiding excessive noise introduced by overly long historical sequences. In the analysis of maximum candidate delay, an appropriate candidate delay range is beneficial for capturing short-term lag effects in the actual propagation process, while too small or too large a candidate range may affect the accuracy of time delay estimation. In the analysis of cross-regional balance coefficient, appropriately enhancing inter-regional propagation messages can improve the model's ability to characterize cross-regional epidemic propagation, but excessively strong cross-regional propagation weights may weaken the role of the region's own historical state. Therefore, reasonably setting the above parameters helps to achieve a balance between regional evolution and cross-regional delayed propagation.

[0093] In summary, this embodiment further demonstrates that the delayed-sensing graph neural delayed differential equation structure proposed in this invention can effectively improve the accuracy and stability of continuous prediction of multi-regional epidemics. Ablation experiments show that heterogeneous propagation delay modeling, delayed-sensing propagation graph learning, continuous history encoding, and graph-coupled delayed differential equation evolution are all indispensable technical components of this invention. Case prediction results show that the method of this invention can generate prediction curves that more closely reflect the actual case change trends on different epidemic datasets. Parameter sensitivity analysis shows that the method of this invention maintains good prediction robustness under changes in key hyperparameters. Therefore, this invention can provide reliable technical support for early warning of multi-regional epidemics, judgment of transmission trends, and public health emergency decision-making.

[0094] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the invention. This application is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the claims.

[0095] It should be understood that the present invention is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is defined only by the appended claims.

Claims

1. A method for continuous prediction of propagation risk based on graph neural delay differential equations, characterized in that, The method includes: S1, Input Multi-Regional Epidemic Surveillance Sequence: Obtain and input time-series data containing historical epidemic surveillance records for N regions; S2, Epidemic Transmission Lag Estimation: Transform the original monitoring values ​​in step S1, calculate the logarithmic growth sequence, calculate the lead-lag correlation between the source region and the target region under different candidate lags within the candidate lag set, and estimate the inter-regional transmission lag accordingly; at the same time, estimate the intra-regional time lag, and construct the epidemic transmission lag prior based on the inter-regional transmission lag and the intra-regional time lag. S3, Delay-aware propagation graph construction: Based on the delay prior obtained in step S2, combined with the continuous historical representation of each region, the learnable region embedding, and the learnable delay embedding, calculate the delay-aware relationship scores corresponding to the target region, source region, and candidate delay; normalize the delay-aware relationship scores, and construct the delay-aware propagation graph based on the normalized delay-aware relationship scores. S4, Graph-coupled neural delay differential equation evolution: Based on the delay-aware propagation graph obtained in step S3, a continuous history function is constructed and initialized with the hidden state of the predicted starting point; in continuous time, the delayed state within the region and the delayed propagation state between regions are aggregated according to the delay-aware propagation graph to obtain the delayed message of the target region, and a graph-coupled neural delay differential equation is constructed to realize the continuous time evolution of the hidden state of the multi-region epidemic. S5, Continuous prediction of multi-regional transmission risk: Read out the future hidden state obtained from step S4 to obtain the predicted logarithmic number of cases, and obtain the predicted number of cases through inverse transformation; finally, output the multi-regional epidemic prediction results for multiple future time steps.

2. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 1, characterized in that, In step S1, the time series data is represented as ,in, , indicating the first A vector of monitoring values ​​for all regions at each observation time point. Represents the monitoring value vector The Each component, i.e., the region In the Epidemiological monitoring records at each observation time point.

3. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 2, characterized in that, In step S2, the specific method for estimating the epidemic transmission time lag is as follows: Step 2.1, Calculation of Lag Correlation: First, calculate the lag correlation of the input in step S1. The region in the first Raw monitoring values ​​at each observation time step Perform a logarithmic transformation to obtain the logarithmic case sequence. Log-growth sequence is calculated based on the logarithmic difference between adjacent observation time steps. ; Let the candidate delay set be: , in, Indicates the maximum candidate propagation delay; for the target area Heyuan area In candidate delay The following calculates the lead-lag correlation: , in, Indicates the region From the 2nd to the 1st A historical logarithmically growing subsequence at each time step Indicates the target area From the The first to the second A time-step lag log-grown subsequence; This represents the Pearson correlation coefficient, used to measure the source region. Changes and target area Postponed The degree of linear correlation between the changes after each time step; Step 2.2, Inter-regional propagation time delay estimation: For propagation time delay estimation between different regions, when At that time, in the candidate delay set Within the source region, the candidate delay that maximizes the lead-lag correlation is selected as the source delay. To the target area Inter-regional propagation time lag: , The corresponding correlation is then nonnegated and used as the reliability strength of the propagation delay: in, Indicates nonnegativity; Step 2.3, Time Delay Estimation within the Region: For time self-delay estimation within the same region, when At that time, a candidate delay set that excludes zero delay is used. and in Within the region, the candidate delay that maximizes the correlation between the current state of change and its historical state of change is selected as the region. Time delay within the region: , Meanwhile, the corresponding correlation is nonnegated and used as the reliability strength of the time delay within this region: Step 2.4, Unified Time-Lag Prior Generation: Based on the inter-regional propagation time lag obtained in Step 2.2 and the intra-regional time lag obtained in Step 2.3, a unified epidemic propagation time lag prior is constructed. .

4. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 3, characterized in that, In step S2, the specific method for constructing the delayed sensing propagation map includes the following steps: Step 3.1, Continuous Control Path Construction: Construct a continuous control path for each region's logarithmic case observation sequence input in Step S1; for the... Each region, based on its available observation points Constructing continuous control paths using natural cubic spline interpolation ,in, Indicates the region The actual observation timestamp set; Step 3.2, Continuous history representation encoding: based on the continuous control path The region is obtained by encoding using neural control differential equations. Continuous historical representation Its expression is: in, Represents the time variable of integration. Indicates the region In time The continuous history represents, Indicates based on region The logarithmic case observations were used to obtain the continuous control path in time using natural cubic spline interpolation. The value of , Represents a learnable encoding function Step 3.3, Calculation of delayed perception relationship score: The delayed perception relationship score It is jointly determined by the learnable relation scoring term and the propagation delay prior soft constraint term, and its expression is: in, The parameter is Learnable relational scoring function; Indicates the target area At the starting point of the prediction A continuous historical representation; Indicates the source region At the delayed time point A continuous historical representation; and Representing the target area Heyuan area Learnable region embedding; Indicates candidate delay Learnable delayed embeddings; The weighting coefficients represent the propagation delay prior soft constraints; This represents the Gaussian kernel type time-delay prior bias term; Step 3.4, Generation of Delayed-Aware Propagation Graph: Based on the delayed-aware relationship scores obtained in Step 3.3, construct the delayed-aware propagation graph: , in, Indicates candidate delay The corresponding propagation relationship matrix, matrix elements Indicates the source region The state after candidate delay Then target area Influence weight; For intra-regional delayed dependencies, i.e. At that time, in the region Normalize across the entire range of candidate delays: For inter-regional delayed propagation dependency, i.e. At that time, in the target area Normalization is performed across the entire source region and the candidate delay combination range: in, Indicates the region In candidate delay The regional delay relationship score is below. Indicates the source region In candidate delay Below the target area Inter-regional delayed propagation relationship score; and These represent the summation indexes for the candidate delay and the source region, respectively.

5. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 4, characterized in that, In step S4, the specific method for the evolution of the graph-coupled neural delayed differential equation includes the following steps: Step 4.1, Construction of continuous history function: Based on the region continuous history representation obtained in step S3 Mapping it to the hidden state space of the neural delay differential equation, constructing the region Continuous history function: , in, This represents a learnable linear mapping matrix. Indicates the region The historical hidden state trajectory prior to the prediction starting point is determined; and the region is initialized with the historical function value at the prediction starting point. Future hidden state ; Step 4.2, Delayed State Query: During continuous-time evolution, when a query region is needed... exist When there is a time delay, it depends on whether the query time is earlier than the prediction start time. Obtain the delayed state using different methods: in, This represents the region maintained by the numerical solver. The future hidden state trajectory This indicates an interpolation query operation; Step 4.3, Delayed Message Aggregation: Based on the delay-aware propagation graph obtained in step S3 The target region is obtained by weighted aggregation of the delay propagation states within and between regions. In time Delayed message : , in, and Represents the learnable state transition matrix. This represents the balance coefficient of the inter-regional propagation contribution; the first term represents the intra-regional delayed state aggregation term, and the second term represents the inter-regional delayed propagation state aggregation term. Step 4.4, Construction of the graph-coupled neural delayed differential equation: The current hidden state... Delayed messages and continuous time Input a vector field parameterized by a neural network to construct a region The graph-coupled neural delay differential equation: in, The parameter is The neural network vector field is used to describe the evolution of the hidden state of regional epidemics over continuous time; Step 4.5, Gated Continuous State Update: The neural network vector field A gated update structure is adopted to map the current hidden state and delayed messages to the same hidden space, and adaptively control the speed at which the current hidden state evolves into a delay-aware candidate state. Its expression is: in, Indicates delayed perception candidate state, Indicates the gating coefficient. This represents element-wise multiplication. All are learnable parameters; Step 4.6, Solving for the future hidden state: A numerical solver is used to solve the graph-coupled neural delay differential equation to obtain the future hidden state. A multi-region hidden state sequence at each time step: 。 6. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 5, characterized in that, In step S5, the specific method for continuous prediction of multi-regional propagation risk includes the following steps: Step 5.1, Predicting the logarithmic number of cases: The future hidden state sequence obtained from step S4. In the middle, read the area In the future Hidden states at each prediction time step ,in, Then, the future hidden state is input into the prediction readout network to obtain the region. In the future The predicted logarithm of cases at each predicted time step : in, The parameter is Multilayer perceptron readout network To ensure the non-negativity of the predicted logarithmic number of cases; an inverse exponential transform is performed on the predicted logarithmic number of cases to obtain the region. In the future Predicted number of infected cases at each prediction time step: Step 5.2, Multi-region Prediction Result Output: Execute Step 5.1 for all regions and all future prediction time steps to obtain the future... Multi-regional epidemiological prediction results at each time step: Indicates the first The predicted number of infected cases in all regions at each predicted time step.

7. The method for continuous prediction of propagation risk based on graph neural delay differential equations according to claim 6, characterized in that, Step S5 further includes step 5.3, prediction loss optimization: during the training phase, the actual number of infected cases is... Transformed into the true logarithm of the number of cases: The model is then optimized using the Huber loss function in the logarithmic case space. in, This represents the threshold parameter of the Huber loss function.

8. A storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the continuous prediction method for propagation risk based on graph neural delay differential equations as described in any one of claims 1 to 7.

9. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the continuous prediction method for propagation risk based on graph neural delay differential equations as described in any one of claims 1 to 7.

10. An electronic device, characterized in that, include: processor; as well as Memory for storing the executable instructions of the processor; The processor is configured to execute the propagation risk continuous prediction method based on graph neural delay differential equations as described in any one of claims 1 to 7 by executing the executable instructions.

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