Influenza transmission parameter estimation method based on fusion of three-dimensional variation method and SEIR model

Abstract

The application discloses an influenza propagation parameter estimation method based on a three-dimensional variation method and SEIR model fusion, and belongs to the technical field of infectious disease propagation dynamics modeling. The method comprises the following steps: acquiring continuous week-scale influenza monitoring data of a target region and performing pretreatment; constructing an SEIR propagation dynamics model; taking an initial compartment state and propagation parameters as unified control variables and incorporating them into a three-dimensional variation assimilation framework; constructing a cost function in the form of a logarithmic residual square sum to balance the observation contribution of different epidemic stages; adopting a forward finite difference method to approximate a gradient, and combining an adaptive step size steepest descent method to iteratively solve; applying a compartment non-negative constraint, a total population conservation correction and a propagation parameter boundary projection in the iteration process. The application realizes joint optimal estimation of an initial state and propagation parameters, and improves the adaptation ability and prediction accuracy of the SEIR model to a local influenza epidemic process.

Description

Technical Field

[0001] This invention relates to the field of infectious disease transmission dynamics modeling technology, specifically to an influenza transmission parameter estimation method based on the fusion of three-dimensional variational methods and the SEIR model. Background Technology

[0002] Influenza (flu) is a highly contagious and distinctly seasonal acute respiratory infectious disease that breaks out periodically worldwide each year, posing a persistent threat to public health. Scholars from various countries have conducted extensive research on the transmission dynamics of influenza, revealing the complex patterns of influenza epidemics across different regions and time periods. Infectious disease dynamics models are indispensable for accurately understanding the transmission patterns of influenza and achieving effective prediction.

[0003] Among numerous models, the SEIR (Susceptibility-Exposure-Infection-Recovery) model has become one of the most widely used frameworks due to its ability to characterize the incubation period specific to diseases such as influenza. This model divides the population into four stages: susceptible (S), exposed (E), infected (I), and recovered (R), using a system of differential equations to describe the transitions between these stages. However, the predictive accuracy of the SEIR model is highly dependent on whether its parameters (such as transmission rate and recovery rate) truly reflect the transmission dynamics under specific spatiotemporal conditions, which constitutes the core challenge in its application.

[0004] The applicant has discovered that the prior art has at least the following technical problems: (1) Existing SEIR influenza prediction methods typically use empirical values ​​from literature, historical epidemic season parameters, or single curve fitting results to determine the transmission rate β, latency rate σ, and recovery rate γ, and use the initial state S0, E0, I0, and R0 as pre-given conditions. This approach separates the initial state estimation from the transmission parameter estimation, making it impossible to simultaneously coordinate the dynamic constraints between variables under the same objective function. When the target region or target epidemic season changes, the model is prone to problems such as shifts in the onset time, misalignment of the peak time, and deviations in peak intensity. Even if a single parameter is within a reasonable range from the literature, the combination of parameters may not accurately reflect the local real-time transmission process.

[0005] (2) Influenza transmission parameters are significantly affected by regional climate, population contact patterns, immunization background, surveillance system, and epidemic season characteristics. Existing methods directly adopt empirical parameters from other regions or historical periods, lacking a mechanism for dynamically correcting parameters using local real-time observation data. Cross-regional parameter transplantation may lead to inconsistencies between model curves and actual observation trends, and in some cases, even results with low correlation or opposite trends, making it difficult to provide a reliable basis for local influenza prediction and resource allocation. Summary of the Invention

[0006] The purpose of this invention is to provide an influenza transmission parameter estimation method based on the fusion of three-dimensional variational methods and SEIR models, so as to solve the technical problems in existing SEIR influenza prediction methods, such as the difficulty in coordinating the estimation of initial state and transmission parameters and the prediction bias caused by the cross-regional transfer of empirical parameters.

[0007] To achieve the above objectives, the present invention provides the following technical solution: The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model provided by this invention includes the following steps: S1. Obtain influenza surveillance data for the target area, obtain observation sequences, and perform preprocessing; S2. Construct a SEIR influenza transmission dynamics model, dividing the population into four compartments: susceptible individuals, exposed individuals, infected individuals, and recovered individuals. Use transmission rate, latency rate, and recovery rate as transmission parameters to describe the transfer process between compartments, and use the weekly number of new infections as the model output. S3. Set the initial chamber state and the initial guess values ​​for the propagation parameters; S4. Using the initial guess as background information and the observation sequence as observation constraints, construct a cost function in the form of log residual sum of squares to measure the difference between the model output and the observation data. S5. Using the initial chamber state and propagation parameters as unified control variables, the gradient of the cost function with respect to each control variable is approximated by the forward finite difference method. S6. The steepest descent method combined with an adaptive step size mechanism is used to iteratively update each control variable along the negative gradient direction. S7. Apply constraint projection operations to the control variables during the iteration process. The constraint projection operations include non-negative constraints on the warehouse state, corrections for the conservation of total number of people, and boundary projections of propagation parameters. S8. Determine whether convergence has been achieved based on the preset iteration termination criterion. If the criterion is met, output the optimal initial chamber state and propagation parameter estimate. Otherwise, return to step S5 to continue iteration.

[0008] Furthermore, in step S1, the preprocessing includes: time alignment of the observation sequence, missing value checking, outlier handling, and scale transformation.

[0009] Furthermore, in step S2, the SEIR influenza transmission dynamics model adopts the closed population assumption, that is, it does not consider births, deaths, and population migration, and recovered individuals acquire permanent immunity. The model describes the influenza transmission dynamics process through the following set of differential equations:

[0010] Where S represents susceptible individuals, E represents exposed individuals, I represents infected individuals, R represents recovered individuals, N represents the total population, N = S + E + I + R, β represents the transmission rate, σ represents the latency rate, γ represents the recovery rate, and t represents the continuous time variable.

[0011] Further, in step S4, the cost function in the form of the logarithmic residual sum of squares is:

[0012] Where J(θ) is the cost function to be minimized; θ is the vector of control variables to be estimated; K is the total number of observation time points within the assimilation window; k is the observation time point index; t k This refers to the k-th observation time. For t k The number of newly observed cases is obtained from monitoring data at any given time; t is obtained by integrating the SEIR model under parameter θ control. k The time-based model outputs the number of new infections; ε is a small constant to prevent logarithmic singularity, with a value of 1.0.

[0013] Further, in step S5, the formula for approximating the gradient using the forward finite difference method is:

[0014] in, Let i be the i-th control variable. =10 -6 For small perturbation step sizes, each perturbation step requires one forward integration of the SEIR model.

[0015] Furthermore, in step S6, the parameter update formula for the steepest descent method combined with the adaptive step size mechanism is as follows:

[0016] in, The adaptive step size for the nth iteration; the step size adjustment strategy is: the initial step size is 10. -5 If the current iteration causes the cost function to increase, then reduce the step size and backtrack to the parameters before the iteration; if the cost function continues to decrease, then appropriately increase the step size, with an upper limit of 10. -3 .

[0017] Furthermore, in step S7, the total population conservation correction is as follows: if the relative deviation between S0+E0+I0+R0 and the total population N after iteration exceeds 1%, then the states of the four cells are scaled proportionally to restore the sum to N.

[0018] Furthermore, in step S8, the iteration termination adopts multiple criteria: the gradient norm is less than 10. -12 The parameter change is less than 10.-8 The decrease in cost function is less than 10. -8 Or reach the maximum number of iterations of 3000.

[0019] Based on the above technical solution, the embodiments of the present invention can produce at least the following technical effects: (1) This invention integrates a three-dimensional variational data assimilation method with the SEIR model, directly incorporating local observation sequences into the objective function. This allows the model to synchronously update the initial cell state and transmission parameters under the combined constraints of background and observation information. Compared to directly transplanting empirical parameters from other regions, this method can more fully reflect the true transmission characteristics of the target region and the target epidemic season, reducing systematic biases caused by differences in monitoring intensity, population structure, immunization background, and epidemic season. The model output includes not only the fitted curve but also interpretable indicators such as transmission rate, incubation rate, recovery rate, and basic reproduction number, providing more localized data support for assessing influenza transmission intensity, predicting peak levels, and determining public health response levels.

[0020] (2) This invention incorporates the initial chamber state and propagation parameters into three-dimensional variational control variables and performs joint optimization under the same objective function, enabling the model to simultaneously coordinate the rise rate, peak time, and fall rate of the epidemic curve. Especially when there is limited early observation data for influenza, this method can more accurately capture the initial conditions of epidemic development by inversely constraining the unobservable E0 and I0 through the observation sequence, thus avoiding continuous deviation in subsequent predictions due to improper initial value settings.

[0021] (3) The present invention adopts logarithmic residual observation terms, which can balance the observation contributions of low epidemic period and high epidemic period, so that the optimization results pay more attention to the dynamic characteristics of the whole process; adopts an adaptive step size mechanism, which moderately accelerates the convergence when the objective function decreases and reduces the update amplitude when the objective function increases, thereby reducing the risk of oscillation and divergence; by applying non-negative constraints and total amount conservation correction to the state of the warehouse, and applying epidemiological reasonable boundaries to the transmission parameters, it avoids the generation of invalid solutions by simply pursuing goodness of fit. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0023] Figure 1 This is a flowchart of the influenza transmission parameter estimation method of the present invention; Figure 2This is a comparison chart of fitting predictions from different parameter sources in embodiments of the present invention. Detailed Implementation

[0024] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. 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. In addition, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0025] The objective of this invention is achieved through the following technical solution: This embodiment provides a method for estimating influenza transmission parameters based on the fusion of three-dimensional variational and SEIR models. The overall technical process is as follows: Figure 1 As shown. The method specifically includes the following steps.

[0026] Step 1: Acquisition and preprocessing of influenza surveillance data.

[0027] Obtain continuous weekly influenza surveillance data for the target area, including the total number of influenza-like illness samples tested and the number of positive samples, and calculate the influenza virus detection positivity rate from these two data. Perform the following preprocessing on the observed sequences: (1) Time alignment: Align each observation record according to the natural week to ensure that the observation sequence is continuous and consistent in the time dimension; (2) Missing value check: Check whether there are missing cycles in the observation sequence, and impute or remove missing values; (3) Outlier handling: Identify and handle outliers in the observation sequence caused by factors such as reporting delays and adjustments to detection strategies; (4) Scale transformation: Perform necessary scale transformation on the observed data according to the population size of the target area to match the scale of the state variables in the SEIR model.

[0028] The observation sequence is obtained after preprocessing.

[0029] Step 2: Construct a SEIR propagation dynamics model.

[0030] The total population N of the target area is divided into four compartments: susceptible individuals (S), exposed individuals (E), infected individuals (I), and recovered individuals (R). The transfer process between compartments is described by the transmission rate β, latency rate σ, and recovery rate γ. The model uses the following set of equations to describe the influenza transmission dynamics:

[0031] Where N = S + E + I + R represents the total population; β is the transmission rate, representing the rate at which a single infected person effectively contacts a susceptible person and causes infection per unit time; σ is the latency rate, equal to the reciprocal of the average latency period, used to describe the speed at which an exposed person becomes infected; γ is the recovery rate, equal to the reciprocal of the average infection period, used to describe the speed at which an infected person becomes a recovered or removed person; and t is a continuous-time variable. The model adopts the closed population assumption, i.e., it does not consider births, deaths, and population migration, and recovered individuals acquire permanent immunity. The basic reproduction number R0 is a key indicator for measuring the transmissibility of a pathogen, defined as the average number of secondary cases generated by a single infected person in a fully susceptible population, calculated as: R0 = β / γ. The model output is the weekly new infection count I. model The observed variable is the number of new cases per week (I). obs .

[0032] Step 3: Set the initial guess value.

[0033] Based on common knowledge of influenza epidemiology, the initial guess value is set as: S0=N 1, E0=0, I0=1, R0=0, β=0.3 days -1 σ=0.5 days -1 (Corresponding to an average incubation period of 2 days), γ = 0.2 days -1 (Corresponding to an average infection period of 5 days). This initial conjecture provides sufficient trend constraints for the identification of transmission dynamics parameters.

[0034] Step 4: Construct the cost function in the form of the logarithmic sum of squared residuals.

[0035] If ordinary least squares (OLS) is used, which directly minimizes the sum of squares of the differences between simulated and observed values, the residuals from the peak period will dominate the cost function because the number of cases in the early and late stages of an influenza epidemic can differ by more than two orders of magnitude. The model will prioritize fitting the peak period data and be insensitive to earlier growth trends. To overcome this deficiency, this embodiment constructs the cost function J(θ) in the form of the logarithmic sum of squared residuals, and its mathematical expression is:

[0036] Where J(θ) is the cost function to be minimized; θ is the vector of control variables to be estimated, θ=(S0,E0,I0,R0,β,σ,γ) K represents the total number of observation time points within the assimilation window; k represents the sequence number of the observation time point; t k This refers to the k-th observation time. For t k The number of newly observed cases is obtained from monitoring data at any given time; t is obtained by integrating the SEIR model under parameter θ control. kThe model outputs the number of new infections at each time step; ε is a small constant to prevent logarithmic singularity, with a value of 1.0. The essential advantage of logarithmic residuals is that the number of cases may differ by several orders of magnitude in the early and late stages of a disease outbreak. If ordinary least squares (absolute error) is used, the residuals at the peak will dominate the entire cost function, and the model will prioritize fitting the peak and be insensitive to the early growth trend. The logarithmic transformation converts absolute error into relative error, keeping the contribution of each stage to optimization balanced, and ensuring that the model accurately captures the entire dynamic process of the outbreak's onset, peak, and decline.

[0037] Step 5: Unify the control variables and gradient calculation.

[0038] The seven variables S0, E0, I0, R0, β, σ, and γ are combined into a unified control variable θ = (S0, E0, I0, R0, β, σ, γ). Joint optimization is performed within a three-dimensional variational assimilation framework: using the initial guesses from step three as background information and the observation sequence obtained from step one as observation constraints, the optimal estimation of the control variables is achieved by minimizing the cost function J(θ).

[0039] Since the SEIR model is a nonlinear system and lacks an analytical adjoint model, the gradient calculation is approximated using the forward finite difference method:

[0040] in, Let i be the i-th control variable. =10 -6 For small perturbation step sizes, each perturbation step requires one forward integration of the SEIR model.

[0041] Step Six: Steepest Descent Method and Adaptive Step Size Update.

[0042] The parameter update uses the steepest descent method, and the update formula is:

[0043] That is, iterating along the negative gradient direction gradually reduces the cost function to a local optimum.

[0044] Adaptive step size The mechanism is the key numerical technique in this embodiment: initial step size α = 10 -5 If the current iteration step causes the cost function to increase (J) n+1 >J n If the cost function continues to decrease, then reduce the step size and backtrack to retry; if the cost function continues to decrease, then accelerate appropriately; the upper limit of the step size is set to 10. -3 This is to prevent oscillations or divergence. The purpose of this mechanism is to achieve a dynamic balance between convergence speed and iterative stability without manual parameter tuning.

[0045] Step 7: Constrain projection operation.

[0046] The following biological hard constraints are imposed on the updated control variables: (1) Non-negative constraint of the compartment state: ensure that S0, E0, I0 and R0 are all non-negative values; (2) Total number of people conservation correction: S0+E0+I0+R0=N. If the relative deviation after iteration exceeds 1%, the state of the four compartments is scaled proportionally to restore the sum to N. (3) Projection of transmission parameters: Based on prior knowledge of influenza epidemiology, the parameter boundaries are set as β∈[0.1,2.0], σ∈[0.25,1.0] (corresponding to incubation period of 1 to 4 days) and γ∈[0.14,0.25] (corresponding to infection period of 4 to 7 days). Parameters that exceed the boundaries are directly projected to the nearest boundary value.

[0047] The purpose of constrained projection is to limit the search space to a biologically reasonable region, avoiding invalid solutions with no epidemiological significance that may result from simply pursuing goodness of fit.

[0048] Step 8: Convergence judgment and result output.

[0049] Iteration termination is based on multiple criteria: gradient norm is less than 10. -12 (The flat area has been reached) and the parameter change is less than 10. -8 (Parameter convergence), cost function decrease less than 10 -8 (Cost convergence), or reaching the maximum number of iterations of 3000 (to prevent infinite loops). If any criterion is met, the iteration terminates, outputs the optimal initial chamber state (S0*,E0*,I0*,R0) and propagation parameter (β,σ*,γ) estimates, and calculates the basic regeneration number R0=β / γ* accordingly; otherwise, return to step five to continue iterating.

[0050] The method of this embodiment was applied to influenza season surveillance data in the target area and compared with the SEIR model that directly transplants empirical parameters from other areas. The results are as follows: Figure 2 As shown in the figure. Experiments show that the parameter combination estimated by the method of the present invention can maintain a high consistency between the model curve and the local observed trend. It is superior to the empirical parameter scheme in terms of the onset time, peak time, peak intensity and decline stage of the epidemic. Moreover, the optimization process converges stably, and the output transmission rate, latency rate, recovery rate and basic reproduction number are all within the reasonable range of epidemiology, which verifies the effectiveness and reliability of the method of the present invention.

[0051] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for estimating influenza transmission parameters based on the fusion of a three-dimensional variational method and a SEIR model, characterized in that, Includes the following steps: S1. Obtain influenza surveillance data for the target area, obtain observation sequences, and perform preprocessing; S2. Construct a SEIR influenza transmission dynamics model, dividing the population into four compartments: susceptible individuals, exposed individuals, infected individuals, and recovered individuals. Use transmission rate, latency rate, and recovery rate as transmission parameters to describe the transfer process between compartments, and use the weekly number of new infections as the model output. S3. Set the initial chamber state and the initial guess values ​​for the propagation parameters; S4. Using the initial guess as background information and the observation sequence as observation constraints, construct a cost function in the form of log residual sum of squares to measure the difference between the model output and the observation data. S5. Using the initial chamber state and propagation parameters as unified control variables, the gradient of the cost function with respect to each control variable is approximated by the forward finite difference method. S6. The steepest descent method combined with an adaptive step size mechanism is used to iteratively update each control variable along the negative gradient direction. S7. Apply constraint projection operations to the control variables during the iteration process. The constraint projection operations include non-negative constraints on the warehouse state, corrections for the conservation of total number of people, and boundary projections of propagation parameters. S8. Determine whether convergence has been achieved based on the preset iteration termination criterion. If the criterion is met, output the optimal initial chamber state and propagation parameter estimate. Otherwise, return to step S5 to continue iteration.

2. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S1, the preprocessing includes: time alignment of the observation sequence, missing value checking, outlier handling, and scale transformation.

3. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S2, the SEIR influenza transmission dynamics model adopts the closed population assumption, that is, it does not consider births, deaths, and population migration, and recovered individuals acquire permanent immunity. The model describes the influenza transmission dynamics process through the following set of differential equations: ; Where S represents susceptible individuals, E represents exposed individuals, I represents infected individuals, R represents recovered individuals, N represents the total population, N = S + E + I + R, β represents the transmission rate, σ represents the latency rate, γ represents the recovery rate, and t represents the continuous time variable.

4. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S4, the cost function in the form of the logarithmic residual sum of squares is: ; Where J(θ) is the cost function to be minimized; θ is the vector of control variables to be estimated; K is the total number of observation time points within the assimilation window; k is the observation time point index; t k This refers to the k-th observation time. For t k The number of newly observed cases is obtained from monitoring data at any given time; t is obtained by integrating the SEIR model under parameter θ control. k The time-matter model outputs the number of new infections; ε is a small constant to prevent logarithmic singularity, with a value of 1.

0.

5. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S5, the formula for approximating the gradient using the forward finite difference method is as follows: ; in, For the i-th control variable, =10 -6 For small perturbation step sizes, each perturbation step requires one forward integration of the SEIR model.

6. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S6, the parameter update formula for the steepest descent method combined with the adaptive step size mechanism is as follows: ; in, The adaptive step size for the nth iteration; the step size adjustment strategy is: the initial step size is 10. -5 If the current iteration causes the cost function to increase, then reduce the step size and backtrack to the parameters before the iteration; if the cost function continues to decrease, then appropriately increase the step size, with an upper limit of 10. -3 .

7. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S7, the total population conservation correction is as follows: if the relative deviation between S0+E0+I0+R0 and the total population N after iteration exceeds 1%, then the states of the four cells are scaled proportionally to restore the sum to N.

8. The influenza transmission parameter estimation method based on the fusion of three-dimensional variational method and SEIR model according to claim 1, characterized in that, In step S8, the iteration termination is based on multiple criteria: the gradient norm is less than 10. -12 The parameter change is less than 10. -8 The decrease in cost function is less than 10. -8 Or reach the maximum number of iterations of 3000.

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