A kernelized data-driven approach for analyzing and predicting brucellosis in Inner Mongolia

Abstract

Brucellosis remains a persistent public health concern in Inner Mongolia, exhibiting complex spatiotemporal transmission, pronounced periodicity, and strong nonlinear dynamics. To address these challenges, a novel hybrid modeling framework—kernelized Hankel dynamic mode decomposition with sparsity promotion- long short-term memory (KHDMDsp-LSTM) —is proposed for analysis and prediction of epidemic dynamics. The framework’s core innovation is the KHDMDsp, which linearizes nonlinear epidemic processes by embedding low-dimensional system states into a high-dimensional feature space via a hybrid Mercer kernel, enabling extraction of diverse dynamical modes. Kernel parameters are adaptively optimized using Bayesian optimization, and an elastic net regularization strategy promotes sparsity and interpretable mode selection. An integrated LSTM network captures nonlinear temporal dependencies across multiple cities, enhancing long-term prediction. Applied to 11 years of monthly brucellosis data from 12 cities, the framework reconstructs and predicts epidemic dynamics with high fidelity, revealing stable, gradually decaying transmission concentrated in central and eastern Inner Mongolia. Compared with baseline models, KHDMDsp-LSTM demonstrates superior reconstruction and long-horizon prediction, highlighting the potential of combining kernel-based Koopman spectral analysis with deep learning as a robust, data-driven approach for infectious disease modeling and prediction.

Graphical abstract

Public summary

• •A hybrid kernel-based data-driven framework is proposed for high-dimensional linearized modeling.
• •Elastic net regularization balances model sparsity and accuracy, enabling the selection of key Koopman modes.
• •Combining with LSTM, merging Koopman interpretability with deep learning’s nonlinear predictive capability.
• •Empirical analysis of Inner Mongolia brucellosis data validates the framework’s practicality.

Introduction

Brucellosis is a zoonotic disease caused by Brucella bacteria, transmitted primarily through contact with infected animals, consumption of unpasteurized animal products, or inhalation of contaminated aerosols—especially in occupational settings such as farming and animal husbandry. Infected individuals often present with fever, fatigue, night sweats, and musculoskeletal pain. Without timely treatment, the disease may progress to chronic stages, affecting the heart, liver, and spleen, and causing complications such as osteomyelitis and meningitis. Beyond its health burden, brucellosis poses a major threat to the livestock industry, particularly among cattle, sheep, and pigs, by reducing productivity and causing significant economic losses.^1,2,3 These impacts are especially severe in rural China, where animal husbandry remains a critical livelihood. Over the past two decades, the disease has exhibited clear spatial expansion, underscoring the need for more effective surveillance and control. According to data from the Chinese Center for Disease Control and Prevention (CDC), from 2010 to 2015, China reported an average of 45,000 cases annually, with the highest burdens in western and northwestern regions.

Inner Mongolia, one of China’s primary livestock-producing regions, has long been a hotspot for brucellosis. As noted in the literature by Pappas et al.,⁴ factors such as high animal density, frequent interregional livestock movement, and inadequate vaccination coverage have contributed to the persistent transmission of Brucella bacteria. Although control measures—including expanded vaccination campaigns, strengthened quarantine protocols, and public health education—led to a decline in incidence after 2013, a resurgence has been observed in recent years, particularly since 2018. Ongoing challenges—including limited healthcare access, low public awareness, and underdeveloped veterinary infrastructure—continue to impede effective disease control. Given these conditions, Inner Mongolia serves as a representative case for developing and validating advanced modeling frameworks for infectious disease dynamics, with insights that may be generalizable to other endemic regions.

Classical compartmental models, such as the susceptible-infected-recovered framework, have long been fundamental tools for analyzing infectious disease dynamics.⁵ By partitioning the population into compartments and modeling disease transmission through differential equations, these models offer valuable insights for disease research and intervention design.^6,7,8,9,10 However, their effectiveness is often limited by simplifying assumptions—such as homogeneous mixing, time-invariant parameters, and constant transmission rates—that rarely hold in real-world settings. These limitations are particularly pronounced in modeling brucellosis, where incomplete animal infection data and the rarity of human-to-human transmission further compromise model reliability and interpretability.¹¹ Moreover, as these models are based on first principles, they typically focus on qualitative and stability analysis of real-world problems. Accurately characterizing and predicting practical dynamics often requires highly complex models, which are difficult to solve and limit the feasibility of precise forecasting.

With the rise of big data and advanced computational tools, data-driven modeling has become a powerful approach for understanding complex dynamical systems. Unlike traditional methods grounded in first principles, data-driven approaches infer system dynamics directly from observational data, eliminating the need for explicit mechanistic assumptions.^12,13 These methods are particularly well suited to high-dimensional, nonlinear, and data-rich scenarios, and have found successful applications across fields such as physics, engineering, biology, and finance. In the context of epidemiology, data-driven modeling offers distinct advantages when the underlying mechanisms are poorly understood or difficult to parameterize.^14,15 By learning patterns directly from real-world data, these models can enhance predictive accuracy and support more informed public health decision-making. With the rapid advancement of machine learning, particularly deep learning, mainstream approaches increasingly leverage the high complexity of deep neural networks to efficiently extract patterns and regularities from vast amounts of unstructured data. Such methods have shown outstanding capability in capturing hidden nonlinear relationships within complex datasets and have been widely applied to predictive modeling tasks across a broad range of disciplines.^{16,17,18,19,20} However, the limited interpretability of deep neural networks can pose significant challenges for their practical application in public health planning and intervention.

To overcome the limitations of traditional modeling approaches, dynamic system identification has emerged as a powerful and interpretable framework for analyzing complex, time-varying processes. Among these methods, dynamic mode decomposition (DMD) has gained prominence for its ability to extract coherent spatiotemporal structures directly from time series data. Its strength lies in decomposing system evolution into interpretable dynamic modes, each associated with distinct spatial patterns and temporal behavior.²¹ Originally developed in fluid dynamics,²² DMD approximates system evolution via a best-fit linear operator obtained through singular value decomposition (SVD). The Koopman operator framework further generalizes DMD by lifting nonlinear dynamics into an infinite-dimensional linear space,^23,24,25 significantly broadening its applicability. DMD and its variants have achieved success across diverse domains. For instance, Maliyov et al. employed DMD to accelerate time propagation in the Boltzmann transport equation,²⁶ Avila and Mezić applied it to traffic flow prediction,²⁷ and Velagar et al. utilized optimized DMD in atmospheric chemistry.²⁸ Other notable developments include Wormell’s theoretical analysis of extended DMD in chaotic systems,²⁹ and Fukuma et al.’s Grassmann-kernel-based enhancement for neural decoding.³⁰ In epidemiology, DMD has been used to uncover spatiotemporal transmission modes,³¹ and inspired integrated modeling approaches.³² However, standard DMD struggles with strongly nonlinear dynamics. Kernel DMD (KDMD) addresses this by projecting data into a high-dimensional reproducing kernel Hilbert space (RKHS), allowing linear approximations of nonlinear systems.³³ KDMD preserves the interpretability of DMD while greatly improving its representational capacity.

Recent advances in deep learning have empowered recurrent neural networks (RNNs) to model temporal dependencies in epidemiological time series. Unlike feedforward networks, RNNs retain memory across time steps, making them well suited for sequential prediction tasks. However, standard RNNs face the vanishing gradient problem, which hinders their ability to capture long-range dependencies.³⁴ Long short-term memory (LSTM) networks address this limitation through gated mechanisms that support stable information flow over extended time horizons.^35,36 In infectious disease modeling, LSTM networks offer two notable advantages: capturing long-term temporal dynamics³⁷ and adapting to sudden changes driven by interventions or behavioral shifts.³⁸ Empirical studies have shown that LSTM models outperform traditional approaches in identifying critical transmission patterns, as demonstrated in COVID-19 and influenza surveillance.^39,40

Building on the complementary strengths of mechanistic modeling, machine learning, and dynamic system identification, this study proposes a hybrid modeling framework (kernelized Hankel dynamic mode decomposition with sparsity promotion- LSTM [KHDMDsp-LSTM]) that is broadly applicable to the analysis and forecasting of complex epidemic dynamics. The framework employs hybrid Mercer kernel lifting to reveal latent linear structures underlying nonlinear transmission processes, combined with Hankel embedding to extract diverse dynamical modes. An elastic net-based sparse optimization is then applied to identify dominant modes, enabling low-dimensional and interpretable representations of epidemic dynamics directly from data. To further enhance long-term forecasting and capture latent temporal dependencies, an LSTM network is integrated into the framework. By combining the mode decomposition and interpretability of kernelized Hankel DMD (KHDMD) with the strong predictive power of LSTM networks, the proposed approach offers a robust and effective tool for understanding and predicting the spatiotemporal spread of epidemic.

Materials and methods

Building upon prior work on DMD for modeling brucellosis dynamics in China,⁴¹ this study develops an enhanced framework to capture the nonlinear, periodic, and high-dimensional nature of disease transmission. We first construct a hybrid kernel tailored to brucellosis patterns and integrate it with time delay embedding to formulate the KHDMD model. To improve interpretability and generalization, we further incorporate sparsity-promoting regularization, yielding the KHDMDsp method. The following sections detail the kernel design, KHDMD derivation, and sparse extension.

Hybrid kernel construction and optimization

To approximate the Koopman operator without an explicit observation function, the KHDMDsp framework integrates HDMDsp with kernel methods, enabling an implicit nonlinear mapping into a high-dimensional feature space. To further capture the complex transmission dynamics of brucellosis in Inner Mongolia—characterized by strong nonlinearity, periodicity, and spatial heterogeneity—a hybrid kernel is constructed by combining three complementary components.

• (1
  )Polynomial kernel: models nonlinear dependencies.

  $$K_{\text{poly}}\left(\mathit{x},\mathit{y}\right)={\left({\mathit{x}}^T\mathit{y}+1\right)}^d\text{,}$$

  where d is the polynomial degree.
• (2
  )Gaussian (radial basis function, RBF) kernel: captures local smooth variations.

  $$K_{\text{gauss}}\left(\mathit{x},\mathit{y}\right)=\exp \left(-\frac{\parallel \mathit{x}-\mathit{y}{\parallel }^2}{{\mu }^2}\right)\text{,}$$

  where μ controls the bandwidth and smoothness of the kernel.
• (3
  )Fourier kernel: captures periodic patterns.

  $$K_{\text{fourier}}\left(\mathit{x},\mathit{y}\right)=\sum _{k=1}^n\sum _{j=1}^{\dim \left(\mathit{x}\right)}\cos \left(\frac{2\pi k\left(x_j-y_j\right)}{p}\right)\text{,}$$

  where n is the number of frequency components and p is the fundamental periodic parameter.

These components are linearly combined to form a convex hybrid kernel:

$$\begin{array}{l}K\left(\mathit{x},\mathit{y}\right)={\alpha }_1·K_{\text{poly}}\left(\mathit{x},\mathit{y}\right)+{\alpha }_2·K_{\text{gauss}}\left(\mathit{x},\mathit{y}\right)+{\alpha }_3·K_{\text{fourier}}\left(\mathit{x},\mathit{y}\right)\text{,}\end{array}$$

where α₁+α₂+α₃ = 1 and α_i > 0.

According to Mercer’s theorem,⁴² the composite kernel is constructed to satisfy Mercer’s condition—namely, it is symmetric and positive semi-definite—ensuring that it defines a valid inner product in a RKHS. The kernel hyperparameters—including the polynomial degree, Gaussian bandwidth, number of Fourier components, and individual kernel weights—are tuned via Bayesian optimization, which minimizes the mean absolute error (MAE) over a series of sliding window prediction tasks (detailed in Material D of the supplemental information). Furthermore, considering that the sample data in this paper consists of monthly statistics aggregated annually, p = 12 was set. Following this procedure, the final optimized hybrid kernel parameters take: d = 1, μ = 2.006, n = 3, α₁ = 0.058, α₂ = 0.267, α₃ = 0.676. The dominance of the Fourier kernel reflects the strong periodicity in brucellosis incidence, the Gaussian kernel captures localized nonlinear fluctuations, and the relatively small weight of the polynomial kernel suggests weaker long-term trends. By integrating these complementary components, the hybrid kernel substantially enhances the representational capacity of the Koopman approximation. This leads to improved prediction accuracy and better generalizability of the KHDMDsp model, making it well suited for modeling disease dynamics and supporting public health decision-making.

KHDMD

To extract spatiotemporal dynamics from complex nonlinear epidemiological data, this paper developed a KHDMD framework and applied it to human brucellosis data in Inner Mongolia. The raw observation matrix is defined as:

$${\mathit{X}}_{n\times m}^{\text{raw}}=\left[{\mathit{x}}_1,\dots ,{\mathit{x}}_m\right]\text{,}$$

where each column ${\mathit{x}}_i\in {\mathbb{R}}^n$ represents the prevalence vector across n locations at time step i.

The main steps of the KHDMD framework are summarized as follows.

• (1
  )Remove the mean from observable data to highlight volatility dynamics:

  $${\tilde{\mathit{x}}}_i={\mathit{x}}_i-\overline{\mathit{x}},\overline{\mathit{x}}=\frac{1}{m}\sum _{i=1}^m{\mathit{x}}_i\text{.}$$
• (2
  )Construct Hankel matrices using a time delay of s = 12 to enhance numerical robustness:

  $${\mathit{X}}^{\text{Hankel}}=\left[{\mathfrak{x}}_1,\dots ,{\mathfrak{x}}_{m-s+1}\right],{\mathfrak{x}}_j={\left[{\tilde{\mathit{x}}}_j^{\top },\dots ,{\tilde{\mathit{x}}}_{j+s-1}^{\top }\right]}^{\top }\in {\mathbb{R}}^{ns}\text{.}$$
• (3
  )Combine the observation function $\mathit{\phi }\left(\mathfrak{x}\right)={\left[{\phi }_1\left(\mathfrak{x}\right),\dots ,{\phi }_{N_k}\left(\mathfrak{x}\right)\right]}^{\top }$ to map ${\mathfrak{x}}_j$ onto a high-dimensional linear space and construct the adjacent time-shift matrix:

  $${\mathit{H}}_1=\left[\mathit{\phi }\left({\mathfrak{x}}_1\right),\dots ,\mathit{\phi }\left({\mathfrak{x}}_{m-s}\right)\right],{\mathit{H}}_2=\left[\mathit{\phi }\left({\mathfrak{x}}_2\right),\dots ,\mathit{\phi }\left({\mathfrak{x}}_{m-s+1}\right)\right]\text{,}$$

  where each ${\phi }_k:{\mathbb{R}}^{ns}\to \mathbb{R}$ is a scalar observable.
• (4
  )Approximate linear relationships in high-dimensional feature space can be expressed as:

  $${\hat{\mathit{H}}}_2\approx {\hat{\mathit{H}}}_1\mathcal{B}\text{,}$$

  where ${\hat{\mathit{H}}}_1={\mathit{H}}_1^{\top },{\hat{\mathit{H}}}_2={\mathit{H}}_2^{\top }$.
• (5)Following the standard DMD method employed in the papers by Schmid²² and Tu,²² a low-order approximation $\hat{\mathcal{B}}$ of the high-dimensional linear operator $\mathcal{B}$ is obtained via SVD. The Koopman spectral are then extracted through eigenvalue decomposition for model reconstruction and prediction.

However, the above process relies on an explicit observable function φ(·), which is often difficult to specify in practice—particularly for complex nonlinear systems such as epidemic transmission. This poses significant challenges for directly constructing $\mathcal{B}$ and analyzing its spectral. To overcome this limitation, this study adopts a kernel-based approach that employs a Mercer kernel κ(x,y) = φ(y)^⊤φ(x), thereby implicitly mapping the state space data into a higher-dimensional feature space. This formulation enables the direct construction of the following kernel matrices from data:

$${\mathit{K}}_1={\hat{\mathit{H}}}_1{\mathit{H}}_1,{\mathit{K}}_2={\hat{\mathit{H}}}_2{\mathit{H}}_2\text{,}$$

where ${\left({\mathit{K}}_1\right)}_{ij}=\kappa \left({\mathfrak{x}}_i,{\mathfrak{x}}_j\right),{\left({\mathit{K}}_2\right)}_{ij}=\kappa \left({\mathfrak{x}}_i,{\mathfrak{x}}_{j+1}\right)\text{.}$

Building on the theoretical framework established by Williams and co-workers,^33,43 this approach enables a full Koopman spectral decomposition—including eigenvalues μ_k, eigenfunctions ψ_k, and modes γ_k—without requiring explicit knowledge of the observable function. The resulting spectral components offer a compact and interpretable representation of the system’s dynamics. Given an initial state ${\mathfrak{x}}_1$, the system evolution can be expressed as:

$${\mathfrak{x}}_{t+1}\approx \sum _{k=1}^{N_z}{\mu }_k^t{\mathit{\gamma }}_k{\psi }_k\left({\mathfrak{x}}_1\right)\text{,}$$ (Equation 1)

where N_z denotes the number of modes retained.

KHDMD with sparsity promotion

To extract dominant dynamical structures from complex nonlinear systems while improving interpretability, we extend the KHDMD framework by incorporating sparsity-promoting regularization, resulting in the KHDMDsp method. This enhancement suppresses redundant or insignificant modes, yielding a more parsimonious and robust representation of system dynamics.

Our formulation is builds upon the sparsity-promoting DMD framework originally proposed by Jovanović et al.,⁴⁴ its extension to Koopman-invariant subspaces by Pan et al.,⁴⁵ and its recent application to data-driven epidemiological modeling by Liu and Sun.⁴¹ Leveraging these developments, KHDMDsp is employed here to investigate the spatiotemporal evolution of human brucellosis in Inner Mongolia. Compared with traditional decomposition methods, KHDMDsp offers the following advantages.

• •Nonlinear expressiveness: by utilizing kernel-induced feature spaces, KHDMDsp captures complex nonlinear and multiscale behaviors beyond the reach of classical DMD.
• •Model simplicity: the kernel trick avoids manual specification of basis functions, simplifying the modeling process.
• •Sparse interpretability: elastic net regularization selects only the most influential Koopman modes, enhancing interpretability and computational efficiency.

To identify the dominant dynamical structures, we formulate a regularized optimization problem that balances reconstruction accuracy and sparsity. Specifically, the Koopman amplitude vector is optimized through an elastic net penalty term, which is formulated as follows:

$$\rho \theta \parallel ·{\parallel }_1+\frac{1}{2}\rho \left(1-\theta \right)\parallel ·{\parallel }_2^2\text{,}$$

where the regularization parameter ρ determines the trade-off between reconstruction fidelity and sparsity, and the homotopy parameter θ∈[0,1] is introduced to tune the dominant penalty term, enabling a smooth transition between L₁ norm (promoting sparsity) and L₂ norm (promoting smoothness).⁴¹ Incorporating the Koopman spectral expansion in (Equation 1), the optimization problem is expressed as:

$${\tilde{\mathit{D}}}_{\psi }=\arg \underset{{\mathit{D}}_{\psi }}{\min }\frac{1}{2}\parallel \mathit{X}-\mathbf{\Gamma }·\text{diag}\left({\mathit{D}}_{\psi }\right)·{\mathit{V}}_{\text{and}}{\parallel }_F^2+\rho \left(\theta \parallel {\mathit{D}}_{\psi }{\parallel }_1+\frac{\left(1-\theta \right)}{2}\parallel {\mathit{D}}_{\psi }{\parallel }_2^2\right)\text{,}$$

with the variables defined as follows.

• •$\mathit{X}=\left[{\mathfrak{x}}_1,\dots ,{\mathfrak{x}}_{t+1}\right]\in {\mathbb{R}}^{ns\times \left(t+1\right)}$: Hankel type of the system’s observed state matrix.
• •$\mathbf{\Gamma }=\left[{\mathit{\gamma }}_1,\dots ,{\mathit{\gamma }}_{N_z}\right]\in {\mathbb{R}}^{n\times N_z}$: the Koopman mode matrix.
• •$\text{diag}\left({\mathit{D}}_{\psi }\right)=\text{diag}\left({\psi }_1\left({\mathfrak{x}}_1\right),\dots ,{\psi }_{N_z}\left({\mathfrak{x}}_1\right)\right)\in {\mathbb{R}}^{N_z\times N_z}$: the diagonal matrix of Koopman eigenfunction evaluations at the initial state ${\mathfrak{x}}_1$.
• •
  ${\mathit{V}}_{\text{and}}\in {\mathbb{C}}^{N_z\times \left(t+1\right)}$ denotes the Vandermonde matrix constructed from Koopman eigenvalues ${\left\{{\mu }_k\right\}}_{k=1}^{N_z}$, represented as:

  $${\mathit{V}}_{\text{and}}=\left[\begin{array}{cccc}1& {\mu }_1& \cdots & {\mu }_1^t\\ 1& {\mu }_2& \cdots & {\mu }_2^t\\ ⋮& ⋮& \ddots & ⋮\\ 1& {\mu }_{N_z}& \cdots & {\mu }_{N_z}^t\end{array}\right]\text{.}$$
• •∥·∥_F denotes the Frobenius norm, defined as: $\parallel {\mathit{A}}_{m\times n}{\parallel }_F=\sqrt{{\sum }_{i=1}^m{\sum }_{j=1}^n{\left|A_{ij}\right|}^2}\text{.}$

This problem is efficiently solved using the alternating direction method of multipliers,⁴⁶ which guarantees convergence and numerical stability. Finally, we fix the sparsity structure of the amplitudes vector and determine only the non-zero amplitudes. This step is formulated as a constrained optimization problem:

$$\underset{{\tilde{\mathit{D}}}_{\psi }}{\min }\parallel \mathit{X}-\mathbf{\Gamma }·\text{diag}\left({\tilde{\mathit{D}}}_{\psi }\right)·{\mathit{V}}_{\text{and}}{\parallel }_F^2\text{,}$$

subject to ${\mathit{E}}^T{\tilde{\mathit{D}}}_{\psi }=0\text{,}$where E encodes the locations of zero amplitudes.

Results and discussion

In this section, we employ the KHDMDsp framework to analyze the spatiotemporal dynamics of human brucellosis across 12 cities in Inner Mongolia, using monthly case data from January 2010 to December 2020 provided by the China CDC. Figure 1 illustrates the temporal evolution of reported cases at the national level, within Inner Mongolia as a whole, and across its constituent cities, providing a comprehensive overview of both temporal trends and regional disparities in disease prevalence during the study period.

Figure 1.

Brucellosis case trends in China and Inner Mongolia (2010–2020)

(A) Brucellosis cases in China, showing periodic fluctuations.

(B) Brucellosis cases in Inner Mongolia, exhibiting regional trends.

(C) City-level brucellosis cases in Inner Mongolia, revealing spatial variations.

Spatial diffusion characteristics of brucellosis

As shown in Figures 1A and 1B, the total annual number of reported brucellosis cases in China increased steadily from 2010 to 2014, followed by a sharp decline between 2014 and 2018. In contrast, Inner Mongolia experienced a more consistent downward trend throughout this period. From 2018 to 2020, however, both Inner Mongolia and the national total experienced a simultaneous resurgence, suggesting a synchronized rise in disease prevalence. Table S1 summarizes the proportion of monthly brucellosis cases in Inner Mongolia relative to the national total, providing a quantitative depiction of the dynamic relationship between regional and national epidemics. The results reveal that during 2010–2012 and 2018–2020, the total number of cases in Inner Mongolia consistently accounted for more than 25% of the national total, underscoring the significant impact of Inner Mongolia’s epidemic dynamics on the overall national pattern. More importantly, compared with the higher proportion observed before 2012, this value declined markedly after 2018, indicating a gradual shift in the geographic distribution of brucellosis outbreaks across China. Moreover, Figure 1 shows that, while Inner Mongolia experienced a sustained decline in reported cases from 2011 to 2015, national case numbers continued to rise during the same period, further supporting the notion of a progressive spatial expansion of brucellosis transmission beyond Inner Mongolia.

Figure 1C illustrates the spatiotemporal distribution of brucellosis incidence across the 12 cities of Inner Mongolia, providing a clear visual reference for examining intra-regional transmission dynamics. In the digital figure, each city is represented by a line with a distinct color intensity—lighter shades correspond to higher incidence levels in the early years, while darker shades indicate initially lower incidence. Prior to 2012, cities associated with lighter colored lines exhibited significantly higher case numbers, reflecting their dominant contribution to the regional disease burden. However, this pattern shifted after 2018: cities that were initially characterized by darker colored lines experienced a relative increase in case counts. This reversal highlights a spatial redistribution of brucellosis within Inner Mongolia, with the disease burden gradually shifting from historically high-incidence areas to regions previously less affected.

Sparse eigen-spectra reflect brucellosis stable decay

Unlike conventional mode selection strategies that prioritize modes with the largest amplitudes, the KHDMDsp algorithm is specifically designed to retain modes that most strongly influence the system’s temporal evolution. This ensures that the selected modes are not only large in magnitude but also dynamically relevant over time. The following relative loss matrix is employed to assess the relationship between model sparsity and predictive performance, and to guide the selection of regularization parameters:

$${\mathrm{\Pi }}_{\text{loss}}=\sqrt{\frac{\mathit{J}\left({\tilde{\mathit{D}}}_{\psi }\right)}{\mathit{J}\left(0\right)}}\times 100\%=\sqrt{\frac{\parallel \mathit{X}-\mathbf{\Gamma }·\text{diag}\left({\tilde{\mathit{D}}}_{\psi }\right)·{\mathit{V}}_{\text{and}}{\parallel }_F^2}{\parallel \mathit{X}{\parallel }_F^2}}\times 100\%\text{.}$$

Figure 2A illustrates how the number of retained modes N_z and the corresponding loss Π_loss vary with the regularization strength ρ. As expected, increasing ρ enhances sparsity at the cost of higher loss. Figure 2B reveals a nonlinear inverse relationship between N_z and Π_loss, with several inflection points. The loss drops sharply as N_z increases from 0 to 26, indicating that these modes capture the dominant dynamics. A notable inflection point appears at N_z = 8, achieving approximately 50% reconstruction accuracy. Beyond this, performance gains diminish until a sudden 18.31% drop in error occurs between N_z = 98 and N_z = 100, suggesting that these two modes encode important latent dynamics. Consequently, N_z = 100 is selected as the optimal truncation point, with the corresponding ρ inferred from Figure 2A.

Figure 2.

Effect of regularization on retained mode and relative loss

(A) Variation of retained mode N_z (right axis) and loss Π_loss (left axis) as a function of regularization parameter ρ.

(B) Relationship between Π_loss and N_z, with the key turning points highlighted.

(C–H) Eigenvalue spectra under varying sparsity conditions.

Figures 2C–2H shows the eigenvalue spectra generated by KHDMDsp (solid dots) under increasing levels of sparsity. Eigenvalues inside the unit circle correspond to decaying dynamics, while those outside indicate potential instability. As regularization increases, unstable modes are gradually eliminated, indicating that the brucellosis dynamics in Inner Mongolia are fundamentally stable and not prone to large-scale outbreaks.

The following section examines the function of the homotopy parameter in controlling the trade-off between the primary regularization terms, thereby influencing the model’s sparsity and predictive performance. To evaluate its effect, the time series from January 2010 to December 2019 is used for model training, while the final year (January–December 2020) is reserved for out-of-sample testing. This data split is consistently applied throughout the study to assess predictive performance. Model accuracy is evaluated using MAE and root mean-square error (RMSE), defined as follows:

$$\text{MAE}\left(\tilde{\mathit{Y}},\hat{\mathit{Y}}\right)=\frac{1}{m·n}\sum _{i=1}^m\sum _{j=1}^n\left|{\tilde{\mathit{Y}}}_{ij}-{\hat{\mathit{Y}}}_{ij}\right|,\text{RMSE}\left(\tilde{\mathit{Y}},\hat{\mathit{Y}}\right)=\sqrt{\frac{1}{m·n}\sum _{i=1}^m\sum _{j=1}^n{\left({\tilde{\mathit{Y}}}_{ij}-{\hat{\mathit{Y}}}_{ij}\right)}^2}\text{,}$$

where $\tilde{\mathit{Y}}\in {\mathbb{R}}^{m\times n}$ and $\hat{\mathit{Y}}\in {\mathbb{R}}^{m\times n}$ denote the ground truth and predicted values, respectively.

Figure 3 illustrates the variation in the number of retained modes and the model’s predictive error on both the training and testing sets as functions of the homotopy parameter, with detailed numerical values provided in Table S2. As θ increases from 0 to 1, the regularization gradually shifts from L₂-dominant (dense solutions) to L₁-dominant (sparse solutions), leading to a progressive reduction in the number of retained modes and thus enhanced sparsity. Correspondingly, the training error increases noticeably with θ, reflecting the loss of model flexibility as sparsity intensifies. Although the testing error remains higher than the training error, its fluctuation amplitude is relatively smaller, suggesting that the model maintains reasonable generalization capability across the homotopy range. Considering that the primary objective of this study is to capture the dominant dynamic structures of the system, θ = 0.2 is selected as an appropriate trade-off between sparsity and predictive accuracy for the final implementation.

Figure 3.

Effect of the homotopy parameter θ on retained modes number N_z and training/testing errors

The x axis corresponds to the homotopy parameter θ. The left y axis indicates the number of retained modes (N_z) and the training error, while the right y axis corresponds to the testing error.

Dominant modes reveal cycles and hotspots of brucellosis transmission

Figure 4 illustrates representative dynamic modes extracted by the KHDMDsp model, capturing the dominant spatiotemporal features of brucellosis transmission in Inner Mongolia. Specifically, Figure 4A shows a short-term mode with a quarterly oscillation (period = 3.925 months), while Figures 4B and 4C reflect cycles with periods of 12.105 and 10.977 months, respectively, corresponding to annual and near-annual oscillations. Figure 4D represents a long-term mode with a period equal to the full data span (132.051 months). Among these, all except Figure 4C—which was among the additional modes retained when increasing N_z from 98 to 100—belong to the initial set of 26 modes selected under sparsity constraints.

Figure 4.

Spatiotemporal dynamics of the dominant modes extracted by the KHDMDsp model

The blue and red lines represent disease trends in HulunBuir and Hinggan League, respectively. Mode growth/decay rates are determined by the modulus of the corresponding eigenvalues, and oscillation periods are computed from their imaginary parts. The characteristics of the selected modes are:

(A) Decay rate = 0.937, period = 3.925 months.

(B) Decay rate = 0.960, period = 12.105 months.

(C) Growth rate = 1.018, Period = 10.977 months

(D) Decay rate = 0.992, period = 132.051 months.

Spatially, these modes reveal persistent oscillatory patterns concentrated in central Inner Mongolia, particularly in Hinggan League, Ulanqab, Chifeng, Tongliao, and Xilingol League. These areas appear to serve as epidemiological epicenters, significantly influencing regional transmission dynamics. Temporally, the long-period mode in Figure 4D (decay rate = 0.992) indicates a gradual but steady decline in brucellosis prevalence over the 11-year period, suggesting that long-term transmission is being effectively controlled through current public health measures. In contrast, the short-period modes (Figures 4A–4C) highlight localized oscillatory behavior, particularly in Tongliao, Chifeng, and Hinggan League, which have experienced repeated outbreaks. In most other regions, however, disease activity shows a consistent decline. This spatial heterogeneity aligns with observed epidemiological patterns, where outbreaks often emerge and intensify in northeastern and central areas before potentially spreading southward and eastward.

Beyond characterizing transmission dynamics, the identified short-term (3–12 months) and long-term (≈11 years) oscillatory modes provide mechanistic insights into the underlying drivers of brucellosis transmission. The short-term cycles align closely with livestock breeding, trading, and slaughtering activities, which shape seasonal transmission risk. For instance, lambing and calving seasons in late winter and early spring increase human contact with infectious materials, leading to pronounced spring and early summer peaks. A secondary rise often occurs in autumn during periods of intensive livestock trade and slaughter, further amplifying transmission potential. In contrast, the long-term declining mode reflects the accumulated impact of sustained control measures, including vaccination, surveillance, and movement regulation.

The eastward shift of transmission hotspots toward Tongliao, Chifeng, and Hinggan League likely mirrors evolving structural features of the livestock industry, including eastward concentration of breeding activities, active interregional trade, favorable ecological conditions, and heterogeneous control capacity. Uneven vaccination coverage further contributes to the formation of high-risk growth points that can sustain or rekindle local transmission.

From a public health perspective, these spatiotemporal modes provide clear intervention windows and geographic priorities. Vaccination campaigns can be strategically scheduled to precede seasonal peaks—particularly during spring breeding and trading periods—while enhanced surveillance should focus on emerging eastern hubs during high-risk months. Strengthening biosecurity measures at trade nodes, improving local control capacity, and promoting regional joint prevention and control can further mitigate epidemic resurgence.

Model reconstruction and prediction

KHDMDsp for reconstructing and predicting brucellosis data

The top of Figure 5 depicts the heatmap of brucellosis in Inner Mongolia reconstructed and predicted using the KHDMDsp method. Here, reconstruction refers to the model’s approximation of the training data (January 2010 to December 2019), while prediction refers to its extrapolation to the testing period (January to December 2020).

Figure 5.

Data reconstruction and prediction heatmap and curves for selected cities based on the KHDMDsp framework

(A) Original case data for different cities.

(B) KHDMDsp-reconstructed and predicted case data.

(C) Scaled absolute error between observed and predicted cases.

(D–K) Numerical simulation of brucellosis data in typical cities of Inner Mongolia based on the KHDMDsp framework.

The heatmap reveals minimal reconstruction error for the training data, indicating that the KHDMDsp method effectively captures the dominant spatiotemporal modes underlying brucellosis transmission. This strong reconstruction ability allows for a more accurate characterization of transmission dynamics and enhances the interpretability of disease patterns. For the predicted data, the model successfully reproduces the key transmission dynamics but exhibits moderate quantitative deviations, suggesting that further refinement is required to enhance forecasting precision. To rigorously evaluate model performance, we employ the scaled absolute error (SAE) metric from the paper by Avila and Mezić.²⁷ As shown in the curve plots of Figure 5C, the spatiotemporal distribution of SAE remains low across most regions, demonstrating the model’s overall effectiveness. Nevertheless, Wuhai and Alxa League exhibit relatively higher SAE values due to their extremely low incidence levels (0.87 and 2.55 monthly cases on average, respectively), which amplify normalized deviations. Temporal analysis further reveals a gradual increase in SAE toward the later stages of the dataset, underscoring a key limitation of the KHDMDsp model in long-term forecasting. These findings are further illustrated by the representative time series trajectories shown in the curve plots in Figure 5.

To further evaluate the stability and robustness of the proposed model, an uncertainty quantification analysis was performed to examine the influence of kernel parameter variations on the model outcomes. As described in Material D of the supplemental information, the kernel parameters in our model were optimized within predefined parameter ranges using a Bayesian optimization framework based on a Gaussian process surrogate model. This surrogate efficiently explores the parameter space and identifies near-optimal configurations that balance model accuracy and generalization.

To quantify prediction uncertainty arising from kernel parameter selection, we performed multiple independent optimization runs, each initialized with different random seeds within the same parameter bounds. The corresponding sets of near-optimal parameters were used to reconstruct and forecast the time series, and the ensemble mean and variance of these results were employed to construct 95% confidence intervals. As illustrated in Figure 5, the trajectories exhibit confidence ranges of approximately [–84.04, 233.47]. These intervals suggest that, while the model outcomes show moderate sensitivity to kernel parameter changes, particularly in regions of pronounced seasonal variation, the overall dynamical patterns remain consistent. The confidence bands mainly influence the amplitude of local oscillations, whereas the reconstructed and predicted mean trajectories remain in close agreement with the observed data. Indicating that the essential temporal evolution of brucellosis incidence is robustly captured across multiple kernel configurations.

In this context, the relatively wide confidence bounds may be interpreted as reflecting the intrinsic variability of nonlinear epidemic dynamics rather than instability of the modeling framework. The consistency in dominant modes and the preservation of key spatiotemporal trends across different kernel realizations further demonstrate that the KHDMDsp framework provides a reliable and interpretable representation of the underlying dynamics.

Error comparison of multiple models

Figures 6 and 7 present the reconstruction and prediction results of brucellosis dynamics in Inner Mongolia using multiple modeling approaches. These include the LSTM network from RNN, the seasonal autoregressive integrated moving average (SARIMA) model from statistical time series methods, and two DMD approaches: standard DMD and optimized DMD (optDMD).

Figure 6.

Data reconstruction and prediction curves for selected cities based on LSTM network and SARIMA model

(A–H) Numerical simulation of brucellosis data in typical cities of Inner Mongolia based on the LSTM network.

(I–P) Numerical simulation of brucellosis data in typical cities of Inner Mongolia based on the SARIMA model.

Figure 7.

Data reconstruction and prediction curves for selected cities based on standard DMD model, optDMD model, and combined model

(A–H) Numerical simulation of brucellosis data in typical cities of Inner Mongolia based on the DMD model and optDMD model.

(I–P) Numerical simulation of brucellosis data in typical cities of Inner Mongolia based on the combined model.

To provide a quantitative comparison, we evaluated the performance of the proposed KHDMDsp model against four benchmark models—DMD, optDMD, LSTM, and SARIMA—using MAE and RMSE as evaluation metrics. The detailed error comparison across the five models is presented in Table 1. To ensure a fair and meaningful comparison, we standardized the configurations of the benchmark models so that all methods were evaluated under comparable conditions. This unified framework allows us to objectively assess the performance of KHDMDsp relative to the other four approaches.

Table 1.

Error comparison of multiple modeling methods

	Train MAE	Train RMSE	Test MAE	Test RMSE
DMD	3.064E+01	4.561E+01	6.388E+01	8.682E+01
optDMD	6.007E+00	1.213E+01	5.387E+01	8.419E+01
SARIMA	2.116E+01	3.103E+01	5.030E+01	6.599E+01
LSTM	1.527E+01	2.356E+01	2.340E+01	3.382E+01
KHDMDsp	3.534E+00	5.533E+00	5.355E+01	7.588E+01

For the benchmark models, optDMD was calibrated with 80 modes and a relative loss of Π_loss = 11.15%, and the standard DMD was evaluated using the same number of modes. Because the modes of optDMD and KHDMDsp capture different dynamical characteristics, relative loss was adopted as a common reference for comparison, allowing KHDMDsp errors to be assessed under comparable conditions.

The LSTM model was constructed using a 12-month sliding window as input and the 13th month as the prediction target, generating 1,440 input-output pairs across 12 cities. Ninety percent of the data were used for training and 10% for validation. The network consists of two LSTM layers with 68 and 64 units, respectively (the first layer optimized via Bayesian search), followed by dropout layers (15% and 10%) to reduce overfitting. A fully connected layer with ReLU activation ensures non-negative outputs, followed by a regression layer. The model was trained using the Adam optimizer with a Bayesian-optimized learning rate of 2.247 × 10⁻³, L₂ regularization (λ = 0.0005), and a learning rate decay of 30% every 50 epochs.

The SARIMA model for each city’s monthly brucellosis data was established using a dynamic parameter selection strategy. Its basic form is SARIMA (p,d,q) (P,D,Q)₁₂, where the non-seasonal orders (p,d,q) and seasonal orders (P,D,Q) are not fixed but automatically selected from candidate models based on the Bayesian information criterion (BIC). Specifically, p and q range from 0 to 2, P and Q from 0 to 1, and differencing orders d and D take values 0 or 1, with a fixed seasonal period of 12 months. The modeling process first assesses data stationarity to determine whether differencing is required, evaluates seasonal strength to guide seasonal parameter selection, and then identifies the model with the lowest BIC for each city. Fitted models generate reconstructed values and forecasts, and residuals are subjected to white noise and seasonal diagnostic tests to verify model adequacy.

Overall, the results in Table 1 clearly demonstrate the strong modeling capability of the proposed KHDMDsp method. Among all five approaches, KHDMDsp achieves the lowest training MAE and RMSE, indicating its superior ability to accurately reconstruct the intrinsic spatiotemporal dynamics of brucellosis transmission. Although its test errors are at a comparable level with SARIMA and optDMD, the consistently better reconstruction performance highlights its strength in extracting dominant dynamical modes with high fidelity. This suggests that KHDMDsp not only fits the historical epidemic trajectories more precisely but also provides a clearer interpretation of the underlying disease dynamics, offering valuable insights into transmission mechanisms and forming a solid basis for subsequent predictive modeling.

Combined model

The comparative error analysis clearly shows that the KHDMDsp model exhibits outstanding capability in reconstructing the intrinsic spatiotemporal dynamics of brucellosis transmission. However, due to the complex and highly nonlinear nature of the epidemic data—driven by multiple environmental, ecological, and socioeconomic factors—its long-term forecasting performance remains limited. In contrast, the LSTM model demonstrates superior generalization and predictive power by effectively capturing nonlinear temporal dependencies across multiple cities.

To leverage the complementary strengths of both methods, this study designed a hybrid KHDMDsp-LSTM framework, in which KHDMDsp identifies the dominant modes and accurately reconstructs the dynamics, while the LSTM provides precise long-term predictions. In the curve plots of the combined model in Figure 7, the portions before the dividing line correspond to the KHDMDsp reconstruction, whereas the portions after the line represent the LSTM predictions. To further demonstrate the framework’s robustness and generalizability, we applied it to the monthly reported influenza case counts from 12 countries and the monthly mean temperature records from the same cities in Inner Mongolia. As shown in Material C of the supplemental information, the model achieved excellent reconstruction and forecasting performance, confirming its applicability to a wide range of time series beyond brucellosis data.

Conclusion

This study proposes a novel hybrid modeling framework, KHDMDsp-LSTM, designed to analyze and predict the nonlinear, periodic, and regionally heterogeneous transmission dynamics of human brucellosis in Inner Mongolia. The core innovation lies in the development of KHDMDsp, a kernel-enhanced sparse DMD model that employs hybrid kernels within the Koopman operator framework to linearize nonlinear systems, enabling the extraction of multiscale temporal patterns and interpretable reconstruction of historical epidemic dynamics.

The model adaptively tunes kernel parameters via Bayesian optimization and incorporates sparsity-promoting regularization to suppress noise while preserving key epidemic dynamics. However, since brucellosis transmission is influenced by environmental, social, and ecological factors, the long-term forecasting capability of KHDMDsp alone remains limited. To address this, an LSTM network is introduced to capture complex nonlinear temporal dependencies across multiple cities, significantly enhancing long-term predictive performance. The resulting KHDMDsp-LSTM hybrid model combines the interpretability of KHDMDsp with the generalization and accuracy of LSTM.

When applied to 11 years of monthly brucellosis data from 12 cities in Inner Mongolia, the model successfully identified dominant periodic cycles and key hotspot regions, including Tongliao, Chifeng, and Hinggan League. The Koopman spectrum revealed slowly decaying dominant modes, suggesting a generally controllable epidemic trend. Furthermore, when applied to the monthly reported influenza case counts from 12 countries and the monthly mean temperature records from the same cities in Inner Mongolia, the hybrid framework also achieved excellent reconstruction and forecasting performance, demonstrating its broad applicability to diverse types of time series beyond brucellosis data.

In conclusion, KHDMDsp-LSTM integrates Koopman spectral analysis with deep learning, providing a data-driven approach that combines interpretability with predictive power. This framework offers valuable support for analyzing epidemic dynamics, as well as for disease monitoring and early warning.

Resource availability

Materials availability

No material was generated in the present study.

Data and code availability

Data and code are available from the corresponding author upon reasonable request.

Funding and acknowledgments

This work was supported by the National Natural Science Foundation of China (12471183 and 12071175) awarded to Y.L., by the National Natural Science Foundation of China (12571592 and 12022113) and the Fundamental Research Program of Shanxi Province (202303021223009) awarded to G.-Q.S. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Author contributions

Y.L. and G.-Q.S. conceived and designed the project. Y.-P.L. performed the study and wrote the paper. All authors contributed to the manuscript and approved the final version.

Declaration of interests

The authors declare no competing interests.

Published Online: January 29, 2026