Response to Reviewer XKTr

We sincerely thank you for your thorough review and positive assessment of our work. We are grateful for your recognition of EARTH's novelty and contributions to epidemic forecasting. We address your questions below:

Weakness 1: No detailed analysis of the computational complexity.

We appreciate this valuable suggestion. The computational complexity of EARTH can be broken down into:

• Epidemic-Aware Neural ODE (EANO): The computational complexity of our neural ODE solver is primarily determined by $O(T_{\text{ODE}} \times (N \times d^2 + |E| \times d))$, where $T_{\text{ODE}}$ represents the average number of solver steps needed for integration, $N$ is the number of regions, $d$ is the hidden dimension, and $|E|$ is the number of transmission edges in our graph. The term $N \times d^2$ comes from node-level feature transformations applied at each solver step, while $|E| \times d$ reflects the message passing operations between connected regions during epidemic propagation.

• Global-guided Local Transmission Graph (GLTG): This component has a theoretical complexity of $O(N^2 \times T_{\text{hist}}^2 + N^2 \times d)$. The first term arises from computing pairwise temporal similarities using Dynamic Time Warping across $N$ regions with historical sequences of length $T_{\text{hist}}$, while the second term corresponds to the generation of the full adjacency matrix with feature transformations. In practice, we reduce this cost using FastDTW for efficient similarity calculation and by maintaining a sparse graph structure that retains only the most relevant region connections.

• Cross-Attention Mechanism: The complexity for our cross-attention operation between epidemic states and global features is $O(N \times d^2)$, which is notably more efficient than typical attention mechanisms that scale with $O(N^2 \times d)$. This efficiency stems from our design choice to constrain attention to the three epidemic states (S,I,R) per region rather than attending across all regions.

We will include this detailed analysis in the revised version.

Weakness 2: Model sensitivity to hyperparameters.

Thank you for this important observation. We have conducted additional experiments to evaluate EARTH's sensitivity to key hyperparameters on the Australia-COVID dataset with a horizon of 5:

These results demonstrate that EARTH performs optimally with hidden dimensions of 64 and a learning rate of 1e-3, which aligns with our main experimental setup in Table 1. We will incorporate these detailed sensitivity analyses in the revised manuscript.

Question 1: How are these missing rates introduced into the data?

Thank you for this question about our experimental methodology. For the controlled experiments in Table 2, we artificially introduced missingness by randomly removing data points from the complete dataset at rates of 10%-40%. This random deletion strategy follows standard practice in the literature for evaluating model robustness to missing data.

Question 2: In Figure 4, are there any isolated nodes with almost no connections to other nodes? If there are, what does this mean?

This is an insightful question. What appears as isolated nodes in Figure 4 is simply a result of our visualization approach - we only display the top-3 highest weighted edges for each region to maintain visual clarity. In our Local Transmission Graph, we first use Dynamic Time Warping to select top-k similar nodes based on temporal epidemic patterns, then adaptively learn the normalized weighted edges during training using the mechanisms described in Equations 9-10, allowing EARTH to automatically determine appropriate information sharing between regions based on epidemic similarity. We will clarify this visualization choice in the revised manuscript.

Response to Reviewer rfdg

We sincerely thank you for your thorough review and hope our responses below will help improve your assessment of our work:

Weakness S1 (Theoretical Claims & Claims And Evidence): Presentation and Design choices of the epidemic compartment

Our approach addresses these concerns in several ways:

1. Multi-dimensional features vs. scalar rates: Unlike traditional SIR models using scalar rates (Sha et al.; Gao et al.), our "detailed representations" claim refers to using multi-dimensional features. This enables more nuanced transmission dynamics through: 1) Higher expressivity for complex spatio-temporal patterns, 2) Greater flexibility for heterogeneous transmission across regions, and 3) Enhanced representation of time-varying dynamics. Our experiments with an EARTH variant using traditional single-value scalar rates showed:

   Model Variant	h=5 (R)	h=5 (P)	h=10 (R)	h=10 (P)
   w/o Feature	178.6	34.24	198.5	44.18
   EARTH	156.8	30.12	177.6	38.62

2. Safeguards against overfitting: We implement three strategies: 1) Temporal cross-validation with training/validation/testing spanning different epidemic waves, 2) Multi-scale integration balancing local mechanics with global patterns via cross-attention, and 3) Time-continuous formulation smoothing noise in discrete observations. These safeguards are validated in Tables 1&2, where EARTH maintains stable performance even with 40% missing data.
3. Theoretical considerations: Our work builds on classical compartmental models (Grassly & Fraser, 2008) but addresses limitations through: 1) Learnable features capturing spatio-temporal heterogeneity vs. fixed scalar rates, 2) Neural ODE formulation adapting to evolving transmission patterns, and 3) Continuous-time approach handling irregular or missing data. These innovations balance epidemiological principles with data-driven flexibility.

Weakness S2 (Claims And Evidence & Essential References): Positioning against relative works

We acknowledge the need for clearer positioning relative to Kosma et al. (2023) and similar works. While these combine ODEs/ML with epidemic models, our key contribution lies in: 1) Integrating neural ODEs with GNNs through our GLTG mechanism for continuous evolution of node features and edge weights, 2) Adaptively learning connections through DTW and integrating global/local information via cross-attention, and 3) Demonstrating superior performance with missing data, overcoming limitations of fixed-parameter approaches.

Weakness S3 (Experimental Designs): Experimental Choices not well-justified/missing

We appreciate these important observations about experimental design clarity:

1. Horizon lengths: We selected forecast horizons based on: public health decision-making needs, alignment with previous works, and balancing prediction accuracy with utility. Additional experiments on US-Region with different historical windows and larger horizons:

   Method	Win	h=20	h=25	h=30
   STGODE	20	1836	2153	2607
   EpiColaGNN	20	1645	1947	2378
   EARTH	20	1528	1812	2219
   STGODE	40	1682	1976	2391
   EpiColaGNN	40	1476	1763	2134
   EARTH	40	1374	1642	1983

2. Solver type: We use Runge-Kutta 4th order (rk4) with absolute tolerance 1e-9 and relative tolerance 1e-7 for balance between numerical stability and efficiency.

3. Standard deviations: Results are averaged over 5 runs with different random seeds. We will add variance information in the revised manuscript.

4. Downsampling: Random sampling was used to create missing data points, simulating real-world reporting patterns.

Weakness S4: Computational Analysis

The computational efficiency of EARTH can be analyzed as:

• Epidemic-Aware Neural ODE: Complexity $O(T_{\text{ODE}} \times (N \times d^2 + |E| \times d))$, where $T_{\text{ODE}}$ is solver steps, $N$ is regions, $d$ is hidden dimension, and $|E|$ represents edges. $N \times d^2$ corresponds to node-level transformations while $|E| \times d$ captures cross-region exchange.

• Global-guided Local Transmission Graph: Traditional construction would incur $O(N^2 \times T_{\text{hist}}^2 + N^2 \times d)$ complexity, with first term from DTW computation and second from adjacency matrix generation. We implement key optimizations: (1) FastDTW reducing temporal similarity calculation from $O(T_{\text{hist}}^2)$ to $O(T_{\text{hist}})$, (2) one-time DTW matrix computation and reusage during training, (3) retaining only top-k connections per region leveraging epidemic transmission sparsity where $|E| \ll N^2$, and (4) compressed sparse matrices with memory scaling as $O(|E|)$.

• Cross-Attention Mechanism: Our design constrains attention to three epidemic states per region, achieving $O(N \times d^2)$.

These optimizations enable EARTH to avoid $O(N^2)$ complexity, supporting large-scale epidemic forecasting applications.

Response to Reviewer uXQM

We sincerely thank you for your positive assessment of our work and for recognizing the innovation in integrating neural ODEs with epidemiological models. We appreciate your thorough evaluation and answer your questions below:

Cons: On applying neural ODEs to epidemiology

While neural ODEs have been applied in various domains, EARTH goes beyond straightforward application:

• Integration with epidemic mechanism: Our Network SIR-inspired architecture (Equations 5, 11) explicitly models the transition dynamics between susceptible, infectious, and recovered populations.

• Integration of GNN with neural ODEs: Our approach uniquely combines graph neural networks with neural ODEs through the GLTG mechanism. This integration allows the neural ODE to operate on evolving graph structures where both node features and edge weights change continuously in time. The dynamic transmission patterns captured by our graph-based ODE enable more realistic modeling of how disease spreads across regions compared to standard neural ODEs that operate on fixed graphs or no graph structure at all.

• Flexibility for irregular data: Unlike conventional time series models, our continuous formulation naturally handles the irregular reporting and missing data common in epidemic monitoring.

Our ablation studies validate these design choices, with significant performance drops when these specific components are removed.

Question: Dynamic integration of global and regional patterns

EARTH integrates global and regional patterns through:

• Semantic connections via DTW: We identify regions with similar epidemic trajectories using Dynamic Time Warping rather than relying solely on geographic proximity.

• Adaptive transmission learning: Global trends guide regional transmission via a dynamic graph structure that evolves throughout the epidemic timeline, capturing how policies and behaviors change transmission patterns.

• Multi-scale modeling: We capture both local disease dynamics and cross-regional dependencies within a unified framework.

• Continuous-time formulation: By modeling in continuous time, EARTH handles irregular observation intervals and can forecast at arbitrary time points.

This approach enables effective forecasting even when regional reporting patterns change, a common challenge in epidemic monitoring.

Response to Reviewer Q6Uq

We thank you for your positive assessment and for recognizing the importance of our work in epidemic modeling. We address your concerns below:

Question & Weakness: Only evaluated for COVID-19, with potential limitations for heterogeneous datasets and different diseases.

We address this concern from several perspectives:

• Dataset selection: We focused on COVID-19 and influenza datasets due to their availability, spatiotemporal coverage, and real-world significance. This enables fair comparison with prior works [1,2], evaluating EARTH's effectiveness. [1]: Epidemiology-aware Deep Learning for Infectious Disease Dynamics Prediction. CIKM 2023. [2]: PEMS: Pre-trained Epidemic Time-series Models. ArXiv 2023.

• Disease-agnostic design: EARTH's architecture is generalizable across epidemic types through: (a) SIR-inspired neural ODE framework capturing universal transmission-recovery dynamics. (b) Dynamic graph structure adapting to varying contact patterns. (c) Continuous-time formulation accommodating diverse incubation periods and transmission characteristics.

• Orthogonal Contributions: We recognize potential improvements like heterogeneous population flow modeling through Heterogeneous GNN. These represent incremental improvement rather than fundamental limitations, as our contribution lies in the integration of GNN with neural ODEs.

• Cross-disease validation: We tested EARTH on dengue fever using OpenDengue dataset, which exhibits dramatic fluctuations (e.g., 160,265 cases in 2010 declining to 25,503 in 2017 from COLOMBIA). Results validate robustness:

Concern 1 (Methods And Evaluation Criteria & Theoretical Claims): The cases with external interventions.

On EARTH's robustness to transmission dynamics changes:

• Continuous-time modeling: Unlike discrete models assuming fixed patterns, EARTH's ODE formulation adapts to changing dynamics by modeling underlying disease propagation mechanisms rather than statistical patterns.

• Capturing pandemic phases: EARTH models distinct epidemic phases through: (a) Time-dependent parameterization allowing varying transition rates across stages, (b) Context-aware graph structure evolving differently during various phases (lockdown vs. reopening), (c) Global features capturing regime changes across regions.

• External factors integration: EARTH's design allows exogenous incorporation, for example, (a) Mobility data as edge weights reflecting contact patterns, (b) Government interventions as additional node features, (c) Vaccination rates through susceptible population parameters modulation.

While robust to moderate shifts, unprecedented disruptions remain challenging for any data-driven approaches. We can explore methods like change-point detection or causal intervention modeling.

Concern 2 (Claims And Evidence:): Limited explanation of neural ODEs advantages.

Neural ODEs offer key advantages for epidemic modeling: naturally incorporate epidemic principles within ODE framework, fusing mechanistic understanding with data-driven flexibility; operate in continuous time, handling irregular reporting without interpolation errors; ensure physical consistency while modeling complex patterns beyond traditional compartmental models. Ablation studies show 12.4% RMSE degradation when replacing neural ODE with standard GNN.

Concern 3 (Claims And Evidence): Lack of analysis for "detailed representations" claim

EARTH uses multi-dimensional feature vectors for finer granularity disease dynamics versus conventional single-value methods. Figure 4 shows this capability, revealing semantic relationships between regions with similar epidemic trajectories.

Concern 4 (Experimental Designs Or Analyses): Potential overfitting.

We use temporal cross-validation with training/validation/testing spanning different epidemic phases, ensuring generalization. Our model incorporates dropout in GNN layers and L2 regularization. Table 2 shows stability even with 40% missing data, demonstrating robustness.

Concern 5 (Experimental Designs Or Analyses): There may be different metrics and further sensitivity analyses.

We conducted additional experiments with CORR (Pearson's Correlation Coefficient) and sensitivity analysis. EARTH performs well across metrics and remains stable with different parameters:

Suggestion 1: Essential References Not Discussed

We will incorporate suggested references in our revised manuscript.