EARTH: Epidemiology-Aware Neural ODE with Continuous Disease Transmission Graph
Authors: Guancheng Wan
Zewen Liu
Xiaojun Shan
Max SY Lau
B. Aditya Prakash
Wei Jin
Venue: ICML 2025 poster (2025)
Keywords: Graph Learning
Epidemiology
Public Health
Primary area: applications->health_medicine
TL;DR: The paper presents EARTH, a framework combining neural ODEs with epidemic mechanisms to improve forecasting by capturing continuous disease transmission dynamics and integrating global trends guided graphs.

Abstract: Effective epidemic forecasting is critical for public health strategies and efficient medical resource allocation, especially in the face of rapidly spreading infectious diseases. However, existing deep-learning methods often overlook the dynamic nature of epidemics and fail to account for the specific mechanisms of disease transmission. In response to these challenges, we introduce an innovative end-to-end framework called Epidemiology-Aware Neural ODE with Continuous Disease Transmission Graph (EARTH) in this paper. To learn continuous and regional disease transmission patterns, we first propose EANO, which seamlessly integrates the neural ODE approach with the epidemic mechanism, considering the complex spatial spread process during epidemic evolution. Additionally, we introduce GLTG to model global infection trends and leverage these signals to guide local transmission dynamically. To accommodate both the global coherence of epidemic trends and the local nuances of epidemic transmission patterns, we build a cross-attention approach to fuse the most meaningful information for forecasting. Through the smooth synergy of both components, EARTH offers a more robust and flexible approach to understanding and predicting the spread of infectious diseases. Extensive experiments show EARTH superior performance in forecasting real-world epidemics compared to state-of-the-art methods. The code is available at https://github.com/GuanchengWan/EARTH.

EARTH: Epidemiology-Aware Neural ODE with Continuous Disease Transmission Graph
Guancheng Wan 12
Zewen Liu
Xiaojun Shan
Max S. Y. Lau
B. Aditya Prakash 5 Wei Jin
Abstract
Region
Irequiar Intervals
Regional Interactions
Effective epidemic forecasting is critical for pub- lie health strategies and efficient medical re- source allocation, especially in the face of rapidly spreading infectious diseases. However, exist- ing deep-learning methods often overlook the dy- namic nature of epidemics and fail to account for the specific mechanisms of disease transmis- sion. In response to these challenges, we intro- duce an innovative end-to-end framework called Epidemiology-Avee Neural ODE with Continu- Disease Transmission Graph (EARTH) in this paper. To learn continuous and regional disease transmission patterns, we first propose EANO, which seamlessly integrates the neural ODE ap- proach with the epidemic mechanism, consider- ing the complex spatial spread process during epidemic evolution. We also introduce GLTG to model global infection trends and leverage these signals guide local transmission dynam- ically. To accommodate both the global coher- ence of epidemic trends and the local nuances of epidemic transmission patterns, we build a cross- attseatttnnion approach to fuse the most mean- ingful information for forecasting, EARTH offers a more robust and flexible approach to understand- ing and predicting the spread of infectious dis- eases. Extensive experiments show EARTH su- perior performance in forecasting real-world epi- demics compared to state-of-the-art methods. The code is available at https:/ / //gtttu...../ggg/ com/ GuanchengMan/EAae EEARTH.
Region
Continuous-tine
Epidemic System
1) II)
III)
Learn
Learn Trend and Dynamic Regional Pattern
Discase
WIOD
Transmission Pattern
Excellent and Robust Epidemic Forecasting
Figure I. Problem illustration. Considering both evolution of re- gional correlation signals and irregular sampling observation intervals facts, we focus on the continuous-time epidemics. But existing solutions to D) learn disease transmission patterns with epidemic mechanism and IT) address missing states. Additionally, they omit to III) learn global trends caused by external factors (e-g., lockdowns) while developing dynamic regional transmission.
1. Introduction
The COVID-19 19 pandemic has resulted in millions of deaths and significant economic losses worldwide, severely disrupt- ing social and economic systems (Martin al., 2023; Pak et al., 2020). To address these challenges, there is grow- ing interest in epidemiological models, which are crucial for effective public health strategies and efficient medical resource allocation (Liu et al., 2024; Cm, 2020). Traditional models, such as the SIR model and its variants (Dehning et al., 2020), rely on mathematical differential equations to simulate disease spread but often depend on oversimplified assumptions (Funk et al., 2018; Kondratyev, 2013; Yang et al., 2023). To enhance performance, deep learning mod- els like Neural Networks (Madden et al., 2024; Rodríguez et al., 2023) or Graph Neural Networks (GNNs) (Zhang et al., 2024e;b) have been explored. These models effec- tively represent interactions between entities (e.g., regions) as graphs, capturing the spatial spread of disease through
Department of Computer Science, Emory University, USA 'Dearrmeent Computer Science, University California, Los Angeles 'Dpaarmeet of Electrical and Computer Engineering, University of California, San Diego 'Department of Biostatistics and Bioinformatics, Emory University, USA College of Comput- ing, Georgia Institute of Technology, USA. Correspondence to: Wei Jin <wei.jin cecijiinmmmry.cuu.. emory.edu>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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EARTH: Epidemiology-Aware Neural ODE with Continuous Discase Transmission Graph
message-passing mechanisms.
the global infection trend and learn dynamic regional transmission patterns during continuous evolution?
Nevertheless, given the dynamic nature of epidemic sys-
tems, existing work often neglects the dynamic evolution of regional interactions. For example, changes in people's 's be- havior (e.g., lockdowns) at a specific time step of one region, will greatly reduce the spread to surrounding regions in the following period. Existing efforts generally predict future epidemic profiles by modeling regional interactions from the whole series while overlooking these dynamic changes during the evolution. Furthermore, irregular sampling ob- servation intervals are not considered. For instance, some regions may be unable to conduct routine reporting in the early stages of an epidemic due to limited resources. Cur- rent work simplifies this scenario by learning only regular intervals which is impractical in the real world. The problem illustration is detailed in Figure 1.
To address the identified questions, we propose an inno- vative and end-to-end framework for continuous-time epi- demic modeling: Epidemiology-Awwe AaRe ODE with Continu- ous Disease Transmission GrapHl (EARTH). To address ques- tion I) and facilitate we revisit the classic compartmental models (i.e., SIR). In order to surpass previous efforts (Rodríguez et al., 2023) and fully leverage the expressive ability of neural networks, we propose a neural ODE- based Network SIR (Brede, 2012) to implicitly capture the continuous evolution of the re- gional propagation graph. To overcome the challenge II) we propose to initialize disease state features and feed them into a proposed Epidemic-A
weweemioooogyiifoommon transparency,
Neural ODE (EANO) module to learn inherent epidemic transmission pattem.. Moreover, we attempt to achieve the target III. We first obtain a long-range view of epidemic progression and estab- lish a relationship with regions that share similar develop- ment patterns. To further consider dynamic regional trans- mission, we develop an innovative Global-guided Local Transmission Graph (GLTG). Specifically, we fuse global trend indicative features for different regions with GNN. Then they utilized to generate more fine-graie fine-grained locally dynamic transmission graphs, which guide our EANO dis- ease spreading during the evolution. Finally, we develop a cross- mechanism to accommodate both the global coherence of epidemic trends and the local nuances of dis- ease transmission patterns. We conjecture that these two components together make EARTH competitive method for epidemic forecasting.
To tackle the aforementioned issues, neural ordinary differ- ential equation (NODE) (Chen et al., 2018; Poli et al., 2021; Liu et al., 2025) stands out as a powerful approach to model- ing the continuous-tine system. Therefore, in this work, we take inspiration from NODE and focus on the continuous- time epidemic system, capturing the intricate dynamics more accurately. However, directly incorporating the neural ODE with the epidemic system faces nontrivial challenges. Firstly, it does not explicitly learn epidemic mechanisms and fails to provide insights to decision-makers. This motivates us to think: 1) How can we generally combine the neural ODE approaches with epidemic mechanism? ? Some work (Arik et al., 2020; Mežnar et al., 2021) proposes hybrid models trying to combine the epidemic mechanism and deep learn- ing methods. However, due to the limited observations of all disease states (e.g., lacking data on susceptible individ- uals) in the real world, these models are not flexible and are unable to learn inherent epidemic mechanisms. In the meanwhile, some existing neural ODE approaches from other fields (Luo et al., 2023) fail in addressing this problem either. Thus, the following question naturally emerges: II) How can we learn continuous disease transmission under limited observations more flexibly? Apart from the afore- mentioned locally subtle spreading patterns, epidemics also exhibit a global infection trend. From a more macroscopic perspective, the infection trend can be seen as a longer-range and often multi-regional overall direction. This global sig- nal impacts and changes the disease propagation, resulting in different spatial transmission patterns at different times. For instance, global political vaccination trends significantly alter local spatial transmission patterns. In regions with high vaccination rates, the spread of the epidemic slows down, and a "herd immunity" effect may even occur (Chauhan et al., 2023). However, previous work usually considers static geographic graphs or only learns the graph without ac- counting for the continuous evolution of global signals. This raises another intriguing question: III) How can we model
Our principal contributions are summarized as follows:
We are the first harmonize the neural ODE with the epi- demic mechanism, developing innovative framework considering the time-continuous nature of epidemic dy- namics while learning inherent disease spreading patterns. We further consider global epidemic trends and learn dy- namic regional transmission patterns during continuous evolution within the end-to-end model.
3
By integrating global coherence and local dynamics via a cross-attention mechanism, we achieve superior results on multiple epidemic forecasting datasets including COVID- 19 and influenza-like illness.
2. Related Work
2.1. Epidemic Forecasting
Epidemic forecasting plays a crucial role in predicting the spread and impact of infectious diseases, enabling timely and effective public health interventions (Emanuel et al., 2020; Fine, 2015; Terris, 1993). Traditional models like the SIR (Susceptible-Infeee Infectious Infectious-Recovereed model (Hethcote, 2000) use differential equations to describe disease dynam-
2

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EARTH: Epidemiology-Aware Neural ODE with Continuous Discase Transmission Graph
ies but often rely on oversimplified assumptions (Dehning et al., 2020; Caals et al., 2017). Recent advances incorpo- rate deep learning methods like Graph Neural Networks (GNNs) (Dai et al., 2022; Wan et al.. 2024b), which better capture the complex interactions and spatial dependencies in disease spread from data-driven perspectives (Deng et al., 2020; Yu et al., 2023). However, these deep learning meth- ods neglect the dynamic nature of the epidemic system. The issues of regional correlation signals and irregular sampling observation intervals remain unresolved, hindering the ac- curate capture of real-world epidemics. Therefore, in this work, we propose a general framework by seamlessly inte- grating epidemic mechanisms into Neural ODE, capturing the complex evolution of continuous-time epidemics.
disease transmission graph. Since the epidemic system is time-varying, we first attempt to associate each region with the time-corresponding latent variable Zu (t) by a parameter- ized ODE (t) == (t)(t)/dt = 4 (zi t; Z, which
depicts the region-specific dynamic trajectory for series. Thus we can derive temporal dynamics at T for all regions (TT) E RN as follows:
Z(T) Z(0) + 1.¹ 4/x (0.:)
(I)
Here the denotes the time manipulation function parame- terized by 0z. In our framework, we leverage multi-layer perceptrons (MLP) for the time modeling module ,), by default. Furthermore, for different regions, diverse and complex processes of infectious disease transmission ex- ist. Inspired by Neural Controlled Differential Equations (NCDE) (Kidger et al., 2020), we exploit a continuous path Qu for each region v, which reformulates Equation (1) as:
2.2. Graph Neural Networks
Graph Neural Networks (GNNs) (Hamilton et al., 2017; Veličković et al., 2017; Huang et al., 2023; Wan et al., 2024a; Zhang et al., 2024a;d; Wan al.. 2025b) are widely recog- nized for processing non-Euclidean data structures, such as traffic networks (Wu et al., 2019). They update node repre- sentations by aggregating information from neighbors via message-passing (Zhang et al., 2024c; Zhang et al.; Chen et al., 2024a; 2023). Many studies have used GNNs for epidemic modeling (Sha et al., 2021; Wang et al., 2023), focusing on the spatial relationships in disease spread (Jhun, 2021; La Gatta et al., 2021), but often overlook dynamic transmission patterns. Our approach addresses this by con- centrating on continuous-time epidemic modeling, using GNNs to integrate multi-region global trends and create disease transmission graphs for regional propagation.
Z(T) Z(0)+ + J.' vi (P:: Z(t)
dQ(1) dt
dt.
(2)
Equation (2) transforms the integral problem from Rie- mann integral to a Riemann-Stielljes integral. Q(L) is cre- ated from {t,, x by an interpolation algorithm.
3. Preliminaries
We approach the epidemic forecasting challenge by em- ploying graph-based prediction model. Let G = E) represent the graph, where V denotes a set of nodes com- prising |V| = N regions (e.g., cities or states). The edge set & C V x V represents the geographic links between these regions. The adjacency matrix A E R=** is defined such that Aij = 1 if there is an edge G. E and A.j = O otherwise. The normalized adjacency matrix is given by D-1DD-1/20
2.3. Neural Ordinary Differential Equation
Neural ODEs extend discrete neural networks to continuous- time scenarios, offering superior performance and flexibil- ity (Chen et al., 2018). They have been widely adopted in various fields such as traffic flow forecasting (Fang et al., 2021; Choi et al., 2021), continuous dynamical sys- tems (Chen et al., 2024b; Huang et al.), and recommen- dations (Qin et al., 2024). Recent advancements have in- tegrated GNNs with Neural ODEs, enhancing the model- ing of complex dependencies in graph-structured data (Luo et al., 2023; Wan et al., 2025a). Some works have started to apply Neural ODEs to model classic epidemic spread- ing processes like SIR (Kosma et al., 2023). For more comprehensive overview, readers are referred to the sur- vey in (Liu et al., 2025). However, these approaches often rely on pre-defined epidemic models and do not fully cap- ture the dynamic nature of regional interactions or global trends. In contrast to prior work, we extend this concept to investigate important continuous-time epidemic modeling by developing a framework that harmonizes the neural ODE with epidemic mechanisms while dynamically learning the
Å D 1/2 1//2AD1122 1/2, where the degree matrix D is diago- nal matrix with D Σ, Aij.
Each node corresponds to a region with an associated time series input over a window T, such as infection counts for T weeks. We represent the train- ing data over this period as X = [xi,..... xT] E R The goal to construct a model capable of predicting an epidemiological profile Xy+ at future time point T + h, where h denotes the prediction horizon.
Problem Formulation.
4. Methodology
4.1. Overview
In Epidemic-Aware Neural ODE, we initialize disease states as region-specifie features and build them upon time vari- ables, which are then fed into proposed Network SIR- inspired neural ODE functions to capture local subtle dis- ease transmission patterns. Furthermore, in Global-guided
3

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EARTH: Epidemiology-Aware Neural ODE with Continuous Discase Transmission Graph
Local Transmission Graph we leverage the GNN to ob- tain global trends and evolutionary graphs. Then dynamic graphs are learned during the evolution of epidemies to guide EANO propagation. Ultimately, we proposed cross- attention mechanism to accommodate both local nuances and global coherence for final forecasting. The illustration of the overall framework is detailed in Figure 2.
interpolation algorithm. They are resilient against irregular cases (e.g., unpredictable outbreaks) when implemented in real-life epidemics, providing more responsive model for predicting disease spread. Through the NCDE, we can model these disease states in continuous-time epidemics.
Nevertheless, it directly learns these states independently while considering the high-level spatial spreading pattern within discases. Therefore, we move beyond and designate the ODE function for each state. With well-crafted proce- dures dx. Pi. and Φ, functions, we can learn not only intra- state development but also inter-state interactions within regions. Taking SIR process Equation (3) into considera- tion, Φ should be associated as input S(t) and [(t), given that Φs (0.:: (()))))) Similarly, we then have other two functions o, S(t),, A(t) and Φr (0.,; I(t) Inspired by Network SIR applications (Balcan et al., 2009; Sha et al., 2021), we further incorporate regional correlations in dis- ease transmission into the state's 's updating process. Specifi- cally, these functions are rewritten by:
4.2. Epidemic-Aware Neural ODE
Motivation. Existing deep learning methods neglect the continuous evolution of epidemic systems and do not explic- itly learn about epidemic development. Traditional mech- anistic models attempt understand spreading patterns through ODEs but fail to utilize available data sources and model more complex epidemics fully. Therefore, we aim to combine the advantages of both approaches to enable an Epidemic-Aware Neural ODE framework.
SIR-inspired Neural ODE. In epidemiology. the standard SIR model (Hethcote, 2000) categorizes the population into three distinct groups based on their disease states: suscepti- ble (S) to infection population, currently infectious popula- tion (I), and recovered population (R), with the latter group being immune to both contraction and transmission of the disease. The SIR model, formulated using ODEs (Grassly & Fraser, 2008), describes the epidemic dynamics as follows:
O. dS.(t) W nans Ss.( Σ dt
Wiim
BeN.
(t) Φ dt
W uas (t)ll Σ evutu(t)
W (o)))
N
dR. dt
dS S.l. P
W Neoo I.))
= -8
dt dt S.J. The, dt P
(5)
(3)
dR
here N₂ denotes the neighboring nodes for node in the set V. with Cvu representing the weight of disease transmission intensity between regions v and u. The symbol || signifies the concatenation operation. By substituting the traditional SIR's two simple rates from Equation (3) with the more flexible parameters W Wutans E Rdddd and W Wieco E Rdxdd our model can derive more detailed representations of the disease spread and recovery processes. This adaptability enables the model to adjust to diverse epidemic conditions, reflecting the intricate mechanisms of disease transmission.
3
74.
dt
These equations distribute the total population P across the aforementioned categories. Here, susceptible individ- uals become infectious upon contact with infectious ones, driven by the transmission rate 3 (S- + I). In the mean- while, infectious individuals recover and gain immunity at the recovery rate 7 (1 + R). However, due to resource restrictions and observation limitations, these explicit cases may not always be available in real-world scenarios. To address this issue, we propose utilize neural ODEs to automatically infer these ODE functions in Equation (3) via neural networks in data-driven manner. Specifically, we treat these disease states (S, T, and R) as latent high- dimensional variables. Each state is represented by a matrix ((), and R(!) in R-Nxd xd with d denoting the hidden dimension. In continuous epidemic system, these states intrinsically linked with the time variables. Thus, we utilize the NCDE approach in Equation (2) and model the specific epidemic state C as:
Additionally, incorporating regional correlations through Cou allows the model to account for spatial dependencies, thereby improving its ability to mirror real-world interac- tions. However, the approach still depends on static neigh- borhood relationships and fail capture the dynamic nature of disease transmission as epidemics evolve.
4.3. Global-guided Local Transmission Graph
Motivation. As previously noted, EANO solely accounts for the pre-defined neighborhood connections, neglecting the evolutionary disease interactions. This observation drives us to explore more effective approaches to represent local spatial transmission patterns during the evolution.
C(T)= C(0) + S se (o:o:: ((t)
dz(1) dt. dt dQ(t), dt.
(4)
= =C(0) + l" Φe (0:: (()C)))))))))) we (0:: Z(t)
(0::
di
Here Q(t) are controlling paths for regions given by the
Global Infection Trend, In addition to local transmission
4

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EARTH: Epidemiology-Awars Neural ODE with Continuous Discase Transmission Graph
Φ:
Φr
M
Modelling
Disease Propagatalon
Global-guided Graph
K
Cross
Dynamic Transmission
Attn
D
S
Q H(T)
P
F(T)
Global
Connections
Global Trend
(Z(T)
F(T)
Final Forecast
Initial Stage
Continuous Evolution
Z(T)
Figure 2.
Architecture illustration of Epidemiology-Awarr Neural ODE with Continuous Disease Transmission Graph. EARTH a general and end-to-end framework that can flexibly capture the time-continuous epidemic mechanism. Best viewed color.
dynamics, epidemic systems also exhibit a global signal that represents the overall infection trend. This trend encapsu- lates the broader patterns of infection spread, influenced by factors such as international travel, global health policies, and widespread behavioral changes. For example, during a pandemic, international travel restrictions and lockdowns can significantly curtail cross-border disease transmission, resulting in differing regional infection rates and modify- ing the epidemic's overall trajectory (Demey et al., 2020; Russell et al., 2021). To address this, we propose creating global infection indicator feature for each region, correlated with the corresponding temporal features:
and propagation of information across broader regional scope, resulting in the fused global trend features (t).
Dynamic Regional Transmission. Having established the global infection trends, we leverage this signal to guide local spatial transmission patterns by impacting the regional con- texts in which local transmission occurs. For instance, high global vaccination rates can decrease the pool of susceptible individuals across regions, reducing local transmission op- portunities. We employ module to learn these dynamically evolving patterns based H()::
M, (t) = tanh (H())] + +b)).
H(T) = H(O)) + J."' Fo (0;: she (c: z(t)
dQ(()
(t tanh (H()),, + b2).
(9)
dt. dt
A(t)= =00==((an (
(+)))
(6) Since global here refers to the multi-regional overall di- rection, we employ Dynamic Time Warping (Please see details in Appendix C) to assess the similarity of historical case trends across different regions, thereby extending the original geographic connections:
The concurrent local transmission relationship Á(t) is guided by the global trend, while Equation (9) ensures that the learned pattern does not form completely bidirectional graph. This highlights that inter-regional dissemination is not perfectly symmetrical, capturing the asymmetric nature of spatial interactions in epidemic spread. Additionally, we utilize masking technique to balance the weights of static and dynamic transmission patterns:
1, if Aue 1,
A
1. u A and u E Topp
0. otherwise.
In this context, T Topk (v) denotes the indices of the & most similar nodes to node v. By doing this, establish epidemic-semantic spatial relationship that emphasizes gions with analogous epidemic progression patterns. To pro- mote interactions between regions, we implement residual GNN layer to update the global infection trend:
M((1) = a W (WaA()) + byJ).
(10)
E(I M(()) A (0-M()) A(L).
In Equation (10), Mi(I) continuous mask matrix, and J IN represents the all-ones matrix. The matrix E(() integrates the mask matrix Mi(t), the original adjacency matrix A, and the globally guided pattern A(t) through element-wise Hadamard products, aiming to capture prop- agation patterns between regions in this dynamic system. Once the fused spatial transmission pattern E(t) is obtained,
P9 - D 1/2 AD W))) + H(t)
(8)
Here, D represents the degree matrix corresponding to A, W, is the learnable weight matrix, and denotes the activa- tion function. The GNN layer P2 facilitates the aggregation
5

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EARTH: Epidemiology-Awark Neural ODE with Continuous Discase Transmission Graph
Australia-COUs
-Sggon
US-Staten
Methods
h
=
10 h
h = 15
h
=
5
h
=
10
h
15
h = 5
h = 10
h 15
R
P
P
R. P
R
P
R
P
R.
P
R
P
R.
PP
R.
P
VAR
665.3
83.47
575.2
92.41
502.9
95.39
1151
572.3
1396
701.6
1418
688.3
339.2
90.38
371.3
103.2
402.4
150.7
SARIMA
233.7
38.45
428.4
95.20
445.1
93.60
1142
525.8
1428
715.3
1460
675.4
335.6
92.80
368.2
106.4
406.5
149.2
SIR
235.0
42.50
445.2
104.5
425.6
102.8
1235
575.0
1480
745.0
1565
760.0
338.0
121.5
388.0
112.0
428.0
168.0
LSTM
228.0
39.78
433.6
108.4
432.6
98.09
1173
538.6
1475
736.5
1509
757.6
331.8
93.45
370.0
109.8
411.3
154.9
DCRNN
166.5 514.8
853.7
286.4
1186
404.6
1488
760.9
1443
732.7
1412
710.3
329.4
93.15
334.7
96.90
372.8
142.6
STGCN
833.7
232.6
787.8
227.7
802,1
248.3
1335
678.1
1522
819.2
1638
925.4
304.7
89.32
293.7
85.33
312.5
116.3
ASTGCN
821.5
221.9
765.9
201.3
804.1
254.8
1252
545.2
1478
801.1
1576
821.3
310.2
93.44
290.5
80.99
344.6
123.4
STGODE
310.5
66.32
392.2
91.05
571.3
159.2
1304
668.2
1403
732.1
1577
804.3
345,2
107.8
402.4
120.4
477.3
199.4
STG-NCDE
287.2
49.21
77.92 341.3
479.2
111.2
1284
643.1
1399
691.2
1421
732.1
319.2
94.39
377.6
101.5
421.3
176.7
CNNRNN-Res
1802
624.7
612.6
151.4
622.1
153.1
1190
588.3
1332
642.8
1374
652.1
303.3
86.78
292.1
79.33
333.6
105.4
EpiGNN
210.3
40.12
467.3
120.1
764.2
233.7
1136
534.2
1454
728.9
1444
764.2
288.5
84.32
297.6
84.32
391.6
157.4
CAMul
231.4
44.32
398.2
76.62
634.1
164.7
1145
557.3
1434
703.2
1402
699.2
294.6
88.16
312.8
86.71
325.2
107.5
EINN
206.2
38.19
312.4
64.21
456.9
98.72
1178
571.6
1432
729.1
1489
792.3
321.2
97.91
342.1
100.1
402.7
162.9
ColaGNNN
224.2
55.23
544.8 8
161.6
795,8
258.0
1148
533.6
1524
846.6
1552
856.3
299.1
81.53
283.4
79.12
339.4
120.6
EpiColaGiN
204.3
36.86
345.4
68.39
886.0
296.5
1185
575.7
1341
648.1
1371
666.9
286
83.38
300.9
90.65
375.1
132.5
EARTH
156.8
30.12
177.6
38.62
225.3
56.32
1080
522.4
1244
605.3
1301
647.1
243.2
67.43
277.8
80.43
300.1
104.2
Table 1. Comparison with the state-of-the-art methods on three epidemic forecasting datasets. Best in bold and second with underline.
Begi
Regionn
using designated ODE solver, such as Kunge-Kuttar
((H(L) ,Hott)
H(t) = ODESolver
(12)
dt
Given the overall historical time window T, we ultimately derive the global infection trend HT)) and local disease states M(T) ((t)),((t), R(t)). To integrate both the global coherence of epidemic trends and the local intricacies of discase states, we design a multi-headed cross-attention mechanism to merge the global and local transmission in- formation. Specifically, we use HTT) to guide the fusion of M(T). Given three common sets of inputs: query set Q, key set K, and value set V. we define H as follows:
0
12
Step
Timee
(a) Visualization of Region
(b) Visualization of Region 8
Figure3 3. Visualization of predicted cases. We randomly pick two regions in the Mustralia-COVI dataset with horizon 10. shows that EARTH fits the ground truth well and follows the developing trend epidemics. Better view in enlarged.
H(Q. K, V) = (O, ① l22 @ ... D ONyy (Ny)W,
WW,
QKT Vdz
we use to update the weights ewv in Equation (5), resulting in:
S(Q, K, V) = softmax
V.
(13)
P2p = S(QW KWRR K VWH NT
Φ =
W crass
(t)l Σ evu (t)
MEN,
The p-th head is represented by Qz, and the attention func- tion is denoted as S. The learnable linear mappings include W. Wº, WK and WV. The formulation for the global- local fusion is given by:
6. = Wians
(t)ll Σ
Cou (t) - W (1), (11)
N
reowy (f).
The continuous and regional correlation intensity is denoted by (t). The set Nr represents the neighborhood nodes with nonzero weights in A(l) for the dynamic regional trans- mission of region v. This approach enables GLTG to lever- age the global infection trend signal to guide local spatial transmission patterns, considering diverse external factors in epidemics and extending beyond simple partial transfer.
F(T) = H (Z)(T).M(T)MM
(14)
In Equation (14), F(T) represents the fused feature. Con- ceptually, the global trend feature serves as query, calcu- lating the similarity with each detailed disease state. This method aids in recognizing the semantic epidemic condi- tions and attentively integrating the disease features.
Global and Local Epidemic Fusion. With the differential and integral processes of epidemics established, we deter- mine the global infection trend H(t) for continuous time
6

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EARTH: Epidemiology-Awarr Neural ODE with Continuous Discase Transmission Graph
Australia- Castralia-COVID
US- US-Region
et 2023): R. represents RMSE (Root Mean Square Error),
Missing Rate
h == 5
h = 10
h
1=5 5
10 h =
while P stands for Peak Time Error, which calculates the
R
P
R
P
R
P
R
P
MAE (Mean Absolute Error) focusing only significant peaks in the epidemics using specified threshold. For more details please refer to Appendix B.
40%
173.1 38.02
45.68 190.2
1129 541.9 1267 629.9
30%
168.5 36.42
187.2 44.50
1115 535.4
1265 631.8
20%
162.4
33.44
184.7
42.97
1110 536.2 1262 618.4
Counterparts. We compare ours against several SOTA epidemic forecasting methods: VAR (Song et al., 2020),
10%
158.4
30.95
180.4 40.77
1089 529.6 1259 613.1
SARIMA (Valipour, 2015), SIR (Grassly & Fraser, 2008),
0%
156.8 30.12
177.6 38.62
1080 522.4 1244 605.3
LSTM (Sesti al., 2021), DCRNN (Li et al., 2018),
Table 2. Analysis under irregular conditions on two datasets.
STGCN (Yu et al., 2017), ASTGCN (Guo et al., 2019),
STGODE (Fang et al., 2021), STG-NCDE (Choi et al., 2021), CNNRNN-Res (Wu et al., 2018), CAMul (Ka- marthi et al., 2021), EINN (Rodríguez et al., 2023), Co- laGNN (Deng et al., 2020), EpiGNN (Xie et al., 2022) and EpiColaGNN (Liu et al., 2023).
4.4. Overall Objective
Ultimately, we derive the fused features FT'), which har- monize the global consistency of epidemic trends with the particularities of local health conditions. We then concate- nate these features with the time-corresponding features Z() and employ an MLP parameterized by B to pool the final prediction for region 8:
5.2. Performance
This section addresses QI. To demonstrate the excellent per- formance of our proposed EARTH, we conducted comprehen- sive experiments on various epidemic datasets. We consid- ered multiple baselines, including general spatio-temporal and epidemic forecasting methods, as detailed in Tab. 1. Key observations include: 1) VAR and LSTM are inadequate at capturing complex spatial dependencies, making them less effective. 2) General spatio-temporal methods like STGCN or ASTGCN can capture some regional dependencies but struggle with time development and long-term predictions. 3) ODE-based methods like STGODE can learn complex dynamic systems but do not sufficiently consider epidemic mechanisms. 4) Some epidemic-specific methods achieve excellent results but still struggle to model the evolution of cpidemics. 5) Mechanistic methods like EINN do not succeed in capturing high-level spatial interaction between diseases from different regions. 6) EARTH demonstrates competitive performance across various real-world datasets due to its ability to learn complex epidemic evolution and dynamic regional propagation patterns.
(15) Following previous methods (Deng et al., 2020; Xie et al., 2022), we use the MSF loss to compare the predicted values with the ground truth:
V/ (T.
B N Eyse ΣΣ
(16)
where B denotes the sample size, and i is the sample index. Vi, represents the true value for sample of region v.
5. Experiment
In this section, we comprehensively evaluate our proposed EARTH by answering the main questions:
Q1: Performance. Does EARTH outperforms the existing
state-of-the-art epidemic forecasting methods?
Q2: Resilience. Is EARTH stable on different settings?
Q3: Effectiveness. Are proposed two key components:
Additionally, to visually underscore the superiority of EARTH, we compared predicted cases in the Australia- COVID dataset with a horizon of 10 against different base- lines. The results, shown in Figure 3, indicate that our method more accurately fits the ground truth and follows the trend of epidemic development. We also show the learned graph in our GLTG component in Figure 4, which demon- strates that our method goes beyond geographical connec- tions and obtains global horizons during evolution.
EANO and GLTG both effective?
Q4: Sensitivity. What is the performance of the proposed method with different hyper-parameters?
The answers of Q1-Q4 are illustrated as follows.
5.1. Experimental Setup
Real-worid Datasets. We leverage three datasets exam- ine the validity of our EARTH, to ensure fair and consistent comparisons with prior work (Liu et al., 2023; Kamarthi & Prakash). including COVID-19 and influenza-like illness: Australia-COVID, US- Regions and US-States. Please see Appendix A for dataset details.
5.3. Resilience
This section addresses the question Q2. We conducted two key experiments to evaluate this aspect: 1) We examine the robustness of the method under different irregular condi- tions with range of missing rates, detailed in Tab. 2. The
Implemention Details. We use two metrics following (Liu
7

---

EARTH: Epidemiology-Awar Neural ODE with Continuous Discase Transmission Graph
Australii-COViD
US-Region
Variants
10
R
P
R
P
R
P
R
P
w/o Both
267.4
43.65
322.7
63.04
1235 637.4
1367
675.3
EANO
178.6
36.99
184.5
44.62
1120 538.3
1282
639.2
w GLTG
232.4 40.44
301.2 57.42
1204 579.3
1321
654.3
w/o Dyna. Graph
227.6 38.44
290.0 53.89
1184 572.5
1302
647.0
Glo.
172.4 33.75
1102 541.2
1267
621.3
w Concat
167.9 33.44 190.4 45.71
1132 544.3
1304
612.4
Add
175.2 33,01 194.8 43.29
1149 535.7
1291
603.7
Fully Connected
192.1 39.62 194.7 40.22
1192 564.3
1347
666.0
Figure 4. Learned Regional Graph in GLTG. We visualize top-3 weighted edges for each region the US-States dataset, excluding states with no available data.
Sparse Penalty
160.9 32.60 182.4 59.37
1075
519.2
1271
635,4 beight
156.8 30.12
38.62
1080 522.4
1244
605.3
Table 3. Ablation Study of different variants on two datascts.
outcomes show that EARTH remains robust across different datasets with diverse intervals. Our proposed method consis- tently outperforms other baseline methods, demonstrating its resilience to variable intervals and missing data. 2) We also test the performance of EARTH across different predic- tion horizons as shown in Figure 6. The results indicate that our method can make stable predictions over various hori- zons while learning epidemic mechanisms enables superior long-term prediction compared to other methods.
270
250
330
170
150
(a) Head of Attention Ny
(b) Global Connection
Figure 5. Analysis hyper-parameter. Performance with hyper- parameter Ny and k where red, yellow, and green represent the Australia-COVID, US-States, and US-Region respectively.
5.4. Effectiveness
though too few heads can impair the method's ability to capture diverse information. We also test our method with different values of k in Equation (7). More connections can lead to redundancy in message passing for datasets with fewer regions (e.g., Australia-COVID), In contrast, larger datasets (e.g., US-States) can tolerate more connections. In all cases, global connections are essential for learning global infection trends
The explanation for Q3 is presented in this section. Tab. 3 firstly discusses two key design elements in our method: EANO effectively enhances performance by leveraging the powerful capabilities of Neural ODEs while specifically considering the disease propagation mechanism. Addition- ally, GLTG yields promising results by learning the dynamic regional patterns during the evolution of epidemics.
In addition, we dive into GLTG deeper by considering it without dynamic graphs (w/o Dyna. Graph) or global trends (w/o Glo. Trend). The results show that dynamic graphs play crucial role in modeling disease spatial interactions, while global trends help to capture long-distance informa- tion. Additionally, We provide two variants: one using concatenation (w Concat) and another using addition (w Add) instead of cross- aotss-attentoon. These variants show the effectiveness and necessity of cross-attention, as outper- forms both by selectively emphasizing relevant global and local information. We also examine EARTH under a fully connected regional graph, declaring this will lead to infor- mation redundancy. The variant considering adding sparse penalty loss to the learned graph for encouraging sparsity, will not impact the final performance significantly.
6. Conclusion
In this paper, we propose a novel framework, EARTH, to improve epidemic forecasting performance. By integrat- ing neural ODEs with traditional compartmental models, EANO captures the underlying disease propagation mech- anisms. We also identify global infection trends and intro- duce GLTG to dynamically adjust local transmission pat- terns. Using global-local cross-attention fusion approach, we extract representative features that account for both sub- tle disease states and broader trends. Extensive experiments on real-world epidemic datasets highlight the effectiveness of EARTH, offering valuable insights into combining mecha- nistic models with deep learning for future applications in epidemiology and data science.
5.5. Sensitivity
Impact Statement
This section provides an answer to Q4. As shown in Fig- ure 5, we first investigate the effect of varying the number of attention heads Ny in Equation (13). The results demon- strate overall stability with different numbers of heads, al-
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be
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EARTH: Epidemiology-Awarr Neural ODE with Continuous Discase Transmission Graph
large graphs. In Proceedings the Eleventh International Conference on Learning Representations, 2023.
MM
EARTH
Chen, J., Liu, H., Hopcroft. J. E., and He, K. Leveraging contrastive learning for enhanced node representations in tokenized graph transformers. In Proceedings of the 38th Annual Conference on Neural Information Processing Systems, 2024a.
Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. NeurIPS,
31, 2018.
Figure Analysis on different horizon with four methods.
specifically highlighted here.
Chen, Y., Ren, K., Wang, Y., Fang, Y., Sun, W., and Li, D. ContiFormer: Continuous-Tine Transformer for Irregular Time Series Modeling, 2024b.
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EARTH: Epidemiology-AAwrr Neural ODE with Continuous Discase Transmission Graph
A. Datasets Details
Following previous epidemic forecasting work (Deng et al., 2020; Liu et al., 2023), we exploit three widely-used datasets including COVID-19 and influenza-like illness:
AuSvra Provided by JHU-CSSE', this dataset records daily new COVID-19 cases, including states and 2 territories, from January 2020 to August 2021.
US-Regions. This dataset comprises weekly influenza activity levels for 10 Health and Human Services (HHS) regions, spanning from 2002 to 2017, and offers data on regional influenza patterns over time. US-States. The US-States dataset contains weekly counts of patient visits for influenza-like illness (II.I) across 49 states in the United States from 2010 to 2017, excluding Florida, capturing influenza trends.
B. Implemention Details
In all experimental setups, we set the learning rate to le and use SGD (Robbins & Monro, 1951) as the optimizer with momentum of 0.9 and weight decay of le - 5 (Bi et al., 2025a:b). The default hidden size is 64, and the window size T is 20. Considering that decision-makers need time to allocate prevention resources in epidemic modeling, we set the horizon to 5, 10, and 15, repeat each experiment five times for each dataset record the average results.
C. Details about Dynamic Time Warping
In our method, we use Dynamic Time Warping (DTW) measure similarity between nodes based on historical case trends. The DTW distance between two time series X = {11,, T2, ................................ ..n) and Y = (y1, U2, Vmm) is computed by constructing an n X m cumulative distance matrix D, where each element D(i,j) represents the minimum cumulative distance between
the first elements of X and the first j elements of Y, with d(z,, Uy) = ||x: - yyy as the Euclidean distance. The recursive relation
D(i,)) d( Mj) + min 1,j), (i. 1), D(i 1.j 1)}-
(17)
Finally, Dnn, m) gives the DTW distance between X and Y. Using this DTW metric, define the k-most similar nodes to node as:
Top (v) {u u u DXK, X,) is among the k smallest values}.
(18)
This approach captures alignment in epidemic patterns across regions, even with temporal shifts. By identifying the k-most similar nodes using DTW, we create flexible, semantically meaningful connection structure that enhances model performance.
1.9
13


Reviews:
[{"review_id": "mrsNSpdVu5", "reviewer": "Reviewer_Q6Uq", "summary": "The paper presents EARTH (Epidemiology-Aware Neural ODE with Continuous Disease Transmission Graph), a novel framework for epidemic forecasting that integrates neural ordinary differential equations with epidemiological mechanisms. The authors address challenges in current approaches by modeling the continuous-time nature of epidemics, capturing dynamic regional transmission patterns, and considering irregular sampling intervals. EARTH consists of two key components: an Epidemic-aware Neural ODE (EANO) that captures disease transmission patterns, and a Global-guided Local Transmission Graph (GLTG) that models global infection trends to guide local transmission dynamics. The model uses a cross-attention mechanism to integrate global epidemic coherence with local nuances of disease transmission. Experiments on COVID and influenza datasets demonstrate EARTH's superior performance compared to state-of-the-art methods in forecasting real-world epidemics.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to Reviewer Q6Uq\n\nWe thank you for your positive assessment and for recognizing the importance of our work in epidemic modeling. We address your concerns below:\n\n> `Question & Weakness`: Only evaluated for COVID-19, with potential limitations for heterogeneous datasets and different diseases. \n\nWe address this concern from several perspectives:\n\n- 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.\n\n- 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.\n\n- 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.\n\n- 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:\n\n|Methods|VIETNAM|ARGENTINA|MALAYSIA|COLOMBIA|\n|---|---|---|---|---|\n|SIR|1865|627.9|128.8|753.6|\n|DCRNN|1254|401.3|215.5|432.1|\n|STGODE|1196|383.2|187.4|492.7|\n|ColaGNN|1078|464.6|142.3|304.8|\n|EARTH|921.5|312.5|110.1|261.0|\n\n\n> `Concern 1 (Methods And Evaluation Criteria & Theoretical Claims)`: The cases with external interventions.\n\nOn EARTH's robustness to transmission dynamics changes:\n\n- 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.\n\n- 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.\n\n- 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.\n\nWhile 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.\n\n> `Concern 2 (Claims And Evidence:)`: Limited explanation of neural ODEs advantages.\n\nNeural 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.\n\n> `Concern 3 (Claims And Evidence)`: Lack of analysis for \"detailed representations\" claim\n\nEARTH 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.\n\n> `Concern 4 (Experimental Designs Or Analyses)`: Potential overfitting.\n\nWe 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.\n\n> `Concern 5 (Experimental Designs Or Analyses)`: There may be different metrics and further sensitivity analyses.\n\nWe conducted additional experiments with CORR (Pearson's Correlation Coefficient) and sensitivity analysis. EARTH performs well across metrics and remains stable with different parameters:\n\n|Method|Dropout Rate|Learning Rate|\n|---|---|---|\n|DCRNN|0.1: 0.82, 0.3: 0.83, 0.5: 0.81|1e-4: 0.81, 1e-3: 0.83, 1e-2: 0.80|\n|ColaGNN|0.1: 0.86, 0.3: 0.87, 0.5: 0.84|1e-4: 0.85, 1e-3: 0.87, 1e-2: 0.84|\n|EARTH|0.1: 0.91, 0.3: 0.92, 0.5: 0.90|1e-4: 0.90, 1e-3: 0.92, 1e-2: 0.89|\n\n> `Suggestion 1`: Essential References Not Discussed\n\nWe will incorporate suggested references in our revised manuscript.", "claims_and_evidence": "EARTH's superior performance over existing methods is demonstrated through comprehensive experiments across three datasets (Australia-COVID, US-Regions, US-States) with quantitative metrics (RMSE, Peak Time Error)\n\nThe effectiveness of individual components (EANO and GLTG) is validated through ablation studies showing performance degradation when components are removed\nRobustness to irregular sampling intervals is shown through experiments with different missing rates (0-40%)\n\nLimited explanation of why neural ODEs specifically are better than existing approaches for continuous-time modeling\nThe claim that EARTH captures \"more detailed representations of disease spread\" lacks qualitative analysis or interpretability studies\nPerformance gains across datasets vary significantly, with more modest improvements in some scenarios", "methods_and_evaluation_criteria": "The EARTH model is built on a neural ordinary differential equation (ODE) framework, designed to model disease transmission in a continuous and dynamic manner. Unlike traditional compartmental models such as SEIR, which assume discrete transitions between disease states, EARTH integrates graph-based epidemiological modeling to represent evolving interactions among individuals. This allows for more accurate disease spread simulations. The model constructs a continuous disease transmission graph, capturing real-time interactions between susceptible and infected populations. It leverages deep learning techniques to adapt and refine its predictions based on observed infection data.\n\nThe method involves training the neural ODE using historical epidemiological data and real-world disease progression patterns. The model continuously updates its parameters based on observed changes in infection rates, making it adaptive to different disease dynamics. The framework is designed to be scalable and generalizable, capable of modeling multiple infectious diseases such as COVID-19, influenza.\n\nThe evaluation of EARTH includes comparisons against traditional epidemiological models, such as SEIR and graph-based models, to assess its effectiveness. The model is tested on real-world datasets, including reported cases from multiple infectious diseases, ensuring its applicability beyond theoretical scenarios. Further sensitivity analyses and broader disease databases would help.\n\nThe EARTH model primarily focuses on COVID and operates within homogeneous time periods, meaning it learns patterns based on relatively stable disease transmission dynamics. However, infectious disease spread is often highly dynamic, influenced by factors such as travel, spillover events, and new variants, which can introduce sudden shifts that challenge predictive models.\n\nWhile machine learning excels at pattern recognition, it may struggle when underlying transmission mechanisms change abruptly, such as with novel introductions from travelers or zoonotic spillovers. If the model has not been trained on data that reflects such disruptions, it may fail to capture these shifts accurately.\n\nA potential limitation of EARTH is whether it accounts for heterogeneous transmission periods—for example, distinguishing between pre-COVID, pre- COVIDvaccine, post-vaccine, and variant-driven waves in COVID-19. Additionally, external shocks, such as government interventions, behavioral changes, or superspreader events, may not be fully captured in a data-driven framework unless explicitly modeled.\n\nIt would help to explain how next steps could make this approach more robust, especially as external factors may not be predicted.\nIntegration of external factors like mobility data, vaccination rates, etc can help but this will may depend a lot on the pathogen, so this may not be disease agnostic. Climate shocks, novel spillovers, antimicrobials resistance and travel patterns will make a big difference. If EARTH lacks mechanisms to handle these shifts, it may overfit to past trends and fail to generalize to new outbreaks or changing epidemic conditions. Addressing these issues could enhance its real-world applicability for public health decision-making. \n\nCOVID is a very distinct example, where data was better collected and the mechanisms impacting its spread will not represent other diseases.", "theoretical_claims": "Works with Neural ODEs as an approach building on the SIR ODE framework\nGraph-Based Transmission Modeling: The theoretical foundation incorporates graph neural networks (GNNs) to capture spatial and temporal dependencies in disease spread. The paper provides justification for using graph structures, emphasizing their ability to model heterogeneous interactions in populations.\n\nThe authors discuss the model's ability to generalize across different epidemiological scenarios but will need empirical evidence.\n\nSome limitations\nThe theoretical claims assume consistent data availability and stationary transmission patterns, which may not hold in real-world scenarios where data availability vary greatly, especially in early explosive outbreaks where predictions are most needed, and also with external shocks (e.g., new variants, travel-driven outbreaks).\nThe model assumes that the graph structure remains representative of disease spread over time, but the theoretical justifications for handling dynamic changes in population movement and behavior could be further elaborated.", "experimental_designs_or_analyses": "Comparison with Standard Existing Approaches\n- evaluates EARTH against traditional models like SEIR and graph-based models to assess its predictive performance.\nThe paper does demonstrate improvements over classical compartmental models for COVID datasets.\n\nThis work used Real-World Datasets. The model is tested on real-world datasets for COVID and influenza outbreaks. While these datasets provide meaningful benchmarks, a broader set of infectious disease cases (especially with different transmission dynamics) would enhance the robustness of the findings. COVID and influenza like illnesses create a very different type of outbreak than many others.\n\nThe training process involves learning from historical epidemiological data to improve forecasting. However, potential overfitting is a concern, especially given the homogeneous time periods used in the study.\n\nThe analysis used metrics - RMSE (Root Mean Square Error) - and Peak Time Error, which calculates the\nMAE (Mean Absolute Error) that are useful for comparing outbreak predictions. For many outbreaks, there may be different metrics, as peak may not be a particular", "relation_to_broader_scientific_literature": "We are at a pivotal moment in epidemiologic models. There is a real possibility for being able to advance the field of epidemic modeling. There are certainly many approaches, but this is an important possible avenue.", "essential_references_not_discussed": "Would place this in a wider context\nhttps://www.nature.com/articles/s42256-024-00895-7\nhttps://www.nature.com/articles/s41586-024-08564-w\nhttps://dl.acm.org/doi/10.1145/3650215.3650396\nhttps://ieeexplore.ieee.org/document/10039594\n\nWould also describe this as focused on COVID/flu modeling\nhttps://covid19-projections.com/about/\n\nCovidSim by Neil Ferguson\n\nAgent Based Models", "other_strengths_and_weaknesses": "Strength showed improvements and should be using this approach more and more\n\nWeakness: only evaluated for COVID, may not be robust in face of heterogenous datasets, with abrupt epidemic changes or with different diseases", "other_comments_or_suggestions": "Consider the risks of the model failing to account for outbreaks in other contexts, with other diseases.", "questions_for_authors": "Would address other contexts, other diseases", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "zMQEu2SGea", "reviewer": "Reviewer_uXQM", "summary": "The paper proposes EARTH, an Epidemiology-Aware Neural ODE with a Continuous Disease Transmission Graph, as a novel framework for epidemic forecasting. The authors integrate neural ODEs with epidemiological mechanisms, capturing both continuous-time disease transmission and global infection trends. The Global-guided Local Transmission Graph and cross-attention fusion mechanism are introduced to enhance epidemic forecasting accuracy. Through extensive experiments on real-world datasets (COVID-19 and influenza), EARTH is shown to outperform state-of-the-art methods.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to Reviewer uXQM\n\nWe 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:\n\n> `Cons`: On applying neural ODEs to epidemiology\n\nWhile neural ODEs have been applied in various domains, EARTH goes beyond straightforward application:\n\n- Integration with epidemic mechanism: Our Network SIR-inspired architecture (Equations 5, 11) explicitly models the transition dynamics between susceptible, infectious, and recovered populations.\n\n- 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.\n\n- Flexibility for irregular data: Unlike conventional time series models, our continuous formulation naturally handles the irregular reporting and missing data common in epidemic monitoring.\n\nOur ablation studies validate these design choices, with significant performance drops when these specific components are removed.\n\n> `Question`: Dynamic integration of global and regional patterns\n\nEARTH integrates global and regional patterns through:\n\n- Semantic connections via DTW: We identify regions with similar epidemic trajectories using Dynamic Time Warping rather than relying solely on geographic proximity.\n\n- 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.\n\n- Multi-scale modeling: We capture both local disease dynamics and cross-regional dependencies within a unified framework.\n\n- Continuous-time formulation: By modeling in continuous time, EARTH handles irregular observation intervals and can forecast at arbitrary time points.\n\nThis approach enables effective forecasting even when regional reporting patterns change, a common challenge in epidemic monitoring.", "claims_and_evidence": "The claims in the paper are mostly supported by clear evidence.", "methods_and_evaluation_criteria": "The proposed methods are effective and novel, and the evaluations are justified.", "theoretical_claims": "These claims are supported by experiments rather than formal proofs.", "experimental_designs_or_analyses": "The experimental design is robust and well-executed, providing strong support for the paper’s claims. The authors conduct extensive evaluations on real-world datasets, demonstrating that EARTH significantly outperforms SOTA methods in terms of forecasting accuracy, peak time error, and robustness.\n\nKey strengths include:\n\n-\tComprehensive evaluation across multiple datasets, showcasing the model's ability to handle real-world data effectively.\n\n-\tAblation studies that validate the contributions of the EANO and GLTG and cross-attention mechanism to model performance.", "relation_to_broader_scientific_literature": "Traditional models like SEIR and mechanistic models have been widely used, but they often struggle with real-world complexities. EARTH advances this field by leveraging data-driven learning while maintaining interpretability through epidemiological structures.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Pros:\n\n(a)\tThe integration of neural ODEs with epidemiological models represents a great innovation. By combining disease transmission mechanisms with deep learning, EARTH effectively captures epidemic dynamics beyond traditional mechanistic and deep learning approaches.\n\n(b) EARTH accounts for irregular sampling intervals and missing data, making it highly applicable to real-world epidemic data where reporting can be inconsistent. \n\n(c) The paper provides comprehensive experiments across multiple epidemic datasets, showing that EARTH significantly outperforms previous approaches\n\nCons:\n\n(a)\tThe combination of neural networks and epidemiology: Similar neural network methods, such as neural ODEs, have already been applied in many fields, especially in time series analysis. However, their application to specific issues in epidemiology is not yet widespread.", "other_comments_or_suggestions": "NA", "questions_for_authors": "(a)\tThe motivation section points out that the existing epidemic prediction methods have failed to fully capture the complexity of the dynamic evolution and regional transmission patterns of epidemics, especially when dealing with global infection trends and regional transmission changes. Considering this motivation, could you please explain in detail how to dynamically learn and integrate global and regional transmission patterns through these methods to address the challenges mentioned in the motivation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QiGORLIoUB", "reviewer": "Reviewer_rfdg", "summary": "The authors tackle the problem of effectively forecasting epidemics and propose the so-called EARTH method, which combines an epidemiology-aware neural ODE with a continuous disease transmission graph. More specifically, they leverage a **neural ODE-based component** (EANO) based on the **common epidemic SIR mechanism** to capture spatial spreading during disease evolution while enabling continuous modeling. They also introduce a **GNN-based component that captures global epidemic trends** (GLTS) and a cross-attention mechanism that combines global patterns with local transmission patterns. The proposed method is evaluated on real-world datasets capturing COVID-19 and influenza diseases against other baseline methods, showing performance improvements for few horizon lengths under node regression metrics (such as RMSE).\n\n## update after rebuttal\n\nThe authors have addressed some of my initial concerns, particularly those related to the significance of results and experimental design details. However, critical issues around the theoretical positioning (against the relevant first works in the field), selection and interpretability of learned epidemic rates, and justification for explicit SIR-based modeling remain only partially resolved and should be clearly reflected in the revised manuscript. Time/memory cost comparisons against baselines are still missing and can be significant. I revise my recommendation to weak accept, assuming the authors will incorporate these clarifications and adjustments into the final version.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to Reviewer rfdg\n\nWe sincerely thank you for your thorough review and hope our responses below will help improve your assessment of our work:\n\n> `Weakness S1 (Theoretical Claims & Claims And Evidence)`: Presentation and Design choices of the epidemic compartment\n\nOur approach addresses these concerns in several ways:\n\n1. 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:\n   |Model Variant|h=5 (R)|h=5 (P)|h=10 (R)|h=10 (P)|\n   |-------------|-------|-------|---------|---------|\n   |w/o Feature|178.6|34.24|198.5|44.18|\n   |EARTH|156.8|30.12|177.6|38.62|\n2. 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.\n3. 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.\n\n> `Weakness S2 (Claims And Evidence & Essential References)`: Positioning against relative works\n\nWe 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.\n\n\n> `Weakness S3 (Experimental Designs)`: Experimental Choices not well-justified/missing\n\nWe appreciate these important observations about experimental design clarity:\n\n1. 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:\n\n   |Method|Win|h=20|h=25|h=30|\n   |------|---|----|----|-----|\n   |STGODE|20|1836|2153|2607|\n   |EpiColaGNN|20|1645|1947|2378|\n   |EARTH|20|**1528**|**1812**|**2219**|\n   |STGODE|40|1682|1976|2391|\n   |EpiColaGNN|40|1476|1763|2134|\n   |EARTH|40|**1374**|**1642**|**1983**|\n\n2. 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.\n\n3. Standard deviations: Results are averaged over 5 runs with different random seeds. We will add variance information in the revised manuscript.\n\n4. Downsampling: Random sampling was used to create missing data points, simulating real-world reporting patterns.\n\n> `Weakness S4`: Computational Analysis\n\nThe computational efficiency of EARTH can be analyzed as:\n\n- 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.\n\n- 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 $O(|E|)$. \n\n- Cross-Attention Mechanism: Our design constrains attention to three epidemic states per region, achieving $O(N \\times d^2)$.\n\nThese optimizations enable EARTH to avoid $O(N^2)$ complexity, supporting large-scale epidemic forecasting applications.", "claims_and_evidence": "The following claims of the paper are problematic:\n- Introduction, page 2: *“We are the first to harmonize the neural ODE with the epidemic mechanism [...] patterns”*. The authors claim that they are the first to combine the Neural ODE method with approximate system equations for epidemic spreading. However, it seems that relevant works in this field have already combined ODEs from epidemic compartmental models with advances in neural ODE solvers, such as in [1].\n- Methodology, page 4: *“By substituting the traditional SIR’s two simple rates [...] our model can derive more detailed representations of the disease spread and recovery processes.”*\nThe infection $\\beta$ and recovery $\\gamma$ rates in SIR models are fundamental parameters in epidemic modeling and have been extensively studied to develop realistic simulations of disease spread. However, the authors do not showcase how learning these parameters improves performance.\n\n1. Kosma, C., Nikolentzos, G., Panagopoulos, G., Steyaert, J. M., & Vazirgiannis, M. (2023). Neural ordinary differential equations for modeling epidemic spreading. Transactions on Machine Learning Research.", "methods_and_evaluation_criteria": "This study leverages common baselines and benchmark datasets in the field. Evaluation metrics (point-wise deviation) are also common.", "theoretical_claims": "No formal proofs are provided for different parts of the proposed method. However, this may not be necessary, as the approach is application-driven and builds upon existing modelization concepts.\n\nLearning $\\beta$ and $\\gamma$ without constraints is not theoretically substantiated (e.g., initialization, bounds). Several relevant studies rely on specific choices for these parameters [1,2] to ensure they capture meaningful spreading dynamics. \n\n1. Sha, H., Al Hasan, M., & Mohler, G. (2021, October). Source detection on networks using spatial temporal graph convolutional networks. In 2021 IEEE 8th International Conference on Data Science and Advanced Analytics (DSAA) (pp. 1-11). IEEE.\n2. Gao, F., Zhang, J., & Zhang, Y. (2022, May). Neural enhanced dynamic message passing. In International Conference on Artificial Intelligence and Statistics (pp. 10471-10482). PMLR.", "experimental_designs_or_analyses": "The common experimental design choices with the cited studies in section 5.1 are unclear. More specifically, different horizon lengths/window sizes are used in studies (Liu et al., 2023, Kamarthi & Prakash) and the proposed method, while it seems that a common choice in time series forecasting is to consider multiples of the input window length (e.g., 1W, 2W, 3W,...).", "relation_to_broader_scientific_literature": "The main focus of this paper is to combine neural ODEs on epidemic priors and GNNs to perform more accurate epidemic forecasting on graphs. The contributions are more prominent in terms of experimental results compared to baseline methods for example real-world epidemic datasets, while in terms of architectural design, the authors built upon common concepts and existent modules in time series spatial-temporal modeling.", "essential_references_not_discussed": "Several methods in the area of physics-informed modelization of epidemic spreading (particularly based on compartmental models, e.g., SIR, SEIR) could be included in the introduction/related work of the paper to support the positioning of its main concepts. Some examples:\n- *SIR Neural ODEs on Networks:* Kosma, C., Nikolentzos, G., Panagopoulos, G., Steyaert, J. M., & Vazirgiannis, M. (2023). Neural ordinary differential equations for modeling epidemic spreading. Transactions on Machine Learning Research.\n- *SIR-based Embedding Layers:* Zheng, Y., Jiang, W., Zhou, A., Hung, N. Q. V., Zhan, C., & Chen, T. (2024). Epidemiology-informed Graph Neural Network for Heterogeneity-aware Epidemic Forecasting. arXiv preprint arXiv:2411.17372.\n- *Physics-Informed Neural Networks:* Cai, M., Em Karniadakis, G., & Li, C. (2022). Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant. Chaos: an interdisciplinary journal of nonlinear science, 32(7).\n- *Dynamic Message Passing for SIR combined with GNNs:* Gao, F., Zhang, J., & Zhang, Y. (2022, May). Neural enhanced dynamic message passing. In International Conference on Artificial Intelligence and Statistics (pp. 10471-10482). PMLR.", "other_strengths_and_weaknesses": "*Strengths:*\n- The paper is well-written and easy to follow.\n- The experimental results consider several baseline modes and thorough ablation studies of the proposed method's main components.\n\n*Weaknesses:*\n- **S1** - *Presentation and Design choices of the epidemic compartment:* In real-world scenarios, epidemic data is often noisy. If $\\beta$ and $\\gamma$ are learned directly from data without constraints, the model might overfit short-term trends rather than capturing realistic transmission dynamics. Necessary assumptions to derive this form of equations (3), intuition, and limitation of the compartmental model in practical applications are not discussed.\n- **S2** - *Positioning against works explicitly combining ML and compartmental modes:* Based on the missing references above, the presentation of related works fails to showcase the limitations of existing approaches and design choices on incorporating compartmental models/spreading priors to the loss function/architectural structure of the methods.\n- **S3** - *Experimental Choices not well-justified/missing:* \n1. It is unclear why the authors choose h=5,10,15 as horizon lengths for their experiments, given a historical window of 20 timestamps, which constitutes the task rather easy. A study of the performance impact with larger (>20) forecasting horizons would be interesting.\n2. The type (e.g., runge-kutta) and selection of step of the solver chosen in the neural-ode are not mentioned.\n3. Standard deviations are not mentioned.\n4. The downsampling method followed for synthetically creating irregular timestamps in the datasets is unclear but can have a significant impact on the distortion of the underlying continuous dynamics.\n- **S4** - *Computational Analysis is missing:* The proposed method relies on computationally heavy components, including the neural-ode solver and the dtw used to extract A. In practical applications, for increasing dataset sizes (and graphs’ nodes/edges) and increasing windows/horizons, some methods can become very ineffective regarding time/memory costs.", "other_comments_or_suggestions": "Not applies.", "questions_for_authors": "Based on aforementioned weaknesses, the following aspects need enhancement and further clarification:\n1. S1 - theoretical explanations\n2. S3 - experimental choices \n3. S4 - computational analysis", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uPM37zHblO", "reviewer": "Reviewer_XKTr", "summary": "This paper introduces a new epidemic forecasting framework, EARTH, which combines neural ODEs with traditional compartmental models. The core idea behind EARTH is to address common forecasting challenges such as irregular data sampling and missing values by integrating two main modules: a local transmission model based on Epidemic-Aware Neural ODEs and a Global-guided Local Transmission Graph that incorporates global trends. Through experiments on real-world datasets, like COVID-19 data, the authors demonstrate that EARTH achieves superior performance compared to existing methods, suggesting that it could play a crucial role in enhancing the accuracy of epidemic prediction.\n\n## update after rebuttal\n\nThe authors' responses have addressed my concerns. After reviewing the comments as well as the discussions, I'd like to raise my score to 5.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "# Response to Reviewer XKTr\n\nWe sincerely thank you for your thorough review and positive assessment of our work. We are grateful for your recognition of EARTH's \nnovelty and contributions to epidemic forecasting. We address your questions below:\n\n> `Weakness 1`: No detailed analysis of the computational complexity.\n\nWe appreciate this valuable suggestion. The computational complexity of EARTH can be broken down into:\n\n- **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.\n\n- **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.\n\n- **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.\n\nWe will include this detailed analysis in the revised version.\n\n> `Weakness 2`: Model sensitivity to hyperparameters.\n\nThank 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:\n\n| Hidden Dimensions | 16 | 32 | 64 | 128 | 256 |\n|-------------------|-----|-----|-----|------|------|\n| RMSE | 187.3 | 176.5 | 156.8 | 159.2 | 172.8 |\n| MAE | 42.64 | 36.27 | 30.12 | 31.95 | 38.76 |\n\n| Learning Rate | 5e-5 | 1e-4 | 5e-4 | 1e-3 | 5e-3 |\n|---------------|------|------|------|------|------|\n| RMSE | 179.1 | 167.5 | 160.3 | 156.8 | 163.2 |\n| MAE | 41.56 | 36.28 | 32.41 | 30.12 | 33.95 |\n\nThese 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.\n\n> `Question 1`: How are these missing rates introduced into the data?\n\nThank you for this question about our experimental methodology. \nFor 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.\n\n> `Question 2`: In Figure 4, are there any isolated nodes with almost no connections to other nodes? If there are, what does this mean?\n\nThis 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.", "claims_and_evidence": "The paper’s claims are supported by a variety of evidence that extends beyond just experimental results. 1) Experimentally, the model is evaluated on several real-world datasets—including COVID-19 data—with consistent improvements over baselines in metrics such as RMSE and MAE. 2) Detailed ablation studies demonstrate that both the neural ODE component and the global transmission graph are crucial to achieving high performance, as their removal significantly degrades the results. 3) Additionally, hyperparameter sensitivity tests confirm that the model performs robustly across a range of settings, and clear visualizations of predicted versus actual trends further validate its practical applicability.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are well-suited to the problem. Additionally, using real-world datasets like COVID-19 data with standard metrics such as RMSE and MAE offers a robust framework for evaluation.", "theoretical_claims": "I reviewed the theoretical claims, especially those related to the integration of neural ODEs with classical epidemic models. The manuscript’s mathematical formulation is generally free from notable errors, and the conceptual foundations are sound and well-motivated. While the discussion primarily leverages established principles without extensive formal proofs, the theoretical framework is sufficiently robust, and the empirical results further reinforce its validity. Additionally, including an analysis of computational complexity could provide deeper insights into the scalability and practical applicability of the proposed methods, but this is a minor point.", "experimental_designs_or_analyses": "The experimental designs and analyses in this paper are robust and well thought out. The authors employ multiple real-world datasets, including COVID-19 data, and standard metrics such as RMSE and MAE to evaluate performance. Comprehensive ablation studies in Table 3 effectively isolate the contributions of each model component, while sensitivity analyses  from Figure 5 confirm the stability of the model across various hyperparameters. The clear visualizations in Figure 3 and Figure 4 further support the validity of the results by illustrating how the model captures key trends.", "relation_to_broader_scientific_literature": "In terms of machine learning, the paper draws on work related to neural differential equations and graph neural networks. While these techniques have been applied successfully in other domains, their application to epidemic forecasting is a significant innovation. By leveraging these advanced techniques, the paper contributes to the growing body of research on using deep learning to enhance predictive modeling in epidemiology. This approach also connects to broader trends in the literature about the fusion of classical and modern machine learning techniques to address real-world problems.", "essential_references_not_discussed": "The paper already does a good job of addressing relevant literature. It connects well with existing research in epidemic modeling and machine learning.", "other_strengths_and_weaknesses": "Strengths:\n\n1)\tThe paper introduces an innovative approach by combining neural ODEs with traditional epidemic models and incorporating a Global-guided Local Transmission Graph, which effectively captures both continuous dynamics and spatial transmission patterns.\n\n2)\tThe authors provide clear motivation for integrating machine learning with classical epidemiological frameworks, addressing limitations of traditional models in real-world epidemic forecasting.\n\n3)\tThis work advances the field by demonstrating how modern deep learning techniques can be seamlessly merged with established epidemic models, paving the way for more accurate and dynamic forecasting tools.\n\n4)\tBy bridging the gap between classical epidemiological modeling and contemporary machine learning, the paper makes a significant contribution to computational epidemiology and opens up new research avenues for future advancements in the field.\n\nWeaknesses:\n\n1)\tThere is no detailed analysis of the computational complexity, which could help assess the method's scalability and efficiency in large-scale applications.\n\n2)\tThe model may be sensitive to certain hyperparameters (such as learning rate, hidden layer dimensions, regularization coefficients, etc.). Although the author conducted experiments, different hyperparameter settings may affect the model's generalization ability and stability. Further experiments can explore the impact of hyperparameter adjustment on model performance.", "other_comments_or_suggestions": "Please ref to weaknesses.", "questions_for_authors": "-\tHow are these missing rates introduced into the data? Are they intentionally sampled deletions from the original data (such as random deletions) to simulate missingness, or do they originate from missing data in real-world scenarios?\n-\tIn Figure 4, are there any isolated nodes with almost no connections to other nodes? If there are, what does this mean?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper integrates ideas from epidemic modeling together with neural ODEs and transmission graphs to better forecast disease. In particular, the method integrates various sources of information, including knowledge of epidemic mechanisms and spatial information, in a way that attempts to prioritize the most salient information across multiple sources for prediction. The authors show that the predictions outperform existing methods, though I find much of the contributions empirically focused and catered to forecasting, leaving behind many of the desirable interpretability aspects of mechanistic models that mathematical biologists use (indeed it is not difficult to outperform them in forecast benchmarks). The reviewers all lean positive, with no discussion after prompting, and recommend that there are potentially impactful contributions relevant to the ICML readership
Decision: Accept (poster)