MepoGNN: Metapopulation Epidemic Forecasting with Graph Neural Networks

Abstract

Epidemic prediction is a fundamental task for epidemic control and prevention. Many mechanistic models and deep learning models are built for this task. However, most mechanistic models have difficulty estimating the time/region-varying epidemiological parameters, while most deep learning models lack the guidance of epidemiological domain knowledge and interpretability of prediction results. In this study, we propose a novel hybrid model called MepoGNN for multi-step multi-region epidemic forecasting by incorporating Graph Neural Networks (GNNs) and graph learning mechanisms into Metapopulation SIR model. Our model can not only predict the number of confirmed cases but also explicitly learn the epidemiological parameters and the underlying epidemic propagation graph from heterogeneous data in an end-to-end manner. Experiment results demonstrate our model outperforms the existing mechanistic models and deep learning models by a large margin. Furthermore, the analysis on the learned parameters demonstrates the high reliability and interpretability of our model and helps better understanding of epidemic spread. Our model and data have already been public on GitHub https://github.com/deepkashiwa20/MepoGNN.git.KeywordsEpidemic forecastingHybrid modelMetapopulation epidemic modelGraph Neural NetworksDeep learningCOVID-19

Full text

MepoGNN: Metapopulation Epidemic
Forecasting with Graph Neural Networks
Qi Cao1, Renhe Jiang

1, Chuang Yang1, Zipei Fan1, Xuan Song1,2, and
Ryosuke Shibasaki1
1The University of Tokyo, Tokyo, Japan
2Southern University of Science and Technology, Shenzhen, China
{caoqi,jiangrh,chuang.yang}@csis.u-tokyo.ac.jp;
fanzipei@iis.u-tokyo.ac.jp; {songxuan,shiba}@csis.u-tokyo.ac.jp
Abstract.
Epidemic prediction is a fundamental task for epidemic con-
trol and prevention. Many mechanistic models and deep learning models
are built for this task. However, most mechanistic models have difficulty
estimating the time/region-varying epidemiological parameters, while
most deep learning models lack the guidance of epidemiological domain
knowledge and interpretability of prediction results. In this study, we pro-
pose a novel hybrid model called MepoGNN for multi-step multi-region
epidemic forecasting by incorporating Graph Neural Networks (GNNs)
and graph learning mechanisms into Metapopulation SIR model. Our
model can not only predict the number of confirmed cases but also explic-
itly learn the epidemiological parameters and the underlying epidemic
propagation graph from heterogeneous data in an end-to-end manner.
Experiment results demonstrate our model outperforms the existing mech-
anistic models and deep learning models by a large margin. Furthermore,
the analysis on the learned parameters demonstrates the high reliability
and interpretability of our model and helps better understanding of epi-
demic spread. Our model and data have already been public on GitHub
https://github.com/deepkashiwa20/MepoGNN.git.
Keywords: Epidemic forecasting ·Hybrid model ·Metapopulation epi-
demic model ·Graph neural networks ·Deep learning ·COVID-19
1 Introduction
The coronavirus disease 2019 (COVID-19) pandemic has caused around 500
million confirmed cases and more than 6 million deaths in the global, and it is still
ongoing. Due to this circumstance, epidemic forecasting has been a key research
topic again as it can guide the policymakers to develop effective interventions and
allocate the limited medical resources. Many mechanistic models and deep learning
models have been built for the epidemic prediction task. In particular, human
mobility is seen as one of the most important factors to understand and forecast
the epidemic propagation among different regions. In this study, we employ

Corresponding Author.

2 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
Fig. 1: Illustration of metapopulation epidemic propagation among regions [1].
metapopulation SIR model [1, 2] as the base model for our task, which extends
the most fundamental compartmental model (i.e., SIR [11]) in epidemiology with
metapopulation epidemic propagation. As illustrated in Fig. 1, it divides the total
population under the epidemic into several sub-populations (e.g., by regions).
Each sub-population consists of three compartments,
S
(susceptible individuals),
I
(infectious individuals),
R
(removed individuals, including deaths and recovery
cases), and the human mobility between sub-populations is modeled as a directed
graph. Thus, it can well model the epidemic propagation in a large-scale area.
The metapopulation epidemic models have achieved great success in modeling
and analyzing the propagation of epidemic diseases, such as SARS, H1N1, and
Malaria [3–5].
However, it is always a non-trivial task to build a metapopulation epidemic
model, especially for new emerging epidemics such as the COVID-19 due to
the following reasons. First, the epidemiological parameters in metapopulation
model keep varying from region to region and time to time. As we all know,
the Coronavirus keeps evolving, and the transmissibility and mortality of the
variants (e.g., Alpha, Delta, and Omicron) are significantly different. Besides, the
intervention policies and the human movements also vary over different periods
and regions. Second, due to the mixed factors mentioned above, the epidemic
propagation effects via human mobility in metapopulation model are also difficult
to be obtained or estimated. In the case of prefecture-level prediction in Japan,
we need to collect the large-scale human mobility data of the entire Japan and
obtain the amount of human movements between each pair of prefectures. Then
how to accurately infer the underlying disease propagation network becomes
another intractable task. Third, besides the daily infection data, external features
such as date information (e.g.,
dayof week
) and daily movement change patterns
should also be involved.
To tackle these challenges, we incorporate deep learning modules into metapop-
ulation SIR model to form a novel hybrid epidemic model. Specifically, we first
learn the time/region-varying epidemiological parameters from multiple data
features through a spatio-temporal module, which consists of Temporal Con-
volutional Networks (TCN) and Graph Convolutional Networks (GCN). Next,

MepoGNN: Metapopulation Epidemic Graph Neural Networks 3
we design two types of graph learning module to automatically approximate
the underlying epidemic propagation graph based on the countrywide human
mobility data. Furthermore, we let the learned latent graph be shared by the
spatio-temporal module and the metapopulation SIR module, which further
enhances the model interpretability and reliability. Previous deep learning meth-
ods [6–10] simply treat the epidemic forecasting as time-series prediction task or
spatio-temporal prediction task, which can only output the predicted number of
infections in a pure black-box manner. Recent study [29] involves the classical
epidemic modeling into deep neural networks, however, it does not explicitly
consider the epidemic propagation among regions via metapopulation modeling
like ours, which largely limits the model interpretability for multi-region epidemic
forecasting. To the best of our knowledge, our work is the first hybrid model
that couples metapopulation epidemic model with spatio-temporal graph neural
networks. In summary, our work has the following contributions:
–
We propose a novel hybrid model along with two types of graph learning mod-
ule for multi-step multi-region epidemic prediction by mixing metapopulation
epidemic model and spatio-temporal graph convolution networks.
–
Our model can explicitly learn the time/region-varying epidemiological pa-
rameters as well as the latent epidemic propagation among regions from the
heterogeneous inputs like infection related data, human mobility data, and
meta information in a completely end-to-end manner.
–
We collect and process the big human GPS trajectory data and other COVID-
19 related data that covers the 47 prefectures of Japan from 2020/04/01 to
2021/09/21 for countrywide epidemic forecasting.
–
We conduct comprehensive experiments to validate not only the superior
forecasting performance but also the high interpretability of our model. Our
model and data have already been public on GitHub
https://github.com/
deepkashiwa20/MepoGNN.git.
2 Related Work
The models for epidemic simulation and forecasting can be divided into two
types: mechanistic approaches and deep learning approaches.
Mechanistic approaches are built based on the domain knowledge of epidemi-
ology which employ pre-defined physical rules to model infectious diseases’ trans-
mission dynamics, mainly classical compartmental models [11, 12], metapopulation
models [2, 13–15] and agent-based models [16–18]. The classical compartmental
models simulate the spread of infectious diseases in a homogeneous population
which are unable to model epidemic spread between regions. The metapopula-
tion models assume the heterogeneity of sub-populations and use the human
mobility pattern between regions to model the spread of the epidemic [1, 2]. The
agent-based models directly use the individual-level movement pattern [16, 17]
or trajectories [18] to emulate the contagion process. Our work is related to the
metapopulation model which is most suitable for multi-region epidemic forecast-
ing task. To implement epidemic modeling, it needs to be calibrated first using

4 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
historical observations and use the optimized or manually modified parameters
to make prediction. These efforts are hardly applicable for multi-step forecasting
tasks. The parameters calibration process needs high computational complexity,
especially when facing huge parameter state space [16, 13]. Moreover, in most
mechanistic models, epidemiological parameters keep fixed during forecasting.
The variation of parameters through time is not considered which leads to the
problem of cumulative error on multi-step prediction.
Deep learning approaches have shown excellent performance in the modeling
and forecasting on time series prediction tasks. As a typical time series, several
research efforts utilizing deep learning techniques, such as LSTM [6, 8], have been
conducted for epidemic forecasting over a single region [6,8, 19, 20]. Nevertheless,
the epidemic propagation is often spatially dependent, i.e., co-evolving over re-
gions. Thus, treating epidemic forecasting as a multivariate time-series prediction
task, performing collaborative forecasting over multiple geographical units should
be a more reasonable choice. For such tasks, a key challenge is to model the
complex and implicit spatio-temporal dependencies among the observations, on
which much evidence shows that GNN can perform very well for modeling the
inter-series relationships. A series of state-of-the-art solutions based on GNN have
been proposed for multivariate time-series prediction tasks, such as STGCN [21],
DCRNN [22], GraphWaveNet [23], ColaGNN [9], and CovidGNN [10]. In partic-
ular, ColaGNN [9] and CovidGNN [10] were explicitly designed for the epidemic
prediction. However, these works ignore the domain knowledge of epidemiology
and are hard to interpret from the epidemiological perspective. STAN [19] incor-
porates epidemiological constraints into deep learning models, but it can only
predict infections of a single region. CausalGNN [29] embeds single-patched SIRD
model into GNN for multi-region epidemic forecasting.
Overall, we distinguish our work from existing ones in the following ways:
Compared with the mechanistic models, MepoGNN adopts an end-to-end frame-
work that can predict the dynamic change of epidemiological parameters and
use predicted parameters to produce multi-region and multi-step prediction;
Compared with the deep learning models for the multi-region prediction task,
MepoGNN incorporates the domain knowledge of epidemiology and enhances
the interpretability by combining spatio-temporal deep learning model with
the metapopulation model; Furthermore, MepoGNN can output the prediction
of infections through the metapopulation epidemic model and learn the inter-
pretable epidemiological parameters and the latent graph of epidemic propagation
simultaneously.
3 Problem
In this study, we focus on forecasting the number of daily confirmed cases for multi-
region and multi-step simultaneously. For a single region, the historical daily con-
firmed cases from timestep
t−Tin
+1 to
t
can be represented as x
t−(Tin−1):t∈RTin
.
Then, the historical daily confirmed cases of
N
regions can be denoted as
X
t−(Tin−1):t
=
{
x
t−(Tin−1):t
1,
x
t−(Tin−1):t
2, ...,
x
t−(Tin−1):t
N} ∈ RN×Tin
. Besides the

MepoGNN: Metapopulation Epidemic Graph Neural Networks 5
historical observations, we also incorporate the external factors to form a multi-
channel input as
Xt−(Tin−1):t
=
{
X
t−(Tin−1):t
1,
X
t−(Tin−1):t
2, ...,
X
t−(Tin−1):t
C} ∈
RN×Tin×C
. Details of the input features will be introduced in Section 5.1. Ad-
ditionally, human mobility between regions (static flow data U
∈RN×N
or
dynamic flow data
Ot−(Tin−1):t∈RN×N×Tin
) is used as another type of input.
The prediction target is the daily confirmed cases of
N
regions in next
Tout
timesteps Yt+1:t+Tout ∈RN×Tout . The problem can be formulated as follows:
{X t−(Tin−1):t,U}or {X t−(Tin −1):t,Ot−(Tin −1):t}f(·)
−−−−−−→ Yt+1:t+Tout (1)
4 Methodology
We present Metapopulation Epidemic Graph Neural Networks (MepoGNN),
demonstrated in Fig. 2, for spatio-temporal epidemic prediction. MepoGNN
consists of three major components: metapopulation SIR module, spatio-temporal
module and graph learning module. These three components tightly cooperate
with each other. Graph learning module learns the mobility intensity between
regions as a graph and output it to spatio-temporal module and metapopulation
SIR module. Spatio-temporal module captures the spatio-temporal dependency
to predict the sequences of parameters for metapopulation SIR module. Then,
Fig. 2: Proposed Metapopulation Epidemic Graph Neural Networks (MepoGNN)
for spatio-temporal epidemic prediction.

6 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
metapopulation SIR module takes the learned graph and the predicted parameters
to produce the multi-step prediction of daily confirmed cases.
4.1 Metapopulation SIR Module
SIR model is one of the most fundamental compartmental models in epidemi-
ology, used for modeling the epidemic spread [11]. However, it can only model
the epidemic spread for a homogeneous population, which ignores the epidemic
propagation between sub-populations. Metapopulation SIR model [2] fills this
gap by assuming the heterogeneity of sub-populations and using human mobility
to model the propagation between sub-populations. Metapopulation SIR model,
consists of three compartments for each sub-population:
St
n
for number of sus-
ceptible individuals,
It
n
for number of infectious individuals,
Rt
n
for the number
of recovered or deceased individuals of sub-population
n
at time
t
.
Pn
represents
the size of sub-population
n
which is assumed to be a constant number, where
Pn
=
St
n
+
It
n
+
Rt
n
.
β
is the rate of infection, and
γ
is the rate of recovery and
mortality. Furthermore, it uses
hnm
to represent the epidemic propagation from
sub-population (also called patch)
n
to
m
. The original metapopulation SIR
model [2] is shown as follows:
dSt+1
n
dt =−β·St
n
N
X
m=1
(hmn
Pm
+hnm
Pn
)It
m
dIt+1
n
dt =β·St
n
N
X
m=1
(hmn
Pm
+hnm
Pn
)It
m−γ·It
n
dRt+1
n
dt =γ·It
n
(2)
In this study, we model population of each region as sub-population in metapop-
ulation SIR model. So, the
hnm
can be represented by human mobility between
regions. Because of different characteristics of regions, policy changes with time
and so on, there is spatio-temporal heterogeneity of epidemic spread. In our
model,
β
,
γ
and
hnm
are assumed to vary over time and regions. In addition, to
prevent
β
to be extremely small and make it be in a relatively stable magnitude,
St
n
is omitted from the equations. Thus, we extend the original metapopulation
SIR in Eq. 2 as follows:
dSt+1
n
dt =−βt+1
n
N
X
m=1
(ht+1
mn
Pm
+ht+1
nm
Pn
)It
m
dIt+1
n
dt =βt+1
n
N
X
m=1
(ht+1
mn
Pm
+ht+1
nm
Pn
)It
m−γt+1
n·It
n
dRt+1
n
dt =γt+1
n·It
n
(3)

MepoGNN: Metapopulation Epidemic Graph Neural Networks 7
With predicted
βt+1
n
,
γt+1
n
and
Ht+1
(the epidemic propagation matrix formed
by {ht+1
nm |n, m ∈ {1,2, ..., N }}), S,I,Rcan be updated iteratively:
[St
n, It
n, Rt
n]Eq.(3)
−−−−−−−−−−→
βt+1
n,γt+1
n,Ht+1
[St+1
n, It+1
n, Rt+1
n] (4)
The final prediction output of daily confirmed cases can be formed as:
ˆyt+1
n=βt+1
n
N
X
m=1
(ht+1
mn
Pm
+ht+1
nm
Pn
)It
m
ˆ
Y=


ˆyt+1
1. . . ˆyt+Tout
1
.
.
.....
.
.
ˆyt+1
n. . . ˆyt+Tout
n


N×Tout
(5)
4.2 Spatio-Temporal Module for Epidemiological Parameters
Spatio-temporal module takes the node input features
X ∈ RN×Tin×C
and
the weighted adjacency matrix
A∈RN×N
as input and output the predicted
parameters
β∈RN×Tout
and
γ∈RN×Tout
. We use the spatio-temporal layer
(ST layer) combining Gated TCN and GCN (same as in GraphWaveNet [23])
to capture the spatio-temporal dependency. Gated TCN [24] is used to capture
temporal dependency:
Ql=g(Θ1?Zl+b1)σ(Θ2?Zl+b2) (6)
where
Zl
is input of
l
-th layer,
Θ1
and
Θ2
are temporal convolution kernels, b
1
and b
2
are biases,
g
(
·
) is tanh activation function for output,
σ
(
·
) is sigmoid
function to form the gate,
?
is convolution,

is element-wise product. Next,
we model the regions and the interactions between regions as a graph and use
diffusion graph convolution [22,23] to capture the spatial dependency:
Pf=A/rowsum(A),Pb=AT/rowsum(AT) (7)
˜
Zl=
K
X
k=0
Pk
fQlW1+Pk
bQlW2(8)
where A
∈RN×N
is weighted adjacency matrix,
Pf
is forward transition matrix,
Pbis backward transition matrix, ˜
Zlis output of l-th layer.
Multiple ST layers can be stacked to capture the spatio-temporal dependency
in different scales. We use a gated dense connection to bridge different ST layers.
It can extract important information from previous ST layers and pass it to next
layer:
Dl=(X,if l= 1,
Dl−1+Zl,otherwise. (9)

8 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
Zl+1 =(X,if l= 0,
˜
Zlσ(˜
Zl) + Dl(1 −σ(˜
Zl)),otherwise. (10)
where
Dl
stores the information from previous layers. Then, we concatenate the
output from different layers through skip connections to fuse the information
of different scales. Finally, the parameters
β∈RN×Tout
and
γ∈RN×Tout
are
produced through two fully connected layers, respectively.
Fig. 3: Two types of graph learning: Adaptive and Dynamic.
4.3 Graph Learning Module for Epidemic Propagation
There are two different graphs used in metapopulation SIR module and spatio-
temporal module, respectively. Unlike the trivial method which input two fixed
graphs to each module separately, we make two modules share a single learnable
graph. With the shared learnable graph, the spatial dependency used in spatio-
temporal module would be consistent with epidemic propagation in metapopula-
tion SIR module which can improve the interpretability of our model. Furthermore,
the parameters of graph learning module can be updated by gradients from both
spatio-temporal module and metapopulation SIR module which make learned
graph more realistic.
As shown in Fig. 3, there are two types of graph learning module to deal
with different input data. The first type is adaptive graph learning module which
takes the static flow data (e.g., commuter survey data) as input. Intuitively, we
initialize an adaptive graph
G
with static flow matrix Uand make it learnable
through training. Then, the adaptive graph can be output to spatio-temporal
module (Eq. 7) as
A∈RN×N
and to metapopulation SIR module (Eq. 3) as
H ∈ RN×N×1
(which means we use same
hnm
for all timesteps). The second
type is dynamic graph learning module which takes the dynamic OD flow tensor
as input. Although the OD flow and epidemic spread status are both dynamic,
but they are not necessarily one-to-one temporally corresponding. Considering
the delayed effect, influence of mobility on epidemic spread can be seen as a

MepoGNN: Metapopulation Epidemic Graph Neural Networks 9
weighted average of the given past values (
Tin
days). So, we initialize a learnable
time weight matrix
L∈RTout×Tin
and normalize it as
˜
L
through a softmax
function. The normalized time weight matrix can map the historical dynamic
flow
Ot−(Tin−1):t∈RN×N×Tin
to its influence on future epidemic spread. The
output of
Ht+1:t+Tout ∈RN×N×Tout
and
A∈RN×N
can be calculated as follows:
˜
L=Sof tmax:,j (L) (11)
Ht+1:t+Tout =˜
LOt−(Tin−1):t,A=PTout
i=1 Ht+i
Tout
(12)
Why propose two types of graph learning? Dynamic graph learning
module can illustrate the dynamic change of epidemic propagation. But it re-
quires dynamic flow data which is not available in most cases. To improve the
applicability of our model, we propose adaptive graph learning module to address
this problem. With two types of graph learning module, our model can handle
different situations of data availability in the best way possible.
5 Experiment
5.1 Data
We set 47 prefectures of Japan and 2020/04/01
∼
2021/09/21 (539 days) as our
study area and time period, respectively. The number of daily confirmed cases and
cumulative cases and deaths are collected from the NHK COVID-19 database
3
.
The number of recovered cases is collected from Japan LIVE Dashboard
4
[25]
(original data source is from Ministry of Health, Labour and Welfare, Japan).
The population of each prefecture is collected from 2020 census data. With
above-mentioned data, daily
S
,
I
,
R
of each prefecture can be calculated. Apart
from the number of daily confirmed cases, the input node features also include
daily movement change, the ratio of daily confirmed cases in active cases, and
dayof week
. The movement change data is collected from Facebook Movement
Range Maps
5
. It records the change of people movement range compared to a
baseline period. Because it is not provided at prefecture level, we use population
weighted average to get data at prefecture level. The input static flow data for
adaptive graph learning module is the number of commuters between prefectures,
which is collected from 2015 census data. The input dynamic flow data for
dynamic graph learning module is the daily OD flow data among 47 prefectures,
which is generated from human GPS trajectory data provided by Blogwatcher
Inc.. To mitigate the spatio-temporal imbalance in our data, we use stay put
ratio (ratio of people staying in a single location all day) in Facebook Movement
Range Maps to get the ratio of active users and use it to normalize the OD flow.
Finally, the input features of 47 prefectures are generated as a (539, 47, 4) tensor,
the static flow is a (47, 47) matrix, and the dynamic flow is a (539, 47, 47) tensor.
3https://www3.nhk.or.jp/news/special/coronavirus/data/
4https://github.com/swsoyee/2019-ncov-japan
5https://data.humdata.org/dataset/movement-range-maps

10 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
5.2 Setting
The input time length
Tin
and output time length
Tout
are both set to 14
days which means we use two-week historical observations to do the two-week
prediction of daily confirmed cases. Then, we split the data with ratio 6:1:1 to
get training/validation/test datasets, respectively. The fifth wave of infection in
Japan is included in test dataset to test the model performance on a real outbreak
situation. During training, we use the curriculum learning strategy [26] which
increases one prediction horizon every two epochs starting from one day ahead
prediction. The batch size is set to 32. The loss function is set as MAE (Mean
Absolute Error). Adam is set as the optimizer, where the learning rate is 1e-3 and
weight decay is 1e-8. The training algorithm would either be early-stopped if the
validation error did not decrease within 20 epochs or be stopped after 300 epochs.
PyTorch is used to implement our model. Then experiments are performed on a
server with four 2080Ti GPUs. Finally, we evaluate the performance of model
on 3 days, 7 days, 14 days ahead prediction and overall 14 steps prediction. The
four metrics are used to qualify the performance: RMSE (Root Mean Square
Error), MAE (Mean Absolute Error), MAPE (Mean Absolute Percentage Error)
and RAE (Relative Absolute Error). To mitigate the influence of randomness,
we perform 5 trials for each model and calculate the mean and 95% confidence
interval of results. The used random seeds are 0, 1, 2, 3, 4.
5.3 Evaluation
We implement three classes of baselines to compare and evaluate our model on
epidemic prediction task:
Mechanistic Models: (1) SIR [11]. SIR model is one of most basic com-
partmental models in epidemiology. We use optimized
β
and
γ
of each regions
to produce the prediction. (2) SIR(Copy). Because of weekly periodicity, we
copy the
β
and
γ
of last week to produce the prediction. (3) MetaSIR [2].
Metapopulation SIR model considers the heterogeneity of sub-populations and
models the interaction between sub-populations. We use the commuter survey
data as
H
and optimize
β
and
γ
for each region to produce the prediction. (4)
MetaSIR(Copy). We copy the
β
and
γ
of last week to produce the prediction.
Spatio-temporal Deep Learning Models: (5) STGCN [21]. STGCN is
one of the earliest models which applies GCN and TCN to do spatio-temporal
prediction. (6) DCRNN [22]. DCRNN proposes a variant of GCN, called
diffusion convolution and combines it with gated recurrent unit (GRU) to build
a spatio-temporal prediction model. (7) GraphWaveNet [23]. GraphWaveNet
proposes an adaptive learnable graph and uses GCN and TCN to capture spatio-
temporal dependency. (8) MTGNN [26]. MTGNN uses a graph learning module
to learn spatial correlation and fuse different spatial hops and different TCN
kernels to enhance the model capacity. (9) AGCRN [27]. AGCRN uses GCN
and GRU along with a graph learning module and a node adaptive parameter
learning module to capture spatio-temporal dependency.

MepoGNN: Metapopulation Epidemic Graph Neural Networks 11
Table 1: Performance Comparison with Baselines
3 Days Ahead 7 Days Ahead
Model RMSE MAE MAPE RAE RMSE MAE MAPE RAE
SIR 429.4±23.2 153.9±5.2 83.8±0.7 0.47±0.02 507.5±29.6 191.4±7.7 111.4±3.8 0.57±0.02
SIR(Copy) 248.1 97.4 57.4 0.29 318.5 127.1 67.2 0.38
MetaSIR 336.0±21.6 126.8±3.5 72.2±0.9 0.38±0.01 429.8±25.5 166.9±3.7 92.9±0.8 0.50±0.01
MetaSIR(Copy) 236.5 92.2 54.1 0.28 307.6 120.0 62.7 0.36
STGCN 375.6±18.8 118.6±10.8 45.3±2.8 0.36±0.03 381.1±17.7 128.0±6.6 52.5±3.0 0.38±0.02
DCRNN 305.0±9.8 89.3±4.4 37.3±0.7 0.27±0.01 323.8±15.9 107.6±5.3 47.3±1.4 0.32±0.02
AGCRN 223.5±28.5 80.0±7.8 56.6±13.2 0.24±0.02 253.1±37.7 97.9±7.6 60.8±10.1 0.29±0.02
GraphWaveNet 223.8±46.6 70.6±11.7 35.4±1.2 0.21±0.04 259.9±52.2 89.2±15.2 42.3±1.5 0.27±0.05
MTGNN 297.6±19.2 102.4±6.7 40.6±0.8 0.31±0.02 363.5±37.9 130.9±13.1 49.1±1.7 0.39±0.04
CovidGNN 261.9±55.5 88.4±16.7 43.3±3.8 0.27±0.05 305.4±70.6 116.5±23.8 60.9±5.3 0.35±0.07
ColaGNN 221.7±40.7 72.7±7.2 38.9±1.5 0.22±0.02 300.6±61.2 109.4±16.4 49.3±1.5 0.33±0.05
MepoGNN(Adp) 141.0±7.2 54.3±2.3 34.9±0.8 0.16±0.01 174.6±10.1 69.7±4.2 41.4±1.6 0.21±0.01
MepoGNN(Dyn) 135.9±17.8 52.7±4.6 34.2±0.7 0.16±0.01 160.6±4.5 67.6±1.2 41.7±0.9 0.20±0.00
14 Days Ahead Overall
Model RMSE MAE MAPE RAE RMSE MAE MAPE RAE
SIR 890.2±83.8 314.5±16.9 228.3±11.8 0.94±0.05 595.0±43.5 210.0±9.2 128.2±4.7 0.63±0.03
SIR(Copy) 835.5 332.6 183.2 1.00 539.1 190.2 102.7 0.57
MetaSIR 766.1±58.5 279.1±8.2 177.4±4.5 0.84±0.02 500.4±33.9 182.1±4.4 104.9±1.3 0.55±0.01
MetaSIR(Copy) 786.4 302.7 161.9 0.91 503.7 175.6 92.7 0.53
STGCN 430.2±15.8 159.4±6.0 74.7±3.7 0.48±0.02 389.5±7.9 132.0±2.9 55.6±2.4 0.40±0.01
DCRNN 377.9±11.1 146.0±5.0 69.5±4.0 0.44±0.01 335.0±11.8 112.5±4.5 49.5±1.3 0.34±0.01
AGCRN 390.4±105.8 149.0±11.4 88.0±12.8 0.45±0.03 322.7±136.7 108.0±9.9 67.9±15.6 0.32±0.03
GraphWaveNet 389.8±20.8 144.4±7.3 60.2±4.2 0.43±0.02 294.7±40.9 100.1±11.1 44.7±1.4 0.30±0.03
MTGNN 443.5±15.4 168.3±8.1 68.0±2.9 0.50±0.02 363.2±20.5 130.0±8.3 50.7±1.6 0.39±0.03
CovidGNN 414.7±59.8 177.4±15.9 111.2±6.6 0.53±0.05 329.6±59.8 124.2±19.2 66.9±4.2 0.37±0.06
ColaGNN 388.3±23.2 153.4±10.2 75.5±10.8 0.46±0.03 310.7±31.4 110.2±7.2 51.9±3.7 0.33±0.02
MepoGNN(Adp) 261.1±16.0 105.1±7.3 60.1±3.2 0.32±0.02 196.2±11.3 75.4±4.7 44.0±1.6 0.23±0.01
MepoGNN(Dyn) 253.2±7.5 107.0±3.0 62.0±2.0 0.32±0.01 186.1±5.0 74.3±2.0 44.4±0.8 0.22±0.01

12 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
Table 2: Ablation Study
Graph Model Mean RMSE Mean MAE Mean MAPE Mean RAE
Adaptive w/o glm 209.51±22.70 81.85±6.69 47.51±2.62 0.25±0.02
w/o propagation 203.23±24.70 82.05±8.05 45.84±1.68 0.25±0.02
w/o SIR 318.05±16.30 108.53±5.26 46.07±0.53 0.33±0.02
MepoGNN 196.16±11.33 75.45±4.65 44.02±1.55 0.23±0.01
Dynamic w/o glm 194.50±17.65 76.84±6.04 43.63±1.59 0.23±0.02
w/o propagation 200.55±17.00 80.73±5.54 45.16±1.24 0.24±0.01
w/o SIR 290.78±33.92 102.00±9.93 45.79±1.61 0.31±0.03
MepoGNN 186.07±4.99 74.30±1.99 44.43±0.77 0.22±0.01
GNN-based Epidemic Models: (10) CovidGNN [10]. CovidGNN is
one of the earliest GNN-based epidemic models. It embeds temporal features on
node and uses GCN with skip connections to capture spatial dependency. (11)
ColaGNN [9]. ColaGNN uses the location-aware attention to extract spatial
dependency and uses GCN to integrate the spatio-temporal information.
Performance Evaluation: In Table 1, we compare the performance on three
different horizons and overall performance for multi-step prediction among the
above-mentioned three classes of baseline models and proposed MepoGNN with
two types of graph learning module. Generally, the spatio-temporal deep learning
models and GNN-based epidemic models outperform the mechanistic models,
especially for long horizons. Among all baseline models, GraphWaveNet gets the
best performance. However, our proposed two MepoGNN models get the very
significant improvement over all baseline models. For two types of graph learning
module, dynamic one gets slightly better performance than adaptive one. Fig.
4 compares the 7 days ahead prediction results of Tokyo and Hyogo of the top
two baseline models and MepoGNN model with dynamic graph learning module.
From the prediction results, GraphWaveNet and ColaGNN can not produce
accurate predictions for high daily confirmed cases during the outbreak. This
phenomenon could be explained by different data distributions of daily confirmed
cases in training dataset and test dataset. The test dataset covers the period of
fifth epidemic wave in Japan which is much more severe than previous ones. Deep
learning models have difficulty to predict these high daily confirmed cases that
never happened before the fifth wave. However, with the help of metapopulation
SIR module, our proposed MepoGNN model can handle this problem and make
significantly better prediction for unprecedented surge of cases. This capability is
very crucial for a trustworthy epidemic forecasting model.
Ablation Study: To demonstrate the effect of different components of our
model, we conduct an ablation study for MepoGNN models with two different
graph learning modules, respectively. The variants are as follows: (1) w/o glm:
Remove the graph learning module of MepoGNN model; (2) w/o propagation:
Remove the metapopulation propagation from metapopulaiton SIR module
(which means metapopulation SIR model is reduced to SIR model); (3) w/o

MepoGNN: Metapopulation Epidemic Graph Neural Networks 13
SIR: Remove the metapopulation SIR module completely. Table 2 demonstrates
that all three components can bring significant boost of performance for our
model. Particularly, it is easy to find that the biggest performance drop happens
when removing the metapopulation SIR module. Because the metapopulation SIR
module enables the capability of MepoGNN model to handle the unprecedented
surge of cases.
Fig. 4: Predicted daily confirmed cases of Tokyo and Hyogo with horizon=7.
Fig. 5: 7-day moving average of predicted βof Tokyo with horizon=7.

14 Q. Cao, R. Jiang, C. Yang, Z. Fan, X. Song, R. Shibasaki
Fig. 6: Learned time weight matrix in dynamic graph learning module.
5.4 Case Study
The final output of MepoGNN model is fully produced by metapopulation SIR
module. It brings significant interpretability for our model. We conduct an analysis
for the predicted parameters of metapopulation SIR module to demonstrate the
interpretability. As shown in Fig. 5, we plot weekly average of predicted
β
of Tokyo
at 7 days ahead horizon in validation and test dataset and label major events
and policy changes on timeline.
β
starts to increase when state of emergency
ends and starts to decreases when state of emergency starts.
β
rapidly increases
during Tokyo Olympics, and decreases after it. It demonstrates the predicted
β
is consistent with reality. Fig. 6 shows the learned time weight matrix of
dynamic graph learning module. The most significant time lag of mobility effect
on epidemic spread is 22 days. This result is consistent with a public health
research [28] which states that the effective reproduction number significantly
increased 3 weeks after the nightlife places mobility increased in Tokyo. Although
the used indicator is different from our research, the mechanisms behind time
lag could be similar. Fig. 7 shows the learned graph of adaptive graph learning
module and the difference between it and commuter graph. The learned adaptive
mobility graph keeps the major structure of commuter graph. And the minor
change from initialization can reflect the difference between commuter graph and
spatial epidemic propagation.
6 Conclusion
Since the outbreak of COVID-19, epidemic forecasting has become a key research
topic again. In this study, we propose a novel hybrid model for epidemic forecasting
that incorporates spatio-temporal graph neural networks and graph learning
mechanisms into metapopulation SIR model. Our model can not only predict

MepoGNN: Metapopulation Epidemic Graph Neural Networks 15
Fig. 7: Learned adaptive mobility graph of the 47 prefectures of Japan with log
transformation (left) and its difference with static commuter graph (right).
the number of confirmed cases but also explicitly learn the time/region-varying
epidemiological parameters and the underlying epidemic propagation graph from
heterogeneous data in an end-to-end manner. Then, we evaluate our model by
using real COVID-19 infection data and big human mobility data of 47 prefectures
in Japan. The evaluation results demonstrate the superior performance as well
as the high reliability and interpretability of our model.
Acknowledgment
This work was partially supported by JST SICORP Grant Number JPMJSC2104.
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