Fig 1.
(a) Adjacency matrix, which is undirected with no weights, has been used so far. (b) Weighted directed matrix, originating from commuting data, includes weights and directions to assess infectious disease characteristics.
Fig 2.
The model includes sequence-to-sequence combinations of a diffusion GCN and gated recurrent unit (GRU) with uncertainty estimation. We feed the historical time series of patient numbers into the encoder. Next, we use its final states to initialize the decoder. The decoder generates a prediction from previous ground-truths or the values predicted by the model using scheduled sampling. Additionally, our model applies the predicted values to our uncertainty estimation method and then outputs the prediction interval.
Table 1.
Main notations.
Table 2.
Regional prediction model performances (averaged across all 47 prefectures in Japan).
Fig 3.
Visualization of the weights of learned localized filters of Eq (2) for (a) GCN+Seq2seq w/ PF, (b) GCN+Seq2seq w/ DD, and (c) GCN+Seq2seq w/ AD against the prediction target node (Nara prefecture, as shown by a star).
The colors represent the weights, i.e., strength of influence of each prefecture on the prediction of the target prefecture. The red prefectures are given assigned larger weights, i.e., they contribute significantly for to predicting the epidemics of in the target prefecture, while blue prefectures are given assigned smaller weights. Note that most prefectures are represented in white for visibility, as their weights are less than 5% of the maximum.
Table 3.
Improvement percentage of our predictive performance compared with LSTM in terms of MAE in the five most and least improved prefectures.
Lower values indicate greater improvement because a lower MAE indicates better performance.
Fig 4.
Prefecture maps that illustrate the improvements of prediction accuracy measured by MAE in each prefecture, where the improvement ratios of GCN+Seq2seq w/ PF against LSTM are represented by colors.
Red denotes improved prefectures and blue denotes degraded prefectures. Prefectures enclosed in red and blue frames denote the five best and worst prefectures in each year, respectively. The small square at the corner of each map shows Okinawa prefecture.
Fig 5.
Time series for Okayama prefecture: (a) two weeks in advance, (b) three weeks in advance, (c) four weeks in advance, and (d) five weeks in advance prediction time series in Okayama.
The blue and green dotted lines indicate the prediction values of compared models. The red line indicates the prediction values of the proposed GCN+Seq2seq w/ PF model. The black Line indicates the actual influenza patients.
Table 4.
Average bandwidth and empirical coverage of the 95% prediction interval found using the proposed method and Zhu’s method.
Fig 6.
Time series with prediction interval of influenza patients (black line).
Predictive values of the two weeks in advance prediction by our proposed model (red line) in the Okayama prefecture. Prediction intervals by (a) Zhu’s method and (b) proposed method. Light blue and dark blue sections show the 95% and 50% prediction intervals, respectively.