Fig 1.
The framework of M-Graphormer.
In the output layer, FC represents the fully connected layer, and ReLU and Sigmoid represent the ReLU activation function and Sigmoid activation function respectively.
Fig 2.
Framework of Spatio-Temporal Layer.
A: Framework of Single-layer ST Extractor. B: Framework of Multi-layer ST Extractor.
Fig 3.
Flow diagram of the metapopulation model (12) showing interactions between patch i and patch j.
Fig 4.
In-degree heat map of Chinese provinces and daily new cases in Jilin and Shanghai in April 2022.
A: Heat map of in-degree in April 2022 for each province in China. B: Daily new cases in Shanghai and Jilin, April 2022.
Fig 5.
Framework of ICTI and dynamic flow prediction.
A: Intra-city travel intensity prediction. B: Dynamic flow (weighted adjacency matrix A) prediction. Here softmax represents the softmax activation function.
Fig 6.
Fitting of daily new case data and time-dependent contact rate inference in Hubei, Beijing, Shanghai, Hunan, Guizhou, and Shanxi provinces based on M-Graphormer.
Fig 6A-6C and 6G-6I represent the fitting results of daily new cases in Hubei, Beijing, Shanghai, Hunan, Guizhou, and Shanxi respectively, where the cyan ‘×’ represents the real reported data, and the red (the green) dotted line indicates that the model inputs 3 days (7 days) of historical data to predict daily new cases in the next 3 days (days). Fig 6D-6F and 6J-6L represent the time-dependent contact rate inferences for Hubei, Beijing, Shanghai, Hunan, Guizhou, and Shanxi, respectively. The highlighted areas in the two types of graphs correspond to the specific periods that we have drawn.
Fig 7.
Map of cumulative daily root-mean-square errors for each province in China in the test set.
The map was drawn in python using the pyecharts package (https://github.com/pyecharts).
Fig 8.
Effective reproduction number curves for each province in China estimated by Eq 14.
The x-axis represents the dates, from September 23, 2022 to December 1, 2022.
Table 1.
Ablation study.
Table 2.
Definition of commonly used parameters of the rate function.
Table 3.
Definition of commonly used parameters of the rate function.
Fig 9.
Contact rate inference, rate function fitting and root mean square error in Beijing, Hunan and Shanxi.
A-C show the inference and fitting results of time-dependent contact rate in Beijing, Hunan and Shanxi, where the blue triangles represent the M-Graphormer inference results on the contact rate, and the dotted lines and realisations represent the rate function group (30) fitting results. D-F show the root mean square errors corresponding to the fitting results of the rate function group (30) in Beijing, Hunan and Shanxi.
Fig 10.
Arrival times of different thresholds κ for the two systems under different parameter groups.
The abscissa represents the serial number of different parameter groups. A-C represent the arrival time from Shanxi to Guangdong at thresholds κ = 1, 10, 100, respectively. D-F represent the arrival time from Shanxi to Beijing at thresholds κ = 1, 10, 100, respectively. The blue ‘×’ represents the arrival time of the uniform system, and the red solid circle represents the arrival time of the non-uniformity system. The insets in A-F show the minimum arrival time. The x-axis denotes the different parameter group numbers and also the numbering of the date sequence in the test set.
Table 4.
Prediction of the actual time of arrival for both systems.