Using a Layered Ensemble of Physics-Guided Graph Attention Networks to Predict COVID-19 Trends

Figures & data

Figure 1. SIRVC model: susceptible-infectious-recovered with vaccinations and inter-community interactions. An additional term moves susceptible individuals into the infectious group through interactions with external communities.

Table 1. Nomenclature.

Graph Attention Network
$\mathrm{l}\mathrm{e}\mathrm{v}$	level of layered graph: country, continent, or world
$\mathcal{V}$ and $\mathcal{E}$	vertex and edge set
$\mathbf{A}\in {\mathbb{R}}^{|\mathcal{V}|\times |\mathcal{V}|}$	weighted adjacency matrix defined by mobility
${\xi }_{u,v}$	the flight count from vertex $u$ to vertex $v$
$\mathrm{I}\mathrm{W}$, $\mathrm{O}\mathrm{W}$	GAT’s input and output window size (in days)
${\mathbf{x}}_v\in {\mathbb{R}}^F$	$F$-dimensional feature of vertex $v$
${\mathbf{h}}_v\in {\mathbb{R}}^F$	$F$-dimensional input hidden feature of vertex $v$
${\mathbf{y}}_v\in {\mathbb{R}}^{F{ }^{\mathrm{\prime }}}$	$F{ }^{\mathrm{\prime }}$-dimensional output of vertex $v$
$\mathrm{\ell }$	layer number of forward propagation of GAT
${\mathcal{L}}_{v,t}(\mathrm{\Theta })$	loss function corresponding to vertex $v$ at epoch $t$
$\mathrm{\Theta }$	the parameters for neural networks
$\rho \in \mathbb{R}$	weight of physical violation vs. GAT error
Pandemic physics model
$N$, $S$, $I$, $R$, and $V$	total, susceptible, infected, recovered, and
	vaccinated population within the community
$\beta \in \mathbb{R}$	the transmission rate constant
$\gamma \in \mathbb{R}$	the recovery rate constant
$\theta \in \mathbb{R}$	the probability of inter-community contact
$\eta \in \mathbb{R}$	the efficacy rate of the vaccine

Figure 2. Architecture and adaptive physics-guided loss function for GAT vertex.

Figure 3. Layered ensemble architecture for multi-scale pandemic prediction ranging from country through global level.

Table 2. Country-level GAT structure.

# vertices	163
# edges	3765
# features per vertex	42
# features per edge	1
# attention layers	2
# forward propagation layers	3

Figure 4. Country SIRVC-GAT prediction (proposed layered ensemble model: (a) USA; (b) UK.

Figure 5. Country SIRVC-GAT prediction with fixed loss: (a) USA; (b) UK.

Figure 6. Country SIRVC-GAT prediction without edge weight: (a) USA; (b) UK.

Figure 7. United States GAT model: final graph layer attention visualization. Edge color corresponds to min-max normalized edge attention.

Table 3. Performance comparison for country-level model variations.

United States
Model	MSE	% Red. MSE	RMSE	% Red. RMSE
STAN	4018014388	–	63388	–
Proposed GAT	92500552	97.70%	9618	84.83%
Fixed loss	1608996203	59.96%	40112	36.72%
No edge weights	2447913117	39.08%	49476	21.95%
United Kingdom
Model	MSE	% Red. MSE	RMSE	% Red. RMSE
STAN	38101859	–	6173	–
Proposed GAT	5821602	84.72%	2413	60.91%
Fixed loss	14371479	62.28%	3791	38.58%
No edge weights	8012915	78.97%	2831	54.14%

Figure 8. Peer learning (Algorithm 1) for the prediction of Covid-19 epidemics: (a) USA; (b) UK.

Figure 9. Middle ensemble layer predictions (Algorithm 2): Continent MLP trained with country SIRVC-GAT prediction: (a) Africa; (b) Asia; (c) Europe; (d) North America; (e) Oceania; (f) South America.

Figure 10. Final ensemble layer prediction: World MLP trained with continent MLP prediction (proposed ensemble model described by Algorithm 2) compared to 1) using randomized country grouping in the middle layer and 2) training without a middle/continent layer.

Table 4. Performance comparison for proposed layered ensemble (Algorithm 2) against baselines.

Model	MSE	% Inc. MSE	RMSE	% Inc. RMSE
Proposed Ensemble (Algorithm 2)	${9.75031}^{11}$	–	987437	–
Without continent layer	${2.78092}^{12}$	185.21%	1667609	68.89%
Randomly assigned countries	${2.96769}^{12}$	204.37%	1722700	74.46%