Fig 1.
The SFNN architecture with input features grouped according to feature type, (maximum) 3 hidden layers of (maximum) dimension 1201, and 1 output layer of dimension 1 for incidence forecasts. Number of hidden layers and hidden layer dimension differ by forecasting window.
Fig 2.
SFNN vs. TSIR model performance measured by Root Mean Squared Error (RMSE) of within-city-standardized log(incidence + 1).
(A) Within-city SFNN RMSE versus TSIR RMSE, colored by log(population), faceted by k-step ahead forecast. (B) Difference between the within-city-standardized RMSE for TSIR and the within-city-standardized RMSE for SFNN; loess regression curves are fitted.
Fig 3.
SHAP values uncover mechanism of spatial hierarchical spread.
The SHAP value measures the relative importance of the incidence of a core city (e.g., London) for making incidence prediction among cities/towns with different population sizes, which can be heuristically treated as the relative importance to the local transmission of measles in a particular city/town. Core city incidence lag features are shown to be more important when predicting incidence for less populous cities/towns. Specifically, the mean relative absolute SHAP value for each of the core city incidence lag features has an inverse relationship with log population. Cities and towns are categorized (on the x-axis) into 10 groups according to the quantiles of their population sizes.
Fig 4.
Test-set 52-step-ahead incidence predictions over time (A) and Inference of seasonal transmission rate (B) in London.
(A) TSIR-PINN test-set 52-step-ahead incidence predictions for London more closely match true incidence, when compared to those for Naive-PINN. (B) PINN parameter values are notably different between TSIR-PINN and Naive-PINN models over 2,500 epochs. The parameter v (black lines) correspond to the R0. Convergence is rapidly achieved when fitting the TSIR-PINN model, while convergence is less clear for the Naive-PINN model. More importantly, the TSIR-PINN model estimates an R0 of 26.8 which is broadly consistent with the literature, while the Naive-PINN estimates an R0 of 5.7.
Table 1.
52-biweek-ahead London measles incidence forecasting performance of TSIR-PINN and Naive-PINN measured by test-MAE and test-correlation.
TSIR-PINN outperforms Naive-PINN by both measures.
Fig 5.
Measles cases in England and Wales.
(A) Cities/towns are colored by log measles incidence on the first biweek of 1961. The England and Wales map is made with Natural Earth vector map data. (B) The seasonal measles trend is apparent across the four most populous cities in England and Wales from 1944 to 1965.