Reservoir computing on epidemic spreading: A case study on COVID-19 cases

A reservoir computing based echo state network (ESN) is used here for the purpose of predicting the spread of a disease. The current infection trends of a disease in some targeted locations are efficiently captured by the ESN when it is fed with the infection data for other locations. The performance of the ESN is first tested with synthetic data generated by numerical simulations of independent uncoupled patches, each governed by the classical susceptible-infected-recovery model for a choice of distributed infection parameters. From a large pool of synthetic data, the ESN predicts the current trend of infection in $5\%$ patches by exploiting the uncorrelated infection trend of $95\%$ patches. The prediction remains consistent for most of the patches for approximately 4 to 5 weeks. The machine's performance is further tested with real data on the current COVID-19 pandemic collected for different countries. We show that our proposed scheme is able to predict the trend of the disease for up to 3 weeks for some targeted locations. An important point is that no detailed information on the epidemiological rate parameters is needed; the success of the machine rather depends on the history of the disease progress represented by the time-evolving data sets of a large number of locations. Finally, we apply a modified version of our proposed scheme for the purpose of future forecasting.

Figure 1

Training and testing scheme using the echo state network. Upper panel (light-red boxes): Input parts. Lower left panel (light blue box): Data set of the output data. Lower right panel (dark-blue box): Predicted-testing data.

Figure 2

Schematic of the ESN and signal variability. (a) Infection data inputs (red signal) fed into the machine. The dashed vertical line (at $t=t_r$) signifies a time limit of data point inputs to the machine. (b) ESN structure: input layer, reservoir, and output layer. The weights of the input and the reservoir once selected are kept fixed throughout the training and testing procedure. (c) Data output of targeted locations or patches (blue signals). Left parts of the dashed vertical lines ($t\le t_r$) are closely mapped with the machine-generated signals at the time of training. Right parts of the vertical lines are predicted data (red circles) from the machine at the time of testing the ESN.

Figure 3

(a)–(e) Snapshots of synthetically generated data versus machine-based data; (${\mathcal{I}}_R^n$) vs (${\mathcal{I}}_M^n$) plot for data taken at time points $t=7$, 14, 21, 28, and 35, respectively. Results of 50 patches are presented here. Five randomly chosen nodes are marked with squares, diamonds, pentagrams, triangles, and $\times \text{'}\mathrm{s}$. At $t=7$, ${\mathcal{I}}_M^n$ and ${\mathcal{I}}_R^n$ are correlated, as they lie on the diagonal line, signifying that the machine can predict the real data efficiently. (b) At $t=14$, two data points (diamond and pentagram) deviate slightly from their original counterparts. The error increases for a longer duration ($t=21$, 28, 35) of forecasting as shown in (c)–(e). However, most of the patches (green circles) lie on the diagonal line. (f)–(j) The infection trend of five randomly chosen nodes are shown for 5 weeks. Data generated by simulation of the SIR model are shown by thick blue lines. The machine-generated data closely predict infection in some of the patches (marked by triangles, $\times \text{'}\mathrm{s}$, and squares) up to 30–35 days. For two patches (marked by pentagrams and diamonds), the machine-generated data deviate after 10 days.

Figure 4

Prediction of COVID-19 data of 10 randomly chosen locations from world COVID-19 data sets. (a) Croatia: infection sharply increases (blue line) at the time of prediction (5 October to 26 October; shaded region). The machine closely predicts (red circles) the trend (blue line) in the shaded region. A zoomed-in version of the shaded region is presented directly below (second row). Similar scenarios are observed for (b) Germany, (e) Italy, and (j) West Bengal. The decreasing trends of (d) Israel, (f) Rajasthan, (g) Telengana, and (h) Tripura are also well captured by the ESN. The ESN also predicts the trend in (i) Uttar Pradesh, but for a shorter time. Interestingly, the machine prediction of the increasing and decreasing trend of infection in (b) Finland is closely matched by the real data.

Figure 5

Generalized approach to future forecasting of selected locations. This scheme enables prediction of the future trend of infection of the target locations for a duration $\tau$ time unit.

Figure 6

Forecasting the future trend of infection of 10 selected locations from 1 May 2021 to 14 May 2021. The gray region in each figure represents the machine-generated data for 100 realizations. The blue line shows the average of each gray region. Red crosses show the real data for the period 1 May 2021 to 6 May 2021.

Figure 7

Forecasting through the Gompertz curve in 10 randomly chosen regions. Four weeks of data are used to standardize the intrinsic parameters of the Gompertz function. Twenty-two days were forecast, from 5 October to 26 October 2020.