Parameter estimation of epidemic spread in two-layer random graphs by classical and machine learning methods

TL;DR

This study addresses the problem of estimating the infection rate parameter $\tau$ in epidemic spread on two-layer random graphs by comparing classical maximum-likelihood estimation with XGBoost and CNN approaches. It uses SIR dynamics simulated via the Gillespie algorithm on graphs comprising a households layer and a second layer that is either scale-free or clique-based, with edge weight $w$ and recovery rate $\gamma=1$. The work contributes a detailed comparison of estimation accuracy across epidemic phases, analyzes the impact of training/test graph structure and additional features, and provides practical guidance on when to favor ML methods over classical estimators. Overall, XGBoost offers the strongest performance, CNN provides robustness at a higher computational cost, and ML methods particularly excel when structural information is incomplete.

Abstract

Our main goal in this paper is to quantitatively compare the performance of classical methods to XGBoost and convolutional neural networks in a parameter estimation problem for epidemic spread. As we use flexible two-layer random graphs as the underlying network, we can also study how much the structure of the graphs in the training set and the test set can differ while to get a reasonably good estimate. In addition, we also examine whether additional information (such as the average degree of infected vertices) can help improving the results, compared to the case when we only know the time series consisting of the number of susceptible and infected individuals. Our simulation results also show which methods are most accurate in the different phases of the epidemic.

Figures (7)

• Figure 1: Proportion of infected vertices for $n=5000$ with (a) scale-free graph as second layer ($p_{\rm u}=0.7$, $p_{\rm tr}=0.3$, $p_{\rm pa}=0$; each curve is the average of 8 trajectories) and (b) with the cliques of fixed size $9$ as second layer (each curve is the average of 35 trajectories)
• Figure 2: Root mean square error in the two-layer graph with scale-free graph as the second layer, as a function of time, according to estimates \ref{['eq:tauhat']}, \ref{['eq:etsio']} and \ref{['eq:etilde']}. ($n=5000$ vertices)
• Figure 3: The performance of the CNN architecture improves as measured by RMSE loss when trained on increasing amount of information about the development of the epidemic spread in time. In this setting information about the number of SI edges is known. The $\tau$ parameter is in the range $[0.3,0.6]$. The performance is validated on a sample of 260 time series for each datapoint.
• Figure 4: Comparing the performance of the CNN model as a function of the progression of the epidemic using different training strategies: 1 - training datapoints sampled with symmetric beta distribution; 2 - same sampling as 1, but larger dataset; 3 - sampling distribution is a beta distribution skewed towards larger values, to include more datapoints with longer time series; 4 - training dataset containing exclusively full-length time series; 5 - same dataset as 4 with more datapoints; 6 - training datapoints sampled with uniform distribution; 7 - for each timepoint a separate model was trained with initial segments of time series ending at each given timepoint. Only S,I,R data were used and the underlying network model had a second layer generated by the polynomial model.
• Figure 5: Left panel: The RMSE as the function of time for different $\tau$ values. $N=5000$ with relaxed complete cliques as second layer. Each curve is the average of $90$ trajectories. Right panel: Comparing the performance of the CNN model as a function of the progression of the epidemic for different values of $\tau$. Only S,I,R data were used and the underlying network model had a second layer of cliques.