Learning epidemic trajectories through Kernel Operator Learning: from modelling to optimal control

TL;DR

This work addresses the need for fast, data-driven forecasting and scenario analysis of epidemic trajectories under NPIs, without heavy calibration of traditional compartmental models. It introduces Kernel Operator Learning with two surrogates, $KOL$-$m$ and $KOL$-$\partial$, and evaluates their performance using Neural Tangent Kernels among others to approximate the state or its derivative as a function of the control $u(t)$, yielding a closed-form surrogate $\bar{\mathcal{G}}(u)(t)$ guided by RKHS theory. Compared to a neural-network-based model-learning method, KOL offers substantially faster training and often superior generalization across SIS, SIR, SEIRD dynamics, and enables rapid optimal-control analyses. The results show that KOL surrogates can reproduce key optimal-control metrics (e.g., eradication times, total infections) with costs comparable to or better than full-model benchmarks, supporting their use for fast policy analyses in epidemiology.

Abstract

Since infectious pathogens start spreading into a susceptible population, mathematical models can provide policy makers with reliable forecasts and scenario analyses, which can be concretely implemented or solely consulted. In these complex epidemiological scenarios, machine learning architectures can play an important role, since they directly reconstruct data-driven models circumventing the specific modelling choices and the parameter calibration, typical of classical compartmental models. In this work, we discuss the efficacy of Kernel Operator Learning (KOL) to reconstruct population dynamics during epidemic outbreaks, where the transmission rate is ruled by an input strategy. In particular, we introduce two surrogate models, named KOL-m and KOL-$\partial$, which reconstruct in two different ways the evolution of the epidemics. Moreover, we evaluate the generalization performances of the two approaches with different kernels, including the Neural Tangent Kernels, and compare them with a classical neural network model learning method. Employing synthetic but semi-realistic data, we show how the two introduced approaches are suitable for realizing fast and robust forecasts and scenario analyses, and how these approaches are competitive for determining optimal intervention strategies with respect to specific performance measures.

Figures (13)

• Figure 1: Kernel Operator Learning (KOL) diagram. Operating among RKHS justifies the reduction of the problem to learning the behaviour of the vector-value function $f^\dagger$, operating between input/output observations (cf. Appendix \ref{['app:kolDer']}).
• Figure 2: $SIR$, $SIS$, $SIRD$ and $SEIRD$ compartmental models. Each model is provided with its respective basic reproduction number ($\mathcal{R}_0$) and the reproduction number depending on the control ($\mathcal{R}_u$).
• Figure 3: Examples of control functions employed to generate the training-testing dataset for the KOL methods.
• Figure 4: ($SIS$ model) Comparison of the prediction errors over 100 control functions with KOL methods trained on 20 batches of size 500, where the control functions are chosen in the mixed training dataset. Bullet points represent outliers whose prediction error is outside the 1.5 IQR (length of the whiskers) of the set of simulations.
• Figure 5: ($SIR$ model) Comparison of the prediction errors over 100 control functions with KOL methods trained on 20 batches of size 500, where the control functions are chosen in the mixed training dataset. Bullet points represent outliers whose prediction error is outside the 1.5 IQR (length of the whiskers) of the set of simulations.

Theorems & Definitions (1)

• Remark : On the positivity preserving property of KOL-m