Shift, scale and restart smaller models to estimate larger ones: Agent based simulators in epidemiology

Abstract

Agent-based simulators (ABS) are a popular epidemiological modelling tool to study the impact of non medical interventions in managing epidemics [1], [2]. They accurately model a heterogeneous population with time and location varying, person specific interactions. Government policies such as partial and location specific lock downs, case isolation, home quarantine, school closures, partially opened workplaces, etc. are easily modelled and ABS allow flexibility to incorporate important pandemic developments over time including variants and vaccines. For accuracy, each person is modelled separately. This however may make computational time prohibitive when the city population and the simulated time are large. We observe that simply considering a smaller representative model and scaling up the output leads to inaccuracies. In this note we focus on the COVID- 19 pandemic and dig deeper into the underlying probabilistic structure of generic ABS to arrive at modifications that allow smaller models to give accurate statistics for larger ones. We exploit the observation that in the initial disease spread phase, the starting infections create a family tree of infected individuals more-or-less independent of the other trees and are modelled well as a multi-type super-critical branching process whose relative proportions across infected population types stabilize soon and thereafter are invariant to shifts in time. Further, for large city population, once enough people have been infected, the future evolution of the pandemic is closely approximated by its mean field limit with a random starting state. We build upon these insights to develop a shifted, scaled and restart based algorithm that accurately evaluates the ABS's performance using a much smaller model while carefully reducing the bias that may arise from just scaling. Our key contributions are: 1) we develop an algorithm by carefully exploiting the closeness of the underlying exposed/infected process (process of number exposed/infected of each type at each time) initially to a branching process, and then the normalised infection process (infection process divided by the city population) to its mean field limit, so that the output from the smaller model accurately matches the output from the larger model in a realistic city setting. 2) we provide theoretical support for the proposed approach through an asymptotic analysis where the population increases to infinity. For brevity, we conduct the analysis in a simpler and yet practically useful setting.