Stochastic Control of a SIR Model with Non-linear Incidence Rate Through Euclidean Path Integral

TL;DR

This paper models COVID-19 spread using a stochastic SIR framework with non-linear infection dynamics, applying a path integral control method to optimize lockdown and vaccination strategies, supported by simulations and UK data analysis.

Contribution

It introduces a novel stochastic control approach using path integrals for a non-linear SIR model, providing insights into optimal pandemic interventions.

Findings

Higher diffusion coefficients stabilize susceptible and recovered populations.

Infection curves tend to ergodicity with increased stochasticity.

Model aligns with UK COVID-19 data from early 2021.

Abstract

This paper utilizes a stochastic Susceptible-Infected-recovered (SIR) model with a non-linear incidence rate to perform a detailed mathematical study of optimal lock-down intensity and vaccination rate under the COVID-19 pandemic environment. We use a Feynman-type path integral control approach to determine a Fokker-Plank type equation of this system. Since we assume the availability of information on the COVID-19 pandemic is complete and perfect, we can show a unique fixed point. A non-linear incidence rate is used because, it can be raised from saturation effects that if the proportion of infected agents is very high so that exposure to the pandemic is inevitable, then the transmission rate responds slower than linearity to the increase in the number of infections. The simulation study shows that with higher diffusion coefficients susceptible and recovery curves keep the downward trends while the infection curve becomes ergodic. Finally, we perform a data analysis using UK data at the beginning of 2021 and compare it with our theoretical results.