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This paper utilizes a stochastic Susceptible-Infected-recovered (SIR) modelwith a non-linear incidence rate to perform a detailed mathematical study ofoptimal lock-down intensity and vaccination rate under the COVID-19 pandemicenvironment. We use a Feynman-type path integral control approach to determinea Fokker-Plank type equation of this system. Since we assume the availabilityof information on the COVID-19 pandemic is complete and perfect, we can show aunique fixed point. A non-linear incidence rate is used because, it can beraised from saturation effects that if the proportion of infected agents isvery high so that exposure to the pandemic is inevitable, then the transmissionrate responds slower than linearity to the increase in the number ofinfections. The simulation study shows that with higher diffusion coefficientssusceptible and recovery curves keep the downward trends while the infectioncurve becomes ergodic. Finally, we perform a data analysis using UK data at thebeginning of 2021 and compare it with our theoretical results.

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