A stochastic model for COVID-19 with a highly effective vaccine

Abstract

We propose a stochastic model for the population dynamics of COVID-19 with vaccine. The model allows for waning immunity. We start off with a deterministic model in terms of ordinary differential equations (ODEs), which afterwards are stochastically perturbed to form a system of stochastic differential equations (SDEs). The ODE system and the SDE system have global positive solutions. We discuss the equilibrium points of the ODE system. For the SDE model we obtain a stability result in terms of almost sure exponential stability theorem for the disease-free equilibrium of the stochastic model. Our theoretical results are illustrated by numerical simulations.