Exponential convergence to a quasi-stationary distribution with applications to birth and death processes

Abstract

In this project we study the exponential convergence of Markov processes to quasi-stationary distributions (QSDs) with applications. Quasi-stationary distributions are useful when it comes to understanding the behavior of stochastic processes which appear to be persistent over a long time period before reaching extinction. A review of the concept of stationarity and ergodicity is given. Next quasi-stationarity is defined. A simple example that illustrates quasi-stationarity is considered- specifically the example of the finite state case. Finally, we choose a Corona Virus model, convert it to a birth and death process, then show that it converges to a particular QSD exponentially, we also choose the compartment of infected persons from the model and show that it is a branching process that also converges to a QSD over time.