On Quasi-stationary Distributions of Stochastic SIS Epidemic Models

We study a stochastic SIS (Susceptible-Infected-Susceptible) epidemic model, in which individu-als in a population are divided into two classes: “Susceptible” to the infection of a disease, and “Infected” by the disease. We assume the population size N is fixed. The state process, which measures the num-ber of infected individuals, is modeled as a continuous time Markov chain (CTMC). The CTMC has an absorbing state 0, and it will eventually die out by reaching the absorbing state. However, it appears to be persistent over a long time period before extinction. Such behavior can be understood by the study of quasi-stationary distributions. In this thesis, our main results show that the CTMC SIS model has a unique quasi-stationary distribution, and when the basic reproduction number is greater than 1, as N → ∞, the quasi-stationary distribution converges to a dirac probability measure concentrated at the equilibrium point of a deterministic SIS model