A Note on Quasi-stationary Distributions of Birth-death Processes and the Sis Logistic Epidemic

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martinez and Picco [4] studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, . . .}. In the case of a birth-death process, one can write down the components of Φ(ν) explicitly for any given distribution ν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre [7] concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.