Summary: In recent years, the incidence of infection with HIV among young homosexual men has been on the rise in developed countries. It is, therefore, of interest to study stochastic models of HIV/AIDS in which HIV infections become endemic in a population of homosexuals. Within a stochastic paradigm, the idea of an endemic equilibrium corresponds to the mathematical concept of a stochastic process converging in distribution to a stationary distribution. In [C. J. Mode and C. K. Sleeman, Math. Biosci. 180, 115–126 (2002; Zbl 1015.92037)], it was conjectured that in a Monte Carlo simulation experiment convergence to a stationary distribution of a Markov chain was being observed. In this paper, a proof of this conjecture is constructed under a reasonable sufficient condition within an abstract framework that resembles a general sub-critical branching process. The stationary distribution turns out to be a mixture of conditionally independent Poisson densities and the mixing measure is that underlying the population process. It is often difficult to decide in a Monte Carlo simulation experiment whether convergence to a stationary distribution is actually being observed, which sometime leads to controversies. Knowing that convergence to a stationary distribution does indeed occur, under a plausible sufficient in the class of models considered in the paper, will provide a firm mathematical basis for avoiding controversies. In the literature on stochastic models of HIV/AIDS, the class of models considered in this paper is often referred to as chain multinomial models.
For the entire collection see [Zbl 1141.92328].