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A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic. (English) Zbl 1050.60076

The authors consider birth-death processes \(\{X(t): 0\leq t<\infty\}\) on the finite space \(\{0,1,\dots,N\}\). Based on work by P. A. Ferrari, H. Kesten, S. Martinez and P. Picco [Ann. Probab. 23, No. 2, 501–521 (1995; Zbl 0827.60061)] in which the existence of quasi-stationary and limiting conditional distributions were studied by characterizing quasi-stationary distributions as fixed points of a transformation \(\Phi\) on the space of probability distributions on \(\{1,2,\dots\}\), the authors give an explicit representation of the components of \(\Phi(v)\) for any given distribution \(v\). The explicit solution is used to show that the transformation \(\Phi\) preserves the likelihood ratio ordering between distributions. A conjecture by R. J. Kryscio and C. Lefèvre [J. Appl. Probab. 26, No. 4, 685–694 (1989; Zbl 0687.92012)] concerning the quasi-stationary distribution of the SIS logistic epidemic model can be shown. The results carry over to the infinite state-space case with some minor adjustments.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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References:

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