Quantitative Risk Stratification in Markov Chains with Limiting Conditional Distributions

Abstract

Limiting conditional distributions exist in most Markov models of progressive diseases and are well suited to represent risk stratification quantitatively. This framework can characterize patient risk in clinical trials and predict outcomes for other populations of risk.

Quasi-Stationary Distributions

“…For diffusions and other continuous-state processes, a good starting point is Steinsaltz and Evans [140] (but see also Cattiaux et al [22] and Pinsky [119]) and for branching processes there is an excellent recent review by Lambert [95,Section 3]. Whilst many issues remain unresolved, the theory has reached maturity, and the use of quasi-stationary distributions is now widespread, encompassing varied and contrasting areas of application, including cellular automata (Atman and Dickman [9]), complex systems (Collet et al [34]), ecology (Day and Possingham [41], Gosselin [63], Gyllenberg and Sylvestrov [68], Kukhtin et al [89], Pollett [122]) epidemics (Nåsell [106,107,108], Artalejo et al [6,7]), immunology (Stirk et al [141]), medical decision making (Chan et al [24]), physical chemistry (Dambrine and Moreau [37,38], Oppenheim et al [112], Pollett [121]), queues (Boucherie [17], Chen et al [25], Kijima and Makimoto [84]), reliability (Kalpakam and Shahul-Hameed [73], Kalpakam [74], Li and Cao [98,99]), survival analysis (Aalen and Gjessing [1,2], Steinsaltz and Evans [139]) and telecommunications (Evans [53], Ziedins [152]).…”

“…A chain is called α-transient if it is not α-recurrent. For later reference we also note at this point that an α-recurrent process is said to be α-positive if for some state i ∈ S (and then for all states i ∈ S) lim t→∞ e αt P ii (t)dt > 0, (24) and α-null otherwise. (Note that a finite absorbing Markov chain is always αpositive, in view of (11).)…”

Learn Quasi-Stationary Distributions of Finite State Markov Chain

“…Quasi-stationary distribution (QSD) is the long time statistical behavior of a stochastic process that will be surely killed when this process is conditioned to survive [ 1 ]. This concept has been widely used in applications, such as in biology and ecology [ 2 , 3 ], chemical kinetics [ 4 , 5 ], epidemics [ 6 , 7 , 8 ], medicine [ 9 ] and neuroscience [ 10 , 11 ]. Many works for rare events in meta-stable systems also focus on this quasi-stationary distribution [ 12 , 13 ].…”

Learn Quasi-stationary Distributions of Finite State Markov Chain

“…Quasi-stationary distribution (QSD) is the long time statistical behavior of a stochastic process that will be surely killed when this process is conditioned to survive [14,29]. This concept has been widely used in biology and ecology [11,21], chemical kinetics [15,18], epidemics [2,13,35], medicine [12] and neuroscience [6,22]. Many works for the rare events in meta-stable systems also focus on this important quantity [17,23].…”