Quasistationarity of continuous-time Markov chains with positive drift

We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity sets in. We will show how this 'quasi' stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of a dual chain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity for absorbing Markov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing us to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always a quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

Further results on the relationship between mu-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on mu-invariant and mu-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the mu-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between mu-invariant measures and quasi-stationary distributions is discussed. (C) 2000 Elsevier Science Ltd. All rights reserved.

Erectile dysfunction in general medicine practice: prevalence and clinical correlates

Erectile dysfunction (ED) is a common problem in general medical practice affecting especially the elderly and those with cardiovascular disease and diabetes mellitus, A study was undertaken by questionnaire distributed to consecutive adult male attendees at 62 general medical practices. 1240 completed questionnaires were available for analysis. The mean age of participants was 56.4 y (range 18 - 91 y). 488 men (39.4%) reported ED: 119 (9.6%) 'occasionally', 110 (8.9%) 'often', and 231 (18.6%) 'all the time' (complete ED). Among 707 men aged 40-69 y 240 (33.9%) reported ED and 84 (11.9%) had complete ED. The prevalence of complete ED increased with age, rising from 2.0% in the 40-49 y age group to 44.9% in the 70-79 y age group. Only 11.6% of men with ED had received treatment. Hypertension, ischaemic heart disease, peripheral vascular disease and diabetes mellitus were frequently associated with ED. 40% of diabetic men aged 60 y or older had ED all the time.

Preliminary experiences with a single-patient trials service in general practice

Objective: To pilot a single-patient trials (SPTs) service in general practice, designed to improve decision-making about long-term medications for chronic conditions. Design: 12-week within-patient, randomised, double-blind, placebo-controlled, crossover comparison of ibuprofen with paracetamol for osteoarthritis, involving three pairs of two-week treatment periods for each participating patient. Setting and patients: Patients attending an academic general practice with a clinical diagnosis of osteoarthritis, with pain of at least a month's duration severe enough to warrant consideration of long-term non-steroidal anti-inflammatory drug (NSAID) use. Main outcome measures: Pain and stiffness; measures of overall arthritis compared with previous fortnight; preference for NSAID at the end of each two-week treatment period; use of escape analgesia; side effects; and management changes as a result of the SPTs. Results: Eight of 14 patients completed SPTs. One was a clear responder to NSAIDs, five were non-responders, and two were indefinite. Of the five who were using NSAIDs before the SPT, two continued and three ceased using them. Clinically useful information assisted decision-making for all eight participants. Medication management changed for six. Conclusions: Single-patient trials can be successfully implemented in general practice and might be a valuable method for GPs to identify patients who respond to medication for chronic stable conditions such as osteoarthritis, in which individual response to medication is variable.