A note on heroin epidemics Giuseppe Mulone a, * , Brian Straughan b a Dipartimento di Matematica e Informatica, Città Universitaria, Viale A. Doria, 6, 95125 Catania, Italy b Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK article info Article history: Received 26 June 2008 Received in revised form 14 January 2009 Accepted 26 January 2009 Available online 6 February 2009 Keywords: Heroin epidemics Equilibria Stability abstract We show that the steady states of the White and Comiskey [E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci. 208 (2007) 312–324.] model of heroin epidemics are stable. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Treatment of heroin users or users of other drugs such as crack cocaine is a costly procedure and is a major burden on the health system of any country. Likewise, treatment of individuals with alcohol problems is a major issue, cf. Hay et al. [3] and Deacon et al. [2]. As Sir Liam Donaldson (see Deacon et al. [2]) notes, ‘alco- hol misuse represents a substantial cost the NHS (the National Health Service, UK) of 1.7 billion pounds sterling. We recognise that more progress needs to be made towards reducing harmful drinking and its impact as a contributor to ill health and inequali- ties in the English regions’. Mathematical modelling is a means to provide a predictive tool for how classes of drug takers behave, and as such, could hopefully become a useful device to aid specialist teams in devising treatment strategies. Two very interesting models have recently been proposed. One for treating heroin users, proposed by White and Comiskey [12], and a similar one for those with alcohol problems, see Sánchez et al. [10] (see also the account of Sánchez et al. [10] work by Ben- edict [1]). Both models divide the mathematical problem into three classes, namely susceptibles, heroin users or alcoholics, and heroin users or alcoholics undergoing treatment. In fact, the two types of model are very similar. The one of Sánchez et al. [10] differs from that of White and Comiskey [12] only in that she assumes the same death/removal rate for each of the three classes, whereas White and Comiskey [12] correctly allow the drug users and those in treatment to have enhanced death rates. It transpires that two key terms are the probability of becom- ing a drug user per unit time, and the probability of a drug user in treatment relapsing to untreated use per unit time. White and Comiskey [12] denote these, respectively, by b 1 and b 3 . The value for b 1 is usually relatively small. For example, the prevalence rate (number of heroin users per 1000 population) in Dublin is approximately 15. In the North East regions of England, around Durham, the prevalence rate varies from 4.385 in rural Northum- berland during 2004/5 to 21.32 in Middlesbrough during 2004/5, taken over samples aged 15–63, see Hay et al. [3]. The corre- sponding figures in 2005/6 are 3.83 in Northumberland and 24.86 in Middlesbrough. Hay et al. [3] also contains many other data including 95% confidence intervals for the above figures. For techniques of measuring such as prevalence rates see e.g. Hay and Gannon [4] or Nordt and Stohler [8]. In view of this, a figure of b 1 ¼ Oð0:02Þ seems reasonable. However, the relapse rate of those in treatment returning to untreated drug taking is, in Dublin, of order of 80–90% of those in treatment. This yields a figure of b 3 ¼ Oð0:8Þ. We do not expect this to vary much in other regions. 2. The White–Comiskey model White and Comiskey [12] have produced an interesting model for the dynamics of heroin users. Their model is based on the equations _ S ¼ K  b 1 U 1 S N  lS; _ U 1 ¼ b 1 U 1 S N  pU 1 þ b 3 U 1 U 2 N ðl þ d 1 ÞU 1 ; _ U 2 ¼ pU 1  b 3 U 1 U 2 N ðl þ d 2 ÞU 2 : 8 > > < > > : ð1Þ 0025-5564/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2009.01.006 * Corresponding author. E-mail addresses: mulone@dmi.unict.it (G. Mulone), brian.straughan@durham. ac.uk (B. Straughan). Mathematical Biosciences 218 (2009) 138–141 Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs