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Heroin epidemics, treatment and ODE modelling. (English) Zbl 1116.92062
Summary: The UN [United Nations Office on Drugs and Crime (UNODC): World Drug Report, Vol. 1: Analysis. UNODC (2005)], EU [European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, 2005. http://annualreport.emcdda.eu.int/en/home-en.html] and the WHO [World Health Organisation (WHO): Biregional Strategy for Harm Reduction, 2005-2009. HIV and Injecting Drug Use. WHO (2005)] have consistently highlighted in recent years the ongoing and persistent nature of opiate and particularly heroin use on a global scale. While this is a global phenomenon, the authors have emphasised the significant impact such an epidemic has on individual lives and on society. National prevalence studies have indicated the scale of the problem, but the drug-using career, typically consisting of initiation, habitual use, a treatment-relapse cycle and eventual recovery, is not well understood.
This paper presents one of the first ODE models of opiate addiction, based on the principles of mathematical epidemiology. The aim of this model is to identify parameters of interest for further study, with a view to informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness. An epidemic threshold value, $$R_{0}$$, is proposed for the drug-using career. Sensitivity analysis is performed on $$R_{0}$$ and it is then used to examine the stability of the system. A condition under which a backward bifurcation may exist is found, as are conditions that permit the existence of one or more endemic equilibria. A key result arising from this model is that prevention is indeed better than cure.

##### MSC:
 92D30 Epidemiology 34D99 Stability theory for ordinary differential equations 91D99 Mathematical sociology (including anthropology)
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##### References:
 [1] United Nations Office on Drugs and Crime (UNODC): World Drug Report, 2005, vol. 1: Analysis. UNODC, 2005. [2] European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, 2005. http://annualreport.emcdda.eu.int/en/home-en.html. [3] World Health Organisation (WHO): Biregional Strategy for Harm Reduction, 2005-2009. HIV and Injecting Drug Use. WHO, 2005. [4] Comiskey, C., National prevalence of problematic opiate use in Ireland, EMCDDA tech. report, (1999) [5] A. Kelly, M. Carvalho, C. Teljeur: Prevalence of Opiate Use in Ireland 2000-2001. A 3-Source Capture Recapture Study. A Report to the National Advisory Committee on Drugs, Sub-committee on Prevalence. Small Area Health Research Unit, Department of Public. [6] O’Brien, M.; Moran, R., Overview of drug issues in Ireland 1997, Drugs research division, health research board, (1997) [7] Corrigan, D., The identification of drugs of abuse in the republic of Ireland during the years 1968-1978, Bull. narcotics, 31, 2, 57, (1979) [8] G. Dean, A. O’Hare, G. Kelly, A. O’Connor, M. Kelly: The Opiate Epidemic in Dublin 1979-1983. The Medico Social Research Board, 73 Lower Baggot Street, Dublin 2, 1983. [9] J. Long et al., Trends in Treated Problem Drug Use in Ireland, 1998 to 2002, Occasional Paper No. 17/2005. DMRD, HRB, 2005. [10] C. Comiskey, G. Cox: Research Outcome Study in Ireland (ROSIE): Evaluating Drug Treatment Effectiveness, Baseline Findings. www.nuim.ie/ROSIE/ResearchHistory.shtml, March 2005. [11] Bailey, N., The mathematical theory of infectious diseases, (1975), Charles Griffin [12] Anderson, R.M.; May, R.M., Infectious diseases of humans, dynamics and control, (1991), Oxford University Press [13] Murray, J.D., Mathematical biology I and II, (2004), Springer [14] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (2000), Springer · Zbl 1302.92001 [15] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29, (2002) · Zbl 1015.92036 [16] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases, (2000), John Wiley & Son, Ltd · Zbl 0997.92505 [17] European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Population Surveys Methodology. Methods and definitions, general population surveys, detailed notes. EMCDDA Statistical Bulletin, 2004. [18] L. Arriola, J. Hyman, Lecture notes, forward and adjoint sensitivity analysis: with applications in Dynamical Systems, Linear Algebra and Optimisation Mathematical and Theoretical Biology Institute, Summer, 2005. [19] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. biosci. eng., 1, 2, (2004) · Zbl 1060.92041 [20] B. Song, Seminar Notes, Backward or Forward at R0 =1, Mathematical and Theoretical Biology Institute, Summer, 2005. [21] F.E. Su et al., “Descartes’ Rule of Signs” Mudd Math Fun Facts. http://www.math.hmc.edu/funfacts.
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