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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.

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) |

### Keywords:

ODE models; mathematical epidemiology; heroin use; reproduction ratio \(R_{0}\); drug-using career; treatment
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\textit{E. White} and \textit{C. Comiskey}, Math. Biosci. 208, No. 1, 312--324 (2007; Zbl 1116.92062)

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### References:

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