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Journal of Applied Mathematics
Volume 2017 (2017), Article ID 1953036, 21 pages
https://doi.org/10.1155/2017/1953036
Research Article

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

1Royal Melbourne Institute of Technology School of Mathematics and Geospatial Sciences, Melbourne, VIC, Australia
2Biomathematics Unit, Department of Zoology, Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel

Correspondence should be addressed to Isaac Mwangi Wangari

Received 12 May 2017; Accepted 26 July 2017; Published 31 August 2017

Academic Editor: Mehmet Sezer

Copyright © 2017 Isaac Mwangi Wangari and Lewi Stone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that “saturates” representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model’s global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation.