Analysis of a heroin epidemic model with saturated treatment function

Wangari, I and Stone, L 2017, 'Analysis of a heroin epidemic model with saturated treatment function', Journal of Applied Mathematics, vol. 2017, 1953036, pp. 1-22.


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Collection: Journal Articles

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Title Analysis of a heroin epidemic model with saturated treatment function
Author(s) Wangari, I
Stone, L
Year 2017
Journal name Journal of Applied Mathematics
Volume number 2017
Article Number 1953036
Start page 1
End page 22
Total pages 22
Publisher Hindawi
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.
Subject Applied Mathematics not elsewhere classified
Keyword(s) backward bifurcation
global stability
geometric approach
DOI - identifier 10.1155/2017/1953036
Copyright notice 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.
ISSN 1110-757X
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