Journal of Applied Mathematics

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Isaac Mwangi Wangari and Lewi Stone

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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.

Article information

J. Appl. Math., Volume 2017 (2017), Article ID 1953036, 21 pages.

Received: 12 May 2017
Accepted: 26 July 2017
First available in Project Euclid: 19 September 2017

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Wangari, Isaac Mwangi; Stone, Lewi. Analysis of a Heroin Epidemic Model with Saturated Treatment Function. J. Appl. Math. 2017 (2017), Article ID 1953036, 21 pages. doi:10.1155/2017/1953036.

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