Analysis of a Heroin Epidemic Model with Saturated Treatment Function
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 r...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2017-01-01
|
| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2017/1953036 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850229415135412224 |
|---|---|
| author | Isaac Mwangi Wangari Lewi Stone |
| author_facet | Isaac Mwangi Wangari Lewi Stone |
| author_sort | Isaac Mwangi Wangari |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1fa8f730125147009fe606fbb5720abb |
| institution | OA Journals |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-1fa8f730125147009fe606fbb5720abb2025-08-20T02:04:14ZengWileyJournal of Applied Mathematics1110-757X1687-00422017-01-01201710.1155/2017/19530361953036Analysis of a Heroin Epidemic Model with Saturated Treatment FunctionIsaac Mwangi Wangari0Lewi Stone1Royal Melbourne Institute of Technology School of Mathematics and Geospatial Sciences, Melbourne, VIC, AustraliaRoyal Melbourne Institute of Technology School of Mathematics and Geospatial Sciences, Melbourne, VIC, AustraliaA 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.http://dx.doi.org/10.1155/2017/1953036 |
| spellingShingle | Isaac Mwangi Wangari Lewi Stone Analysis of a Heroin Epidemic Model with Saturated Treatment Function Journal of Applied Mathematics |
| title | Analysis of a Heroin Epidemic Model with Saturated Treatment Function |
| title_full | Analysis of a Heroin Epidemic Model with Saturated Treatment Function |
| title_fullStr | Analysis of a Heroin Epidemic Model with Saturated Treatment Function |
| title_full_unstemmed | Analysis of a Heroin Epidemic Model with Saturated Treatment Function |
| title_short | Analysis of a Heroin Epidemic Model with Saturated Treatment Function |
| title_sort | analysis of a heroin epidemic model with saturated treatment function |
| url | http://dx.doi.org/10.1155/2017/1953036 |
| work_keys_str_mv | AT isaacmwangiwangari analysisofaheroinepidemicmodelwithsaturatedtreatmentfunction AT lewistone analysisofaheroinepidemicmodelwithsaturatedtreatmentfunction |