From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province

Mathematical Biosciences

Volumn 225, Issue 2, 2010, Pages 132-140

  Nyabadza, Farai ^a   Hove Musekwa, Senelani D ^b  

^a STELLENBOSCH UNIVERSITY   (South Africa)

^b NATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Zimbabwe)

Author keywords

Epidemics; Methamphetamine; Projections; Reproduction number; Stability

Indexed keywords

AFTER INITIATION; BACKWARD BIFURCATION; CENTER MANIFOLD THEORY; EPIDEMIC THRESHOLD; EPIDEMICS; MULTIPLE EQUILIBRIUM; NUMERICAL SIMULATION; SOCIAL IMPLICATION; SUBSTANCE ABUSE;

COMPUTER SIMULATION;

EPIDEMIOLOGY;

DIAMORPHINE; METHAMPHETAMINE;

COMPUTER SIMULATION; DRUG; EPIDEMIC; NUMERICAL MODEL;

ARTICLE; EPIDEMIC; EQUILIBRIUM CONSTANT; HEALTH SERVICE; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; PREVALENCE; PROBABILITY; QUALITATIVE ANALYSIS; RISK FACTOR; SOUTH AFRICA; SUBSTANCE ABUSE; THEORETICAL MODEL;

AMPHETAMINE-RELATED DISORDERS; COMPUTER SIMULATION; DISEASE OUTBREAKS; HEROIN DEPENDENCE; HUMANS; MODELS, STATISTICAL; SOUTH AFRICA;

SOUTH AFRICA;

EID: 77952674014     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2010.03.002     Document Type: Article

References (43)

1. Alexander M.E., Modhadas S.M. Periodicity in an epidemic model with a generalized nonlinear incidence. Math. Biosci. 2004, 189:75.
2. Bacaër N., Ouifki R., Pretorius C., Wood R., Williams B. Modelling the joint TB/HIV epidemic in a South African township. Math. Biol. 2007, 57:557.
3. Bhunu C.P., Garira W., Mukandavire Z., Magombedze G. Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control. J. Thoer. Biol. 2008, 254:633.
4. G. Birkhoff, G.C. Rota, Ordinary Differential Equations, Ginn, Boston 1982.
5. Burattini M.N., Massad E., Coutinho F.A.B. A mathematical model of the impact of crack-cocaine use on the prevalence of HIV/AIDS among drug users. Math. Comput. Model. 1998, 28:21.
6. Castillo-Chavez C., Song B. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 2004, 1:361.
7. Carr J. Applications Center Manifold Theory 1981, Springer-Verlag, New York.
8. City of Cape Town (2009), State of Cape Town: development issues in Cape Town. Available from: http://www.capetown.gov.za.
9. Cui J., Sun Y., Zhu H. The impact of media on the control of infectious diseases. J. Dyn. Diff. Eqn. 2008, 20:31.
10. Cui J., Mu X., Wan H. Saturation recovery leads to multiple endemic equilibria and backward bifurcation. J. Theor. Biol. 2008, 254:275.
11. C. Day, A. Gray, Health and related indicators. Available from: http://www.hst.org.za/uploads/files/chap16_08.pdf.
12. Diekmann O., Heesterbeek J.A.P., Metz J.A.J. On the definition and computation of the basic reproduction ratio R0 in models of infectious diseases in heterogeneous populations. J. Math. Biol. 1990, 28:503.
13. Dushoff J. Incorporating immunological ideas in epidemiological models. J. Theor. Biol. 1996, 180:181.
14. Dushoff J., Huang W., Castillo-Chavez C. Backwards bifurcations and catastrophe in simple epidemic models of fatal diseases. J. Math. Biol. 1998, 36:227.
15. Egami C. Bifurcation analysis of the Nowak-Bangham model in CTL dynamics. Math. Biosci. 2009, 221:33.
16. Feng Z., Huang W., Castillo-Chavez C. On the role of variable latent periods in mathematical models for tuberculosis. J. Dyn. Diff. Eq. 2001, 13:425.
17. Garba S.M., Gumel A.B., Abu Bakar M.R. Backward bifurcation in dengue transmission dynamics. Math. Biosci. 2008, 215:11.
18. Gómez-Acevedo H., Li M.Y. Backward bifurcation in a model for HTLV-1 infection of CD4+ T cells. Bull. Math. Biol. 2005, 67:101.
19. Greenhalgh D., Diekmann O., de Jong M.C.M. Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity. Math. Biosci. 2000, 165:1.
20. Greenhalgh D., Griffiths D. Backward bifurcation, equilibrium and stability phenomena in a three stage extended BRSV epidemic model. J. Math. Biol. 2009, 59:1.
21. Hadeler K.P., van den Driessche P. Backward bifurcation in epidemic control. Math. Biosci. 1997, 146:15.
22. Hethcote H.W. The mathematics of infectious diseases. SIAM Rev. 2000, 42:599.
23. Hofbauer J., Sigmund K. The Theory of Evolution and Dynamical Systems 1988, Cambridge University Press, Cambridge, UK.
24. Kapp C. Crystal meth boom adds to South Africa's health challenges. Lancet 2008, 371:193.
25. Kribs-Zaletta C.M., Velasco-Hetnandez J. A simple vaccination model with multiple endemic states. Math. Biosci. 2000, 164:183.
26. Kribs-Zaletta C.M., Martcheva M. Vaccination strategies and backward bifurcation in an age-since-infection structured model. Math. Biosci. 2002, 177-178:317.
27. LaSalle J.P. The Stability of Dynamical Systems. CBMS-NSF Regional Conference Series in Applied Mathematics 25 1976, SIAM, Philadelphia.
28. Lineberry T.W., M Bostwick J. Methamphetamine abuse: a perfect storm of complications. Mayo Clin. Proc. 2006, 81:77.
29. Lipsitch M., Murray M.B. Multiple equilibria: tuberculosis transmission require unrealistic assumptions. Theor. Pop. Biol. 2003, 63:169.
30. McKetin R., Kelly E., McLaren J. The relationship between crystalline methamphetamine use and methamphetamine dependence. Drug Alcohol Depend. 2006, 85:198204.
31. Morris K., Parry C. South African methamphetamine boom could fuel further HIV. Lancet 2006, 6:47.
32. Mulone G., Straughan B. A note on heroin epidemics. Math. Biosci. 2009, 218:138.
33. C. Parry, Substance abuse trends in the Western Cape: Summary (25/2/05), Alcohol and Drug Abuse Research Unit, Medical Research Council, 2005.
34. A. Plüddemann et al., Monitoring alcohol and drug abuse trends in South Africa, SACENDU Research Brief, vol. 11, 2008.
35. Rossi C. Operational models for the epidemics of problematic drug use: the mover-stayer approach to heterogeneity. Socio-Econ. Plan. Sci. 2004, 38:73.
36. K. Small, Demographic and socio-economic trends for Cape Town: 1996 to 2007, City of Cape Town, 2008.
37. Sharomi O., Gumel A.B. Curtailing smoking dynamics: a mathematical modelling approach. Appl. Math. Comp. 2008, 195:475.
38. F.E. Su et al., " Descartes' Rule of Signs Mudd Math Fun Facts. Available from: http://www.math.hmc.edu/funfacts.
39. van den Driessche P., Watmough J. A simple SIS model with backward bifurcation. J. Math. Biol 2000, 40:525.
40. Wechsberg W.M., Luseno W.K., Karg R.S., Young S., Rodman N., Meyers B., Parry C.D.H. Alcohol, cannibis, and methamphetamine use and other risk behaviors among black and coloured South African Women: a small randomized trial in the Western Cape. Inter. J. Drug Pol. 2008, 19:130.
41. White E., Comiskey C. Heroin epidemics treatment and ODE modelling. Math. Biosci. 2007, 208:312.
42. Wilkinson P. City profile Cape Town. Cities 2000, 17:195.
43. Mukandavire Z., Gumel A.B., Garira W., Tchuenche J.M. Mathematical analysis of a model for HIV-Malaria co-infection. Math. Biosci. Eng. 2009, 6:333.