# American Institute of Mathematical Sciences

2013, 10(3): 843-860. doi: 10.3934/mbe.2013.10.843

## Modelling the role of drug barons on the prevalence of drug epidemics

 1 Department of Mathematical Science, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch 7602, South Africa, South Africa

Received  May 2012 Revised  January 2013 Published  April 2013

Substance abuse is a global menace with immeasurable consequences to the health of users, the quality of life and the economy of countries affected. Although the prominently known routes of initiation into drug use are; by contact between potential users and individuals already using the drugs and self initiation, the role played by a special class of individuals referred to as drug lords can not be ignored. We consider a simple but useful compartmental model of drug use that accounts for the contribution of contagion and drug lords to initiation into drug use and drug epidemics. We show that the model has a drug free equilibrium when the threshold parameter $R_{0}$ is less that unity and a drug persistent equilibrium when $R_{0}$ is greater than one. In our effort to ascertain the effect of policing in the control of drug epidemics, we include a term accounting for law enforcement. Our results indicate that increased law enforcement greatly reduces the prevalence of substance abuse. In addition, initiation resulting from presence of drugs in circulation can be as high as seven times higher that initiation due to contagion alone.
Citation: John Boscoh H. Njagarah, Farai Nyabadza. Modelling the role of drug barons on the prevalence of drug epidemics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 843-860. doi: 10.3934/mbe.2013.10.843
##### References:
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Capasso, "Mathematical Structures of Epidemic Systems: Lecture Notes in Biomathematics," 97, Springer-Verlag, 1993. [6] C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [7] J. P. Caulkins, A. Gragnani, G. Feichtinger and G. Trangler, High and low frequency oscillations in drug epidemics, Int. J. Bifurcat Chaos, 16 (2006), 3275-3289. doi: 10.1142/S0218127406016781. [8] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of Malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [9] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0. [10] A. 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[17] A. Harocopos, L. A. Goldsamt, P. Kobrak, J. J. Jost and M. C. Clatts, New injectors and the social context of injection initiation, Int. J. Drug Policy, 20 (2009), 317-323. doi: 10.1016/j.drugpo.2008.06.003. [18] A. Hoare, D. G. Regan and D. P. Wilson, Sampling and sensitivity tools (SaSAT) for computational modelling, Theor. Biol. Med. Model., 54 (2008). doi: 10.1186/1742-4682-5-4. [19] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-108. [20] D. T. Jamison, R. G. Feachmen, M. W. Makgoba, E. R. Bos, F. K. Baingana, K. J. Hofman and K. O. Rogo, "Disease and Mortality in Sub-Saharan Africa," second edition, 2006. World Bank, Washington D.C. [21] S. B. Karch, M.D., FFFLM, "Drug Abuse Handbook," Taylor & Francis Group, LLC, 2007. [22] J. P. La Salle, "The Stability of Dynamical Systems," Society for Industrial and Applied Mathematics, 1976. [23] J. D. Lloyd, P. M. O'Malley and J. G. Bachman, "Illicit Drug Use and Smoking and Drinking by America's High School Students and College Student and Young Adults 1975-1987," NIDA, 1989. [24] J. H. Lowinson, P. Ruiz, R. B. Millman and J. G. Langrod, "Substance Abuse: A Comprehensive Textbook," Lippincott Williams & Wilkins, 2005. [25] P. Magel and S. Ruan, "Structured Population Models in Biology and Epidemiology," Springer, 2008. [26] G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 208 (2009), 131-141. doi: 10.1016/j.mbs.2009.01.006. [27] NIDA, "Cigarettes and Other Tobacco Products,", 2012. Available from , (). [28] F. Nyabadza and S. D. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province, Math. Biosci., 225 (2010), 134-140. doi: 10.1016/j.mbs.2010.03.002. [29] C. D. H. Parry, A. Plüddemann, A. Louw and T. Leggett, The 3-metros study of drugs and crime in South Africa: Findings and policy implications, Am. J. Drug and Alcohol Abuse, 30 (2004), 167-185. doi: 10.1081/ADA-120029872. [30] K. Pertzer and Habil, Cannabis use trends in South Africa, SAJP, 13 (2008), 126-131. [31] C. Rossi, Operational models for epidemics of problematic drug use: The mover-stayer approach to heterogeneity, Socio-Econ. Plan. Sci., 38 (2004), 73-90. doi: 10.1016/S0038-0121(03)00029-6. [32] C. Rossi, The role of dynamic modelling in drug abuse epidemiology, Bulletin on Narcotics, LIV (2002), 33-44. [33] SACENDU, "Monitoring Alcohol and Drug Abuse Trends in South Africa," SACENDU Research Briefs, 2006, 12. [34] O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modelling approach, Appl. Math. Comp., 195 (2008), 475-499. doi: 10.1016/j.amc.2007.05.012. [35] J. H. Tein and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, B. Math Biol., 72 (2010), 1506-1533. doi: 10.1007/s11538-010-9507-6. [36] M. Tonouchi, Cutting-edge terahetz technology, Nat. Photonics, 1 (2007), 97-105. doi: 10.1038/nphoton.2007.3. [37] UNDCP, "Social Impact on Drug Abuse,", 1995. Copenhagen, (): 6. [38] UNODC, "World Drug Report,", 2009. United Nations, (). [39] UNODC, "Organized Crime and Its Threat to Security: Tackling a Disturbing Consequence of Drug Control,", 2009. Vienna, (): 16. [40] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [41] W. Wang and S. Ruan, Bifurcation in the epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793. doi: 10.1016/j.jmaa.2003.11.043. [42] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.

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##### References:
 [1] D. A. Behrens, J. P. Caulkins, G. Tragler and G. Freichtinger, Optimal control of drug epidemics: Prevention and treatment-but not at the same time, Management Science, 46 (2000), 333-347. [2] D. A. Behrens, J. P. Caulkins, G. Tragler, J. L. Haunschmied and G. Feichtinger, A dynamic model of drug initiation: Implications for treatment and drug control, Math. Biosci. Eng., 159 (1999), 1-20. doi: 10.1016/S0025-5564(99)00016-4. [3] S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model as an example, Int. Stat. Rev., 64 (1994), 229-243. doi: 10.2307/1403510. [4] M. N. Burattini, E. Massad, F. A. B. Coutinho, R. S. Azzevedo-Neto, R. X. Menezes and L. F. Lopes, A mathematical model of the impact of crank cocaine use on the prevalence of HIV/AIDS among drug users, Math Comput. Modelling, 28 (1998), 21-29. [5] V. Capasso, "Mathematical Structures of Epidemic Systems: Lecture Notes in Biomathematics," 97, Springer-Verlag, 1993. [6] C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361. [7] J. P. Caulkins, A. Gragnani, G. Feichtinger and G. Trangler, High and low frequency oscillations in drug epidemics, Int. J. Bifurcat Chaos, 16 (2006), 3275-3289. doi: 10.1142/S0218127406016781. [8] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of Malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296. doi: 10.1007/s11538-008-9299-0. [9] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20 (2008), 31-53. doi: 10.1007/s10884-007-9075-0. [10] A. P. de Andrés, "West Africa Under Attack: Drugs Organised Crime and Terrorism as the New Threats to Global Security,", 2008. UNISCI Discussion Papers, (): 1696. [11] S. S. Everingham and C. P. Rydell, "Modeling the Demand of Cocaine," Drug Policy Research Centre, 1994. [12] S. S. Everingham and C. P. Rydell, "Promising Strategies to Reduce Substance Abuse,", 2000, (). [13] S. M. S. Everingham, C. P. Rydell and J. P. Caulkins, Cocaine consumption in the United States: Estimating past trends and future scenarios, Socio-Econ. Plann. Sci., 29 (1995), 305-314. [14] M. H. Greene and MD, An epidemic assessment of heroin use, AJHP Supplement, 64 (1974), 1-10. [15] H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168. doi: 10.1080/17513750802120877. [16] K. P. Handeler and P. Van Den Driessche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35. doi: 10.1016/S0025-5564(97)00027-8. [17] A. Harocopos, L. A. Goldsamt, P. Kobrak, J. J. Jost and M. C. Clatts, New injectors and the social context of injection initiation, Int. J. Drug Policy, 20 (2009), 317-323. doi: 10.1016/j.drugpo.2008.06.003. [18] A. Hoare, D. G. Regan and D. P. Wilson, Sampling and sensitivity tools (SaSAT) for computational modelling, Theor. Biol. Med. Model., 54 (2008). doi: 10.1186/1742-4682-5-4. [19] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-108. [20] D. T. Jamison, R. G. Feachmen, M. W. Makgoba, E. R. Bos, F. K. Baingana, K. J. Hofman and K. O. Rogo, "Disease and Mortality in Sub-Saharan Africa," second edition, 2006. World Bank, Washington D.C. [21] S. B. Karch, M.D., FFFLM, "Drug Abuse Handbook," Taylor & Francis Group, LLC, 2007. [22] J. P. La Salle, "The Stability of Dynamical Systems," Society for Industrial and Applied Mathematics, 1976. [23] J. D. Lloyd, P. M. O'Malley and J. G. Bachman, "Illicit Drug Use and Smoking and Drinking by America's High School Students and College Student and Young Adults 1975-1987," NIDA, 1989. [24] J. H. Lowinson, P. Ruiz, R. B. Millman and J. G. Langrod, "Substance Abuse: A Comprehensive Textbook," Lippincott Williams & Wilkins, 2005. [25] P. Magel and S. Ruan, "Structured Population Models in Biology and Epidemiology," Springer, 2008. [26] G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 208 (2009), 131-141. doi: 10.1016/j.mbs.2009.01.006. [27] NIDA, "Cigarettes and Other Tobacco Products,", 2012. Available from , (). [28] F. Nyabadza and S. D. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province, Math. Biosci., 225 (2010), 134-140. doi: 10.1016/j.mbs.2010.03.002. [29] C. D. H. Parry, A. Plüddemann, A. Louw and T. Leggett, The 3-metros study of drugs and crime in South Africa: Findings and policy implications, Am. J. Drug and Alcohol Abuse, 30 (2004), 167-185. doi: 10.1081/ADA-120029872. [30] K. Pertzer and Habil, Cannabis use trends in South Africa, SAJP, 13 (2008), 126-131. [31] C. Rossi, Operational models for epidemics of problematic drug use: The mover-stayer approach to heterogeneity, Socio-Econ. Plan. Sci., 38 (2004), 73-90. doi: 10.1016/S0038-0121(03)00029-6. [32] C. Rossi, The role of dynamic modelling in drug abuse epidemiology, Bulletin on Narcotics, LIV (2002), 33-44. [33] SACENDU, "Monitoring Alcohol and Drug Abuse Trends in South Africa," SACENDU Research Briefs, 2006, 12. [34] O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modelling approach, Appl. Math. Comp., 195 (2008), 475-499. doi: 10.1016/j.amc.2007.05.012. [35] J. H. Tein and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, B. Math Biol., 72 (2010), 1506-1533. doi: 10.1007/s11538-010-9507-6. [36] M. Tonouchi, Cutting-edge terahetz technology, Nat. Photonics, 1 (2007), 97-105. doi: 10.1038/nphoton.2007.3. [37] UNDCP, "Social Impact on Drug Abuse,", 1995. Copenhagen, (): 6. [38] UNODC, "World Drug Report,", 2009. United Nations, (). [39] UNODC, "Organized Crime and Its Threat to Security: Tackling a Disturbing Consequence of Drug Control,", 2009. Vienna, (): 16. [40] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [41] W. Wang and S. Ruan, Bifurcation in the epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793. doi: 10.1016/j.jmaa.2003.11.043. [42] E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling, Math. Biosci., 208 (2007), 312-324. doi: 10.1016/j.mbs.2006.10.008.
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