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:
[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. Google Scholar

[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. doi: 10.1016/S0025-5564(99)00016-4. Google Scholar

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[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. Google Scholar

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V. Capasso, "Mathematical Structures of Epidemic Systems: Lecture Notes in Biomathematics,", 97, 97 (1993). Google Scholar

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C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[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. doi: 10.1142/S0218127406016781. Google Scholar

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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. doi: 10.1007/s11538-008-9299-0. Google Scholar

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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. Google Scholar

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S. S. Everingham and C. P. Rydell, "Modeling the Demand of Cocaine,", Drug Policy Research Centre, (1994). Google Scholar

[12]

S. S. Everingham and C. P. Rydell, "Promising Strategies to Reduce Substance Abuse,", 2000, (). Google Scholar

[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. Google Scholar

[14]

M. H. Greene and MD, An epidemic assessment of heroin use,, AJHP Supplement, 64 (1974), 1. Google Scholar

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H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases,, J. Biol. Dyn., 2 (2008), 154. doi: 10.1080/17513750802120877. Google Scholar

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K. P. Handeler and P. Van Den Driessche, Backward bifurcation in epidemic control,, Math. Biosci., 146 (1997), 15. doi: 10.1016/S0025-5564(97)00027-8. Google Scholar

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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. doi: 10.1016/j.drugpo.2008.06.003. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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). Google Scholar

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S. B. Karch, M.D., FFFLM, "Drug Abuse Handbook,", Taylor & Francis Group, (2007). Google Scholar

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J. P. La Salle, "The Stability of Dynamical Systems,", Society for Industrial and Applied Mathematics, (1976). Google Scholar

[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). Google Scholar

[24]

J. H. Lowinson, P. Ruiz, R. B. Millman and J. G. Langrod, "Substance Abuse: A Comprehensive Textbook,", Lippincott Williams & Wilkins, (2005). Google Scholar

[25]

P. Magel and S. Ruan, "Structured Population Models in Biology and Epidemiology,", Springer, (2008). Google Scholar

[26]

G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 208 (2009), 131. doi: 10.1016/j.mbs.2009.01.006. Google Scholar

[27]

NIDA, "Cigarettes and Other Tobacco Products,", 2012. Available from , (). Google Scholar

[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. doi: 10.1016/j.mbs.2010.03.002. Google Scholar

[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. doi: 10.1081/ADA-120029872. Google Scholar

[30]

K. Pertzer and Habil, Cannabis use trends in South Africa,, SAJP, 13 (2008), 126. Google Scholar

[31]

C. Rossi, Operational models for epidemics of problematic drug use: The mover-stayer approach to heterogeneity,, Socio-Econ. Plan. Sci., 38 (2004), 73. doi: 10.1016/S0038-0121(03)00029-6. Google Scholar

[32]

C. Rossi, The role of dynamic modelling in drug abuse epidemiology,, Bulletin on Narcotics, LIV (2002), 33. Google Scholar

[33]

SACENDU, "Monitoring Alcohol and Drug Abuse Trends in South Africa,", SACENDU Research Briefs, 12 (2006). Google Scholar

[34]

O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modelling approach,, Appl. Math. Comp., 195 (2008), 475. doi: 10.1016/j.amc.2007.05.012. Google Scholar

[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. doi: 10.1007/s11538-010-9507-6. Google Scholar

[36]

M. Tonouchi, Cutting-edge terahetz technology,, Nat. Photonics, 1 (2007), 97. doi: 10.1038/nphoton.2007.3. Google Scholar

[37]

UNDCP, "Social Impact on Drug Abuse,", 1995. Copenhagen, (): 6. Google Scholar

[38]

UNODC, "World Drug Report,", 2009. United Nations, (). Google Scholar

[39]

UNODC, "Organized Crime and Its Threat to Security: Tackling a Disturbing Consequence of Drug Control,", 2009. Vienna, (): 16. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. doi: 10.1016/j.jmaa.2003.11.043. Google Scholar

[42]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Math. Biosci., 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar

show all references

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. Google Scholar

[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. doi: 10.1016/S0025-5564(99)00016-4. Google Scholar

[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. doi: 10.2307/1403510. Google Scholar

[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. Google Scholar

[5]

V. Capasso, "Mathematical Structures of Epidemic Systems: Lecture Notes in Biomathematics,", 97, 97 (1993). Google Scholar

[6]

C. Castillo-Chavez and B. Song, Dynamical models of Tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361. doi: 10.3934/mbe.2004.1.361. Google Scholar

[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. doi: 10.1142/S0218127406016781. Google Scholar

[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. doi: 10.1007/s11538-008-9299-0. Google Scholar

[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. doi: 10.1007/s10884-007-9075-0. Google Scholar

[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. Google Scholar

[11]

S. S. Everingham and C. P. Rydell, "Modeling the Demand of Cocaine,", Drug Policy Research Centre, (1994). Google Scholar

[12]

S. S. Everingham and C. P. Rydell, "Promising Strategies to Reduce Substance Abuse,", 2000, (). Google Scholar

[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. Google Scholar

[14]

M. H. Greene and MD, An epidemic assessment of heroin use,, AJHP Supplement, 64 (1974), 1. Google Scholar

[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. doi: 10.1080/17513750802120877. Google Scholar

[16]

K. P. Handeler and P. Van Den Driessche, Backward bifurcation in epidemic control,, Math. Biosci., 146 (1997), 15. doi: 10.1016/S0025-5564(97)00027-8. Google Scholar

[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. doi: 10.1016/j.drugpo.2008.06.003. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[21]

S. B. Karch, M.D., FFFLM, "Drug Abuse Handbook,", Taylor & Francis Group, (2007). Google Scholar

[22]

J. P. La Salle, "The Stability of Dynamical Systems,", Society for Industrial and Applied Mathematics, (1976). Google Scholar

[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). Google Scholar

[24]

J. H. Lowinson, P. Ruiz, R. B. Millman and J. G. Langrod, "Substance Abuse: A Comprehensive Textbook,", Lippincott Williams & Wilkins, (2005). Google Scholar

[25]

P. Magel and S. Ruan, "Structured Population Models in Biology and Epidemiology,", Springer, (2008). Google Scholar

[26]

G. Mulone and B. Straughan, A note on heroin epidemics,, Math. Biosci., 208 (2009), 131. doi: 10.1016/j.mbs.2009.01.006. Google Scholar

[27]

NIDA, "Cigarettes and Other Tobacco Products,", 2012. Available from , (). Google Scholar

[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. doi: 10.1016/j.mbs.2010.03.002. Google Scholar

[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. doi: 10.1081/ADA-120029872. Google Scholar

[30]

K. Pertzer and Habil, Cannabis use trends in South Africa,, SAJP, 13 (2008), 126. Google Scholar

[31]

C. Rossi, Operational models for epidemics of problematic drug use: The mover-stayer approach to heterogeneity,, Socio-Econ. Plan. Sci., 38 (2004), 73. doi: 10.1016/S0038-0121(03)00029-6. Google Scholar

[32]

C. Rossi, The role of dynamic modelling in drug abuse epidemiology,, Bulletin on Narcotics, LIV (2002), 33. Google Scholar

[33]

SACENDU, "Monitoring Alcohol and Drug Abuse Trends in South Africa,", SACENDU Research Briefs, 12 (2006). Google Scholar

[34]

O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modelling approach,, Appl. Math. Comp., 195 (2008), 475. doi: 10.1016/j.amc.2007.05.012. Google Scholar

[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. doi: 10.1007/s11538-010-9507-6. Google Scholar

[36]

M. Tonouchi, Cutting-edge terahetz technology,, Nat. Photonics, 1 (2007), 97. doi: 10.1038/nphoton.2007.3. Google Scholar

[37]

UNDCP, "Social Impact on Drug Abuse,", 1995. Copenhagen, (): 6. Google Scholar

[38]

UNODC, "World Drug Report,", 2009. United Nations, (). Google Scholar

[39]

UNODC, "Organized Crime and Its Threat to Security: Tackling a Disturbing Consequence of Drug Control,", 2009. Vienna, (): 16. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. doi: 10.1016/j.jmaa.2003.11.043. Google Scholar

[42]

E. White and C. Comiskey, Heroin epidemics, treatment and ODE modelling,, Math. Biosci., 208 (2007), 312. doi: 10.1016/j.mbs.2006.10.008. Google Scholar

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