Mathematical Modeling of the Spread of Alcoholism Among Colombian College Students
Main Article Content
Keywords
Alcohol consumption, mathematical modeling, nonlinear dynamical systems, optimal control, basic reproduction number, numerical simulations
Abstract
In this paper, we present a nonlinear mathematical model, describing the spread of high-risk alcohol consumption behavior among college students in Colombia. We proved the existence and stability of the alcohol-free and drinking state equilibrium by means of Lyapunov function and LaSalle’s invariance principle. Also, we apply optimal control to study the impact of a preventive measure on the spread of drinking behavior among college students. Finally, we use numerical simulations and available data provided by the United Nations Office on Drugs and Crime (UNODC) and the Colombian Ministry of Justice to validate the obtained mathematical model.
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References
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[14] R. Ullah, M. Khan, G. Zaman, S. Islam, M. Altaf Khan, S. Jan, and T. Gul, “ Dynamical features of a mathematical model on smoking,” J. Appl. Environ. Biol. Sci, vol. 6, pp. 92–96, 2016.
[15] I. Kwasi, M. Osman, and C. Yang, “ Mathematical model of drinking epidemic,” British Journal of Mathematics and Computer Science, vol. 22, pp. 1–10, 2017.
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[17] B. Buonomo and D. Lacitignola, “ Modeling peer influence effects on he spread of high-risk alcohol consumption behavior,” Ricerche mat, vol. 63, pp. 101–117, 2014. https://doi.org/10.1007/s11587-013-0167-3
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[19] J. Manthey, A. Aidoo, and K. Ward, “ Campus drinking: an epidemiological model,” J. Biol. Dyn., vol. 2, pp. 346–356, 2008. https://doi.org/10.1080/17513750801911169
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[22] P. Van den Driessche and J. Watmough, “ Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci, vol. 180, no. 1–2, pp. 29–48, 2002. https://doi.org/10.1016/S0025-5564(02)00108-6
[23] B. Brown, The world’s youth: adolescence in eight regions of the globe. Cambridge University Press, 2002.
[24] B. Brown, Encyclopedia of adolescence . Academic Press, 2011.
[25] V. Ivaniushina, V. Titkova, and D. Alexandrov, “Peer influence in adolescent drinking behaviour: a protocol for systematic review and meta-analysis of stochastic actor-based modeling studies,” BMJ Open, vol. 9, no. 7, 2019. https://doi.org/10.1136/bmjopen-2018-028709
[26] B. Simons Morton, D. Haynie, A. Crump, S. Eitel, and K. Saylor, “ Peer and parent influences on smoking and drinking among early adolescents,” Health Educ Behav, vol. 28, pp. 95–107, 2001.
[27] C. Martin and M. Hoffman, “ Alcohol expectancies, living environment, peer influence, and gender: a model of college-student drinking,” Journal of college student development, vol. 34, pp. 206–211, 1993.
[28] P. O’Malley and L. Johnston, “ Epidemiology of alcohol and other drug use among American college students,” J. Stud. Alcohol, vol. 63, no. 2, pp. 23–39, 2002. https://www.collegedrinkingprevention.gov/media/ journal/023_039.pdf
[29] S. Betancourth, L. Tacán, and E. Cordoba, “ Alcohol consumption in Colombian university students,” Rev. Univ. Salud, vol. 19, pp. 37–50, 2017.
[30] L. Arriola and J. Hyman, Lectures notes, Forward and adjoint sensitivity analysis: with applications in Dynamical Systems, Linear Algebra and Optimization. Mathematical and Theoretical Biology Institute, 2005.
[31] A. Rachah and D. F. M. Torres, “ Mathematical modelling, simulation and optimal control of the 2014 Ebola outbreak in west Africa,” Discrete dynamics in nature and society, vol. 2015, 2015. https://doi.org/10.1155/2015/842792
[32] H. S. Rodrigues, M. T. T. Monteiro, and D. F. Torres, “Vaccination models and optimal control strategies to dengue,” Mathematical Biosciences, vol. 247, pp. 1 – 12, 2014. https://doi.org/10.1016/j.mbs.2013.10.006
[33] S. Djiomba and F. Nyabadza, “ An optimal control model for Ebola virus disease,” Journal of Biological Systems, vol. 24, no. 01, pp. 29–49, 2016. https://doi.org/10.1142/S0218339016500029
[34] S. Rosa and D. F. M. Torres, “ Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida,” Statistic Opt. Inform. Comput, vol. 6, no. 1, 2018. https://doi.org/10.19139/soic.v6i1.472
[35] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models. Chapman and Hall-CRC, Boca Raton, FL, USA, 2007.
[36] E. Bonyah, M. A. Khan, K. O. Okosun, and J. F. Gomez-Aguilar, “ Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control,” Mathematical Biosciences, vol. 309, pp. 1–11, 2019. https://doi.org/10.1016/j.mbs.2018.12.015
[37] B. Khajji, A. Labzai, A. Kouidere, O. Balatif, and M. Rachik, “ A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control,” Journal of Applied Mathematics, pp. 1–13, 2020. https://doi.org/10.1155/2020/9284698
[38] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes . John Wiley and Sons, 1962.
https://doi.org/10.1016/j.jhep.2012.11.045
[2] D. R. Hannay, “ Student drinking habits,” Alcohol and Alcoholism, vol. 33, no. 5, pp. 552–553, 1998. https://doi.org/10.1093/alcalc/33.5.552
[3] C. N. Brown and A. Gunn, “ Alcohol consumption in a student community,” The Practitioner, vol. 219, pp. 238–242, 1977.
[4] L. Ham and D. Hope, “ College students and problematic drinking: a review of literature,” Clin. Psychol. Rev., vol. 23, no. 5, pp. 719–759, 2003. https://doi.org/10.1016/S0272-7358(03)00071-0
[5] J. Gill, “ Reported levels of alcohol consumption and binge drinking within the UK undergraduate student population over the last 25 years,” Alcohol and Alcoholism, vol. 37, no. 2, pp. 109–120, 2002. https://doi.org/10.1093/alcalc/37.2.109
[6] M. Testa and K. Parks, “ The role of women’s alcohol consumption in sexual victimization,” Aggression and violent behavior, vol. 1, no. 3, pp. 217–234, 1996. https://doi.org/10.1016/1359-1789(95)00017-8
[7] S. Ullman, G. Karabatsos, and M. Koss, “ Alcohol and sexual assault in a national sample of college women,” Journal of interpersonal violence, vol. 14, no. 6, pp. 603–625, 1999. https://doi.org/10.1177/088626099014006003
[8] O. de las Naciones Unidas Contra la Droga y el Delito (UNODC) y el Ministerio de Justicia de Colombia, III Estudio epidemiológico andino sobre consumo de drogas en la población universitaria de Colombia, 2017.
[9] M. d. S. y. P. S. d. C. Ministerio de Justicia-Observatorio de Drogas de Colombia, Ministerio de Educación Nacional, Estudio nacional de consumo de sustancias psicoactivas en población escolar Colombia, 2016.
[10] W. Kermack and A. McKendrick, “ A contribution to the mathematical theory of epidemics,” Proc. Roy. Soc. London. Ser. A 115, no. 772, pp. 700–721, 1927. https://doi.org/10.1098/rspa.1927.0118
[11] F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology . Springer, 2001.
[12] D. J. Daley and J. Gani, Epidemic modelling: an introduction. Cambridge University Press, 1999.
[13] C. Castillo-Garsow, G. Jordan-Salivia, and A. Rodriguez Herrera, “ Mathematical models for the dynamics of tobacco use, recovery, and relapse,” Technical report series BU-1505-M, Cornell University, Ithaca, NY, USA, 2000. https://ecommons.cornell.edu/bitstream/handle/1813/ 32095/BU-1505-M.pdf;sequence=1
[14] R. Ullah, M. Khan, G. Zaman, S. Islam, M. Altaf Khan, S. Jan, and T. Gul, “ Dynamical features of a mathematical model on smoking,” J. Appl. Environ. Biol. Sci, vol. 6, pp. 92–96, 2016.
[15] I. Kwasi, M. Osman, and C. Yang, “ Mathematical model of drinking epidemic,” British Journal of Mathematics and Computer Science, vol. 22, pp. 1–10, 2017.
[16] M. Mancuso, “ A mathematical mode for alcoholism epidemic,” University of Dayton, 2016. https://ecommons.udayton.edu/cgi/viewcontent.cgiarticle=1758&context=stander_posters&httpsredir=1&referer=
[17] B. Buonomo and D. Lacitignola, “ Modeling peer influence effects on he spread of high-risk alcohol consumption behavior,” Ricerche mat, vol. 63, pp. 101–117, 2014. https://doi.org/10.1007/s11587-013-0167-3
[18] G. Mulone and B. Straughan, “ Modelling binge drinking,” Int. J. Biomath, vol. 5, no. 1, 2012.
https://doi.org/10.1142/S1793524511001453
[19] J. Manthey, A. Aidoo, and K. Ward, “ Campus drinking: an epidemiological model,” J. Biol. Dyn., vol. 2, pp. 346–356, 2008. https://doi.org/10.1080/17513750801911169
[20] O. Diekmann and J. Heesterbeek, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Chichester, UK: Wiley, 2000.
[21] O. Diekmann, J. Heesterbeek, and J. Metz, “ On the definition and computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations,” J. Math. Biol, vol. 28, pp. 365–382, 1990. https://doi.org/10.1007/BF00178324
[22] P. Van den Driessche and J. Watmough, “ Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci, vol. 180, no. 1–2, pp. 29–48, 2002. https://doi.org/10.1016/S0025-5564(02)00108-6
[23] B. Brown, The world’s youth: adolescence in eight regions of the globe. Cambridge University Press, 2002.
[24] B. Brown, Encyclopedia of adolescence . Academic Press, 2011.
[25] V. Ivaniushina, V. Titkova, and D. Alexandrov, “Peer influence in adolescent drinking behaviour: a protocol for systematic review and meta-analysis of stochastic actor-based modeling studies,” BMJ Open, vol. 9, no. 7, 2019. https://doi.org/10.1136/bmjopen-2018-028709
[26] B. Simons Morton, D. Haynie, A. Crump, S. Eitel, and K. Saylor, “ Peer and parent influences on smoking and drinking among early adolescents,” Health Educ Behav, vol. 28, pp. 95–107, 2001.
[27] C. Martin and M. Hoffman, “ Alcohol expectancies, living environment, peer influence, and gender: a model of college-student drinking,” Journal of college student development, vol. 34, pp. 206–211, 1993.
[28] P. O’Malley and L. Johnston, “ Epidemiology of alcohol and other drug use among American college students,” J. Stud. Alcohol, vol. 63, no. 2, pp. 23–39, 2002. https://www.collegedrinkingprevention.gov/media/ journal/023_039.pdf
[29] S. Betancourth, L. Tacán, and E. Cordoba, “ Alcohol consumption in Colombian university students,” Rev. Univ. Salud, vol. 19, pp. 37–50, 2017.
[30] L. Arriola and J. Hyman, Lectures notes, Forward and adjoint sensitivity analysis: with applications in Dynamical Systems, Linear Algebra and Optimization. Mathematical and Theoretical Biology Institute, 2005.
[31] A. Rachah and D. F. M. Torres, “ Mathematical modelling, simulation and optimal control of the 2014 Ebola outbreak in west Africa,” Discrete dynamics in nature and society, vol. 2015, 2015. https://doi.org/10.1155/2015/842792
[32] H. S. Rodrigues, M. T. T. Monteiro, and D. F. Torres, “Vaccination models and optimal control strategies to dengue,” Mathematical Biosciences, vol. 247, pp. 1 – 12, 2014. https://doi.org/10.1016/j.mbs.2013.10.006
[33] S. Djiomba and F. Nyabadza, “ An optimal control model for Ebola virus disease,” Journal of Biological Systems, vol. 24, no. 01, pp. 29–49, 2016. https://doi.org/10.1142/S0218339016500029
[34] S. Rosa and D. F. M. Torres, “ Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida,” Statistic Opt. Inform. Comput, vol. 6, no. 1, 2018. https://doi.org/10.19139/soic.v6i1.472
[35] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models. Chapman and Hall-CRC, Boca Raton, FL, USA, 2007.
[36] E. Bonyah, M. A. Khan, K. O. Okosun, and J. F. Gomez-Aguilar, “ Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control,” Mathematical Biosciences, vol. 309, pp. 1–11, 2019. https://doi.org/10.1016/j.mbs.2018.12.015
[37] B. Khajji, A. Labzai, A. Kouidere, O. Balatif, and M. Rachik, “ A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control,” Journal of Applied Mathematics, pp. 1–13, 2020. https://doi.org/10.1155/2020/9284698
[38] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes . John Wiley and Sons, 1962.
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Published
Nov 11, 2020
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Pérez, E. (2020). Mathematical Modeling of the Spread of Alcoholism Among Colombian College Students. Ingeniería Y Ciencia, 16(32), 195–223. https://doi.org/10.17230/ingciencia.16.32.9
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