## Global stability for a binge drinking model with two stages.(English)Zbl 1253.91157

Summary: A more realistic two-stage model for binge drinking problem is introduced, where the youths with alcohol problems are divided into those who admit the problem and those who do not admit it. We also consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers. Mathematical analyses establish that the global dynamics of the model are determined by the basic reproduction number, $$R_0$$. The alcohol-free equilibrium is globally asymptotically stable, and the alcohol problems are eliminated from the population if $$R_0 < 1$$. A unique alcohol-present equilibrium is globally asymptotically stable if $$R_0 > 1$$. Numerical simulations are also conducted in the analytic results.

### MSC:

 91D30 Social networks; opinion dynamics 37N40 Dynamical systems in optimization and economics 91B74 Economic models of real-world systems (e.g., electricity markets, etc.)

### Keywords:

two stage model; binge drinking problem
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### References:

 [1] L. D. Johnston, P. M. OMalley, and J. G. Bachman, “National survey results on drug use from the monitoring the future study, 1975-1992,” National Institute on Drug Abuse, Rockville, Md, USA, 1993. [2] H. Wechsler, J. E. Lee, M. Kuo, and H. Lee, “College binge drinking in the 1990s: a continuing problem. Results of the Harvard School of Public Health 1999 College Alcohol Study,” Journal of American College Health, vol. 48, no. 5, pp. 199-210, 2000. [3] P. M. O’Malley and L. D. Johnston, “Epidemiology of alcohol and other drug use among American college students,” Journal of Studies on Alcohol, vol. 63, no. 14, pp. 23-39, 2002. [4] P. Ormerod and G. Wiltshire, ““Binge” drinking in the UK: a social network phenomenon,” Mind and Society, vol. 8, no. 2, pp. 135-152, 2009. [5] H. Wechsler, J. E. Lee, T. F. Nelson, and M. Kuo, “Underage college students’ drinking behavior, access to alcohol, and the influence of deterrence policies: findings from the Harvard School of Public Health College Alcohol Study,” Journal of American College Health, vol. 50, no. 5, pp. 223-236, 2002. [6] S. Mushayabasa and C. P. Bhunu, “Modelling the effects of heavy alcohol consumption onthetransmission dynamics of gonorrhea,” Nonlinear Dynamics, vol. 66, pp. 695-706, 2011. · Zbl 1242.92041 [7] G. Thomas and E. M. Lungu, “The influence of heavy alcohol consumption on HIV infection and progression,” Journal of Biological Systems, vol. 17, no. 4, pp. 685-712, 2009. · Zbl 1342.92115 [8] L. Deacon, S. Hughes, K. Tocque, and M. A. Bellis, Eds., “Indications of public health in English regions. 8,” Alcohol, Association of Public Health Observatories, York, UK, 2007. [9] G. Hay, M. Gannon, J. MacDougall, T. Millar, C. Eastwood, and N. McKeganey, “Local and national estimates of the prevalence of opiate use and/or crack cocaine use,” in Measuring Different Aspects of Problem Drug Use: Methodological Developments, N. Singleton, R. Murray, and L. Tinsley, Eds., Online report OLR 16/06, London, UK, 2006. [10] M. S. Goldman, G. M. Boyd, and V. Faden, “College drinking, what is it, and what to do about it: a review of the state of the science,” Journal of Studies for Alcohol, vol. 14, pp. 1-250, 2002. [11] F. Sanchez, X. H. Wang, C. Castillo-Chavez, D. M. Gorman, and P. J. Gruenewald, “Drinking as an epidemica simple mathematical model with recovery and relapse,” in Therapists Guide to Evidence-Based Relapse Prevention: Practical Resources for the Mental Health Professional, K. A. Witkiewitz and G. A. Marlatt, Eds., pp. 353-368, Academic Press, Burlington, Vt, USA, 2007. [12] J. L. Manthey, A. Y. Aidoo, and K. Y. Ward, “Campus drinking: an epidemiological model,” Journal of Biological Dynamics, vol. 2, no. 3, pp. 346-356, 2008. · Zbl 1154.92323 [13] A. Cintron-Arias, F. Sanchez, X. H. Wang, C. Castillo-Chavez, D. M. Gorman, and P. J. Gruenewald, “The role of nonlinear relapse on contagion amongst drinking communities,” in Mathematical and Statistical Estimation Approaches in Epidemiology, pp. 343-360, Springer, New York, NY, USA, 2009. · Zbl 1345.92137 [14] A. Mubayi, P. E. Greenwood, C. Castillo-Chávez, P. J. Gruenewald, and D. M. Gorman, “The impact of relative residence times on the distribution of heavy drinkers in highly distinct environments,” Socio-Economic Planning Sciences, vol. 44, no. 1, pp. 45-56, 2010. [15] G. Mulone and B. Straughan, “Modeling binge drinking,” International Journal of Biomathematics, vol. 5, no. 1, Article ID 1250005, 14 pages, 2012. · Zbl 1297.92079 [16] S. Reinberg, “Third of Americans Have Alcohol Problems at Some Point,” 2007, http://www.washingtonpost.com/wp-dyn/content/article/2007/07/02/AR2007070201237.html. [17] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599-653, 2000. · Zbl 0993.92033 [18] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29-48, 2002. · Zbl 1015.92036 [19] V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125, Marcel Dekker, New York, Ny, USA, 1989. · Zbl 0676.34003 [20] X. Tian and R. Xu, “Stability analysis of a delayed SIR epidemic model with stage structure and nonlinear incidence,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 979217, 17 pages, 2009. · Zbl 1183.34130 [21] S. Yuan and B. Li, “Global dynamics of an epidemic model with a ratio-dependent nonlinear incidence rate,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 609306, 13 pages, 2009. · Zbl 1177.37089 [22] J. Li, Y. Yang, and Y. Zhou, “Global stability of an epidemic model with latent stage and vaccination,” Nonlinear Analysis. Real World Applications, vol. 12, no. 4, pp. 2163-2173, 2011. · Zbl 1220.34069
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