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THE DEMON DRINK

Part of: Mathematical sociology

Published online by Cambridge University Press:  02 November 2017

MARK IAN NELSON Open the ORCID record for MARK IAN NELSON [Opens in a new window] ,
PETER HAGEDOORN  and
ANNETTE L. WORTHY
MARK IAN NELSON*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia email nelsonm@member.ams.org, ph247@uowmail.edu.au, annie@uow.edu.au
PETER HAGEDOORN
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia email nelsonm@member.ams.org, ph247@uowmail.edu.au, annie@uow.edu.au
ANNETTE L. WORTHY
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia email nelsonm@member.ams.org, ph247@uowmail.edu.au, annie@uow.edu.au
*
✉nelsonm@member.ams.org
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Abstract

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We provide a qualitative analysis of a system of nonlinear differential equations that model the spread of alcoholism through a population. Alcoholism is viewed as an infectious disease and the model treats it within a sir framework. The model exhibits two generic types of steady-state diagram. The first of these is qualitatively the same as the steady-state diagram in the standard sir model. The second exhibits a backwards transcritical bifurcation. As a consequence of this, there is a region of bistability in which a population of problem drinkers can be sustained, even when the reproduction number is less than one. We obtain a succinct formula for this scenario when the transition between these two cases occurs.

Keywords

alcohol consumptionbackwards bifurcationbinge drinkingcollege studentsepidemiologyequilibriasocial influencestability

MSC classification

Primary: 92D30: Epidemiology
Secondary: 91D30: Social networks 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 135 - 154
Copyright
© 2017 Australian Mathematical Society 

References

Anderson, R. M. and May, R. M., Infectious diseases of humans (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
Bhunu, C. P., “A mathematical analysis of alcoholism”, World J. Modell. Simul. 8 (2012) 124–134; http://www.wjms.org.uk/wjmsvol08no02paper05.pdf.Google Scholar
Brauer, F. and Castillo-Chávez, C., Mathematical models in population biology and epidemiology, 1st edn. Volume 40 of Texts in Appl. Math. (Springer, Berlin, 2001).CrossRefGoogle Scholar
Brauer, F., van den Driessche, P. and Wu, J. (eds), Mathematical epidemiology, Volume 1945 of Lect. Notes in Math. (Springer, Berlin–Heidelberg, 2008).Google Scholar
Buonomo, B. and Lacitignola, D., “Modeling peer influence effects on the spread of high-risk alcohol consumption behavior”, Ric. Mat. 63 (2014) 101–117; doi:10.1007/s11587-013-0167-3.Google Scholar
Cintrón-Arias, A., Sánchez, F., Wang, X., Castillo-Chavez, C., Gorman, D. M. and Gruenewald, P. J., “The role of nonlinear relapse on contagion amongst drinking communities”, in: Mathematical and statistical estimation approaches in epidemiology (eds Castillo-Chavez, C., Bettencourt, L. M. A. and Chowell, G.), (Springer, The Netherlands, 2009) 343–360; doi:10.1007/978-90-481-2313-1_14.Google Scholar
Diekmann, O., Heesterbeek, J. A. P. and Britton, T., Mathematical tools for understanding infectious disease dynamics (Princeton University Press, Princeton, NJ, 2013).Google Scholar
Dulac, H., Points singulieres des équations differentielles, Volume 61 of Mém. Sci. Math. (Gauthier-Villars, Paris, 1934).Google Scholar
Eastwood, C. J., Mahajan, V. and Muller, E., “A nonuniform influence innovation diffusion model of new product acceptance”, Market. Sci. 2 (1983) 273–295; doi:10.1287/mksc.2.3.273.Google Scholar
Gomes, M. G. M., White, L. J. and Medley, G. F., “Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives”, J. Theoret. Biol. 228 (2004) 539–549; doi:10.1016/j.jtbi.2004.02.015.CrossRefGoogle ScholarPubMed
González, B., Huerta-Sánchez, E., Ortiz-Nieves, A., Vázquez-Alvarez, T. and Kribs-Zaleta, C., “Am I too fat? Bulimia as an epidemic”, J. Math. Psych. 47 (2003) 515–526; doi:10.1016/j.jmp.2003.08.002.Google Scholar
Hagedoorn, P., “The impact of media campaigns on the spread of alcoholism”, Undergraduate honours Thesis, School of Mathematics and Applied Statistics, The University of Wollongong, 2015.Google Scholar
Harrison, D., “Australian alcohol consumption at 50-year low, ABS says”. in: The Sydney Morning Herald (7 May 2015).Google Scholar
Hill, A. L., Rand, D. G., Nowak, M. A. and Christakis, N. A., “Infectious disease modeling of social contagion in networks”, PLOS Comput. Biol. 6 (2010) e1000968 doi:10.1371/journal.pcbi.1000968.CrossRefGoogle ScholarPubMed
Hindmarsh, C. S., Jones, S. C. and Kervin, L., “Effectiveness of alcohol media literacy programmes: a systematic literature review”, Health Educ. Res. 30 (2015) 449–465; doi:10.1093/her/cyv015.Google Scholar
Huo, H.-F. and Song, N.-N., “Global stability for a binge drinking model with two stages”, Discrete Dyn. Nat. Soc. 2012 (2012) Article ID: 829386; doi:10.1155/2012/829386.CrossRefGoogle Scholar
Huo, H.-F. and Wang, Q., “Modelling the influence of awareness programs by media on the drinking dynamics”, Abstr. Appl. Anal. 2014 (2014) Article ID: 938080; doi:10.1155/2014/938080.Google Scholar
Huo, H.-F. and Zhu, C.-C., “Influence of relapse in a giving up smoking model”, Abstr. Appl. Anal. 2013 (2013) Article ID: 525461; doi:10.1155/2013/525461.Google Scholar
Jódar, L., Santonja, F. J. and González-Parra, G., “Modeling dynamics of infant obesity in the region of Valencia, Spain”, Comput. Math. Appl. 56 (2008) 679–689; doi:10.1016/j.camwa.2008.01.011.Google Scholar
Jones, J. D., Comer, S. D. and Kranzler, H. R., “The pharmacogenetics of alcohol use disorder”, Alcohol. Clin. Exp. Res. 39 (2015) 391–402; doi:10.1111/acer.12643.Google Scholar
Jordan, D. W. and Smith, P., Nonlinear ordinary differential equations, 2nd edn Oxford Appl. Math. Comput. Ser. (Clarendon Press, Oxford, 1989).Google Scholar
Kribs-Zaleta, C. M., “Sociological phenomena as multiple nonlinearities: MTBI’s new metaphor for complex human interactions”, Math. Biosci. Eng. 10 (2013) 1587–1607; doi:10.3934/mbe.2013.10.1587.Google Scholar
Manthey, J. L., Aidoo, A. Y. and Ward, K. Y., “Campus drinking: an epidemiological model”, J. Biol. Dyn. 2 (2008) 346–356; doi:10.1080/17513750801911169.Google Scholar
Mubayi, A., Greenwood, P. E., Castillo-Ch’avez, C., Gruenewald, P. J. and Gorman, D. M., “The impact of relative residence times on the distribution of heavy drinkers in highly distinct environments”, Socio-Econ. Plann. Sci. 44 (2010) 45–56; doi:10.1016/j.seps.2009.02.002.CrossRefGoogle ScholarPubMed
Mulone, G. and Straughan, B., “Modeling binge drinking”, Int. J. Biomath. 5 (2009) 1250005; doi:10.1142/S1793524511001453.Google Scholar
Mulone, G. and Straughan, B., “A note on heroin epidemics”, Math. Biosci. 218 (2009) 138–141; doi:10.1016/j.mbs.2009.01.006.Google Scholar
Nowak, M. A. and May, R. M., Virus dynamics, 1st edn (Oxford University Press, Oxford, 2000).Google Scholar
Perko, L., Differential equations and dynamical systems, 2nd edn. Volume 7 of Texts in Appl. Math. (Springer, New York, 1996).CrossRefGoogle Scholar
Roberts, M. G., “The pluses and minuses of $r_{0}$ ”, J. R. Soc. Interface 4 (2007) 949–961; doi:10.1098/rsif.2007.1031.Google Scholar
Sánchez, F., Wang, X., Castillo-Chávez, C., Gorman, D. M. and Gruenewald, P. J., “Drinking as an epidemic – a simple mathematical model with recovery and relapse”, in: Therapist’s guide to evidence-based relapse prevention (eds Witkiewitz, K. and Marlatt, G.), (Academic Press, Burlington, MA, 2007) Chapter 16, 353–368; doi:10.1016/B978-012369429-4/50046-X.Google Scholar
Sharomi, O. and Gumel, A. B., “Curtailing smoking dynamics: a mathematical modeling approach”, Appl. Math. Comput. 195 (2008) 475–499; doi:10.1016/j.amc.2007.05.012.Google Scholar
Smith, R., Modelling disease ecology with mathematics, Volume 2 of Differential Equ. Dyn. Syst. (American Institute of Mathematical Sciences, Springfield, MO, 2008).Google Scholar
Walters, C. E., Straughan, B. and Kendal, J. R. , “Modelling alcohol problems: total recovery”, Ric. Mat. 62 (2013) 33–53; doi:10.1007/s11587-012-0138-0.Google Scholar
White, E. and Comiskey, C., “Heroin epidemics, treatment and ODE modelling”, Math. Biosci. 208 (2007) 312–324; doi:10.1016/j.mbs.2006.10.008.Google Scholar