Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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DRINKING AS AN EPIDEMIC: A MATHEMATICAL MODEL WITH DYNAMIC BEHAVIOUR

  • Sharma, Swarnali (Department of Mathematics, Bengal Engineering and Science University) ;
  • Samanta, G.P. (Department of Mathematics, Bengal Engineering and Science University)
  • Received : 2012.01.10
  • Accepted : 2012.04.20
  • Published : 2013.01.30
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Abstract

In this paper we have developed a mathematical model of alcohol abuse. It consists of four compartments corresponding to four population classes, namely, moderate and occasional drinkers, heavy drinkers, drinkers in treatment and temporarily recovered class. Basic reproduction number $R_0$ has been determined. Sensitivity analysis of $R_0$ identifies ${\beta}_1$, the transmission coefficient from moderate and occasional drinker to heavy drinker, as the most useful parameter to target for the reduction of $R_0$. The model is locally asymptotically stable at disease free or problem free equilibrium (DFE) $E_0$ when $R_0$ < 1. It is found that, when $R_0$ = 1, a backward bifurcation can occur and when $R_0$ > 1, the endemic equilibrium $E^*$ becomes stable. Further analysis gives the global asymptotic stability of DFE. Our aim of this analysis is to identify the parameters of interest for further study with a view for informing and assisting policy-makers in targeting prevention and treatment resources for maximum effectiveness.

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References

  1. L. Arriola, J. Hyman, Lecture notes, Forward and adjoint sensitivity analysis: with applications in Dynamical Systems, Linear Algebra and Optimization, Mathematical and Theoretical Biology Institute, Summer, 2005.
  2. F. Brauer, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348 (2008) 433-443. https://doi.org/10.1016/j.jmaa.2008.07.042
  3. F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlay, New York, 2001.
  4. R. J. Braun, R. A. Wilson, J. A. Pelesko, J. R. Buchanan, Applications of small-world network theory in alcohol epidemiology, J. Stud. Alcohol 67 (2007) 591-599.
  5. S. Busenberg, C. Castillo-Chavez, Interaction, pair formation and force of infection terms in sexually transmitted diseases. In: C. Castillo-Chavez (ed.) Mathematical Epidemiology, Lecture Notes in Biomathematics 83 (1989) 280-300.
  6. J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.
  7. C. Castillo-Chavez, W. Huang, Competitive exclusion in gonorrhea models and other sexually-transmitted diseases, SIAM, J. Appl. Math. 56 (1996) 494-508. https://doi.org/10.1137/S003613999325419X
  8. C. Castillo-Chavez, B.Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering,1 (2004) 361-404. https://doi.org/10.3934/mbe.2004.1.361
  9. Centers for Disease Control and Prevention (2008), Alcohol and Public Health. http://www.cdc.gov/alcohol/index.htm. Cited 29 Apr 2008.
  10. Centers for Disease Control and Prevention (2008),Frequently Asked Questions: What do you mean by heavy drinking? http://www.cdc.gov/alcohol/faqs.htm# 10. Cited 11 May 2008.
  11. Centers for Disease Control and Prevention (2008),Frequently Asked Questions: What does moderate drinking means? http://www.cdc.gov/alcohol/faqs.htm# 6. Cited 11 May 2008.
  12. Centers for Disease Control and Prevention (2008), General Information on Alcohol Use and health. http://www.cdc.gov/alcohol/quickstats/generalinfo.htm. Cited 1 May 2008.
  13. G. Chowell, P. W. Fenimore, M. A. Castillo-Carsow, C. Castillo-Chavez, SARS out-breaks in Ontario, HOng Kong and Singapore: the role of diagnosis and isolation as a control mechanism, J. Theor. Biol, 224 (2003) 1-8. https://doi.org/10.1016/S0022-5193(03)00228-5
  14. A.Cintron-Arias, F.Sanchez, X. Wang, C. Castillo-Chavez, D.M.Gorman, P.J.Gruenewald, The role of nonlinear relapse on contagion amongst drinking communities.
  15. College Drinking (2008). http://www.collegedrinkingprevention.gov. Cited 11 May 2008.
  16. S. M. Garba, A. B. Gumel, M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Mathematical Biosciences 215 (2008) 11-25. https://doi.org/10.1016/j.mbs.2008.05.002
  17. H. Hethcote, The mathematics of infectious disease, SIAM Rev. 42 (2000) 599-653. https://doi.org/10.1137/S0036144500371907
  18. H. Hethcote, J. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics, Springer-Verlag, Berlin 56, 1984.
  19. M. Kot, Elements of Mathematical Ecology, Cambridge, Cambridge University Press, 2001.
  20. J. Liu, T. Zhang, Global behaviour of a heroin epidemic model with distributive delays, Applied Mathematics Letters 24 (2011) 1685-1692. https://doi.org/10.1016/j.aml.2011.04.019
  21. A.Mubayi, P.Greenwood, C.Castillo-Chavez, P.J.Gruenewald, D.M.Gorman,On the impact of Relative Residence Times, in highly distinct environments, on the distribution of heavy drinkers, Socio. Econ. Plan. Sci.(In Press).
  22. G. Mulone, B. Straughan, A note on heroin epidemics, Mathematical Biosciences 218 (2009) 138-141. https://doi.org/10.1016/j.mbs.2009.01.006
  23. National Institute of Alcohol Abuse and Alcoholism (2008) Five Year Strategic Plan.http://www.niaaa.nih.gov/publications/srtategicplan/NIAAASTRATEGICPLAN.htm. Cited 29 Apr 2008.
  24. National Institute of Alcohol Abuse and Alcoholism (2008) Frequently Asked Questions for the General Public. http://www.niaaa.nih.gov/FAQs/General-English/default.htm. Cited 29 Apr 2008.
  25. NHS Information Centre. http://www.ic.nhs.uk/wefiles/publications. Cited 26 May 2011.
  26. F. Nyabadza, S. D. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: Modelling substance abuse in a South African province, Mathematical Biosciences 225 (2010) 132-140. https://doi.org/10.1016/j.mbs.2010.03.002
  27. J. Orford, M. Krishnan, M. Balaam, M. Everitt, K. Van der Graaf, University student drinking: the role of motivational and social factors, Drug-Educ. Prev. Polic. 11 (2004) 407-421. https://doi.org/10.1080/09687630310001657944
  28. C. Parry, Substance abuse trends in the Western Capes: Summary (25/2/05), Alcohol and Drug Abuse Research Unit, Medical Research Council, 2005.
  29. E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991.
  30. G. P. Samanta, Dynamic behaviiour for a nonautonomous heroin epidemic model with time delay, J. Appl. Math. Comput. 35 (2009) 161-178.
  31. F. Sanchez, Studies in Epidemiology and Social Dynamics, Dissertation, Cornell University, 2006.
  32. F. Sanchez, X. Wang, C. Castillo-Cahvez, D. M.Gorman, P.J. Gruenwald, Drinking as an epidemic: a simple mathematical model with recovery and relapse,In: K. Witkiewitz, G.A. Marlett,(eds.) Therapists Guide to Evidence- Based Relapse Prevention: Practical Resources for the Mental Health Professional, Academic Press, Burlington (2007) 353-368.
  33. O.Sharomi, C.N. Podder, A. B. Gumel, E. H. Elbasha, J. Watmough, Role of incidence function in vaccine-induced backward bifurcation in some HIV models, Mathematical Biosciences 210 (2007) 436-463. https://doi.org/10.1016/j.mbs.2007.05.012
  34. B. Song, Seminar Notes, Backward or Forward at $R_0$ = 1, Mathematical and Theoretical Biology Institute, Summer, 2005.
  35. P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 (2002) 29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
  36. W. Wang, Backward bifurcation of an epidemic model with treatment, mathematical Biosciences 201 (2006) 58-71. https://doi.org/10.1016/j.mbs.2005.12.022
  37. E.R.Weitzman, A. Flokman, K.L.Folkman, H. Weschler, The relationship of alcohol outlet density to heavy and frequent drinking and drinking related problems among college students at eight universities, Health Place (2003) 1-6.
  38. E. White, C. Comiskey, Heroin epidemics, treatment and ODE modelling, Mathematical Biosciences 208 (2007) 312-324.
  39. X. Zhang, X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl. 348 (2008) 433-443. https://doi.org/10.1016/j.jmaa.2008.07.042

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