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Modeling peer influence effects on the spread of high-risk alcohol consumption behavior. (English) Zbl 1330.92121
Summary: Alcohol dependence is among the main healthy risky behavior due to the high relevance of negative health and social effect. We study a mathematical model, given by nonlinear ordinary differential equations, describing the spread of high-risk alcohol consumption behavior within a community of individuals. We describe the peer-influence effects on alcohol addiction by assuming that susceptibles become heavy drinkers through the mechanism of imitation. We show that the model may exhibit the phenomenon of backward bifurcation. This means that alcohol problems may persist in the population even if the basic reproduction number is less than one. Nonlinear stability analysis of equilibria is also provided.

92D30 Epidemiology
91C99 Social and behavioral sciences: general topics
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Anderson, R.M., May, R.M.: Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford (1991)
[2] Bass, F.M.: A new product growth for model consumer durables. Manag. Sci. 15, 215–227 (1969) · Zbl 1231.91323 · doi:10.1287/mnsc.15.5.215
[3] Beliakin, S.A., Bobrov, A.N., Pliusnin, S.V.: Interdependence between alcohol consumption and mortality from hepatic cirrhosis. Voenno-meditsinskiń≠ zhurnal 330, 48–54 (2009)
[4] Benedict, B.: Modeling alcoholism as a contagious disease: how ”infected” drinking buddies spread problem drinking. SIAM News. 40(3) (2007)
[5] Bhunu, C.P.: A mathematical analysis of alcoholism. World J. Model. Simul. 8, 124–134 (2012) · Zbl 1270.34118
[6] Brauer, F., van den Driessche, P., Wu, J. (eds.): Mathematical epidemiology. Lecture Notes in Mathematics. Mathematical biosciences subseries, vol. 1945, Springer, Berlin (2008) · Zbl 1159.92034
[7] Buonomo, B., Lacitignola, D.: On the dynamics of an SEIR epidemic model with a convex incidence rate. Ric. Mat. 57, 261–281 (2008) · Zbl 1232.34061 · doi:10.1007/s11587-008-0039-4
[8] Buonomo, B., Lacitignola, D.: On the backward bifurcation of a vaccination model with nonlinear incidence. Nonlinear Anal. Mod. Control 16, 30–46 (2011) · Zbl 1271.34045
[9] Buonomo, B., Lacitignola, D.: Forces of infection allowing for backward bifurcation in an epidemic model with vaccination and treatment. Acta Appl. Math. 122, 283–293 (2012) · Zbl 1254.37054
[10] Buonomo, B., Rionero, S.: On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate. Appl. Math. Comput. 217, 4010–4016 (2010) · Zbl 1203.92049 · doi:10.1016/j.amc.2010.10.007
[11] Capasso, V.: Mathematical structures of epidemic systems. Lecture Notes in Biomathematics, vol. 97. Springer, Berlin (1993) · Zbl 0798.92024
[12] Capone, F., De Cataldis, V., De Luca, R.: On the nonlinear stability of an epidemic SEIR reaction-diffusion model. Ric. Mat. 62, 161–181 (2013) · Zbl 1304.35078 · doi:10.1007/s11587-013-0151-y
[13] Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1, 361–404 (2004) · Zbl 1060.92041 · doi:10.3934/mbe.2004.1.361
[14] Diekmann, O., Heesterbeek, J.A.P.: Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation. John Wiley & Sons, Chichester (2000) · Zbl 0997.92505
[15] d’Onofrio, A., Manfredi, P. (eds.): Modeling the interplay between human behavior and the spread of infectious diseases. Springer (2013) · Zbl 1276.92002
[16] Easingwood, C.J., Mahajan, V., Muller, E.: A nonuniform influence innovation diffusion model of new product marketing science. 2, 273–295 (1983)
[17] Garcia-Tsao, G., et al.: Management and treatment of patients with cirrhosis and portal hypertension: recommendations from the Department of Veterans affairs Hepatitis C Resource Center Program and the National Hepatitis C Program. Am. J. Gastroenterol. 104, 1802–1829 (2009) · doi:10.1038/ajg.2009.191
[18] Gill, J.S.: Reported levels of alcohol consumption and binge drinking within the UK undergraduate student population over the last 25 years. Alcohol Alcohol 37, 109–120 (2002) · doi:10.1093/alcalc/37.2.109
[19] Guckenheimer, J., Holmes, P.: Nonlinear oscillations. Dynamical systems and bifurcations of vector fields. Springer-Verlag, Berlin (1983) · Zbl 0515.34001
[20] Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599–653 (2000) · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[21] Huo, H.F., Song N.N.: Global stability for a binge drinking model with two stages. Discret. Dyn. Nat. Soc. 2012 Article ID 829386 (2012) · Zbl 1253.91157
[22] Jin, Y., Wang, W., Xiao, S.: A SIRS model with a nonlinear incidence. Chaos Solitons Fractals 34, 1482–1497 (2007) · Zbl 1152.34339 · doi:10.1016/j.chaos.2006.04.022
[23] Lin, C.W., et al.: Heavy alcohol consumption increases the incidence of hepatocellular carcinoma in hepatitis B virus-related cirrhosis. J. Hepatol. 58, 730–735 (2013) · doi:10.1016/j.jhep.2012.11.045
[24] Mahajan, V., Peterson, R.A.: Models for innovation diffusion. SAGE Publications Inc., (1985) · Zbl 0661.90051
[25] Manthey, J.L., Aidoob, A., Ward, K.Y.: Campus drinking: an epidemiological model. J. Biol. Dyn. 2, 346–356 (2008) · Zbl 1154.92323 · doi:10.1080/17513750801911169
[26] Mulone, G., Straughan, B.: Modelling binge drinking. Int. J. Biomath. 5 Article ID 1250005 (2012) · Zbl 1297.92079
[27] National Institute for Health and Clinical Excellence (NICE) report: alcohol use disorders: diagnosis, assessment and management of harmful drinking and alcohol dependence, CG115. http://www.nice.org.uk/nicemedia/live/13337/53191/53191.pdf (2011). Accessed 12 Sep 2013
[28] O’Malley, P.M., Johnston, L.D.: Epidemiology of alcohol and other drug use among American college students. J. Stud. Alcohol 63, 23–39 (2002)
[29] Polich, J.M., Armor, D.J., Braiker, H.B.: Stability and change in drinking patterns. The course of alcoholism: four years after treatment, pp. 159–200. John Wiley & Sons, New York (1981)
[30] Rionero, S.: $$L\^2$$ L 2 stability of solutions to a nonlinear binary reaction-diffusion system of P.D.E.s. Rend. Mat. Acc. Lincei 16, 227–238 (2005) · Zbl 1150.35012
[31] Rionero, S.: A nonlinear $$L\^2$$ L 2 stability analysis for two species dynamics with dispersal. Math. Biosci. Eng. 3, 189–204 (2006) · Zbl 1090.92039 · doi:10.3934/mbe.2006.3.189
[32] Rionero, S.: A rigorous reduction of the $$L\^2$$ L 2 -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s to the stability of the solutions to a linear binary system of ODEs. J. Math. Anal. Appl. 319, 377–397 (2006) · Zbl 1099.35041 · doi:10.1016/j.jmaa.2005.05.059
[33] Rionero, S.: On the nonlinear stability of the critical points of an epidemic SEIR model via a novel Lyapunov function. Rend. Acc. Sci. Fis. Nat. Napoli 75, 115–129 (2008) · Zbl 1222.34053
[34] Rionero, S.: Stability of ternary reaction-diffusion dynamical systems. Rend. Lincei Mat. Appl. 22, 245–268 (2011) · Zbl 1242.35044
[35] Rionero, S.: Soret effects on the onset of convection in rotating porous layers via the ”auxiliary system method”. Ric. Mat. (2013). doi: 10.1007/s11587-013-0163-7 · Zbl 1305.76113
[36] Sanchez, F., Wang, X., Castillo-Chavez, C., Gorman, D.M., Gruenewald, P.J.: Drinking as an epidemic- a simple mathemathical model with recovery and relapse. In: Witkiewitz, K., Marlatt, G.A. (eds.) Therapist’s guide to evidence-based relapse prevention. Academic Press, New York (2007)
[37] Smith, L., Foxcroft, D.: Drinking in the UK: an exploration of trends. http://www.jrf.org.uk/sites/files/jrf/UK-alcohol-trends-FULL.pdf (2009). Accessed 12 Sep 2013
[38] van den Driessche, P., Watmough, J.: A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540 (2000) · Zbl 0961.92029 · doi:10.1007/s002850000032
[39] van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[40] van den Driessche, P., Watmough, J.: Epidemic solutions and endemic catastrophes. In: Dynamical systems and their applications in biology, Cape Breton Island, NS, 2001. Fields Inst. Commun., vol. 36, pp. 247–257. American Mathematical Society Providence, (2003) · Zbl 1162.92325
[41] Walters, C.E., Straughan, B., Kendal, J.R.: Modelling alcohol problems: total recovery. Ric. Mat. 62, 33–53 (2013) · Zbl 1330.92135 · doi:10.1007/s11587-012-0138-0
[42] Wechsler, H., Lee, J.E., Kuo, M., Lee, H.: College binge drinking in the 1990s: a continuing problem. Results of the Harvard School of Public Health 1999 College Alcohol Study. J. Am. College Health 48, 199–210 (2000) · doi:10.1080/07448480009599305
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