Examination of Stability Analysis of Sakarya and Turkey Scale Alcohol Use Model

This paper is devoted to studying the mathematical model of the alcohol-consuming population. For this purpose, the formulation of the model including the alcohol-consuming population is presented; then the balance points related to non-alcohol use and positive alcohol use are discussed. Hurwitz theorem is used to find the local stability of the model, and Lyapunov function theory is used to investigate the global stability. The same mathematical model with alcohol use is considered for Sakarya and Turkey, individual numerical results are presented, and stability analyzes are examined. Finally, using the numerical data, a simulation is made in Matlab with the Runge-Kutta fourth-order method.

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[1] Brownlee, J. (1909). Certain considerations on the causation and course of epidemics. Proc. R. Soc. Med. 2: 243–258
[2] Brownlee, J. (1912). The mathematical theory of random migration and epidemic distribution. Proc. R. Soc. Edinb. 31: 262–289
[3] Kermack, W.O., McKendrick, A.G. (1927). Contributions to the mathematical theory of epidemics, part 1. Proc. R. Soc. Edinb., Sect. A., Math., 115: 700–721
[4] Song, S., Wang, K., Wang, W. (2008). Dynamics of an HBV model with diffusion and delay. J. Theor. Biol. 253(1): 36–44
[5] Huo, H.F., Ma, Z.P. (2010). Dynamics of a delayed epidemic model with non-monotonic incidence rate. Commun. Nonlinear Sci. Numer. Simul. 15(2): 459–468
[6] McCluskey, C.C. (2010). Complete global stability for an SIR epidemic model with delay distributed or discrete. Nonlinear Anal., Real World Appl. 11(1): 55–59
[7] Ma, Z., Xu, R. (2009). Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal., Real World Appl. 10(5): 3175–3189
[8] Ma, Z., Xu, R. (2009). Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solitons Fractals 41(5): 2319–2325
[9] Cheng, S., Song, X. (2005). A delay-differential equation model of HIV infection of CD4+ T-cells. J. Korean Math. Soc. 42(5): 1071–1086
[10] Guglielmi N., Hairer, E. (2001). Implementing Radau IIA methods for stiff delay differential equations. J. Comput. Math. 67(1): 1–12
[11] Beretta, E., Ma, W., Takeuchi, Y. (2010). Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal., Theory Methods Appl. 42(6): 931–947
[12] Van den Driessche, P., Watmough, J. (2008). Further notes on the basic reproduction number. In: Mathematical Epidemiology. Lecture Notes in Mathematics, vol. 1945, pp. 159–178. Springer, Berlin
[13] D’Onofrio, A., Manfredi, P., Salinelli, E. (2007). Bifurcation thresholds in an SIR model with information-dependent vaccination. Math. Model. Nat. Phenom. 2(1),: 26–43
[14] Yi, N., Zhao, Z., Zhang, Q. (2009). Bifurcations of an SEIQS epidemic model. Int. J. Inf. Syst. Sci. 5(3–4): 296–310
[15] Anwar, M.N., Fathalla, A.R. (2012). Qualitative analysis of delayed SIR epidemic model with saturated incidence rate. Int. J. Differ. Equ. 2012, Article ID 408637
[16] Hethcote, H.W. (1976). Qualitative analyses of communicable disease models. Math. Biosci. 7: 335–356
[17] Alzahrani, E., Zeb, A. (2020). Stability analysis and prevention strategiesa of tobacco smoking model,/ doi.org./10.1186/s13661-019-01315-1
[18] Okongo, O.M. (2015). The local and global stability of the disease free eguilibrium in a coinfection model of HIV7AIDS, Tuberculosis and malaria, IOSR Journal of Mathematics/ISSN:2319-764X.Volume 11.,pp 1-13
[19] Bhattacharjee, A. (2015). A transmission model for HIV/AIDS in the presence of treatment, IOSR Journal of Mathematics/ISSN:2319-764X.Volume 11.,pp. 72-80
[20] Momani, S., Zaman, G., Zeb, A. (2013). Dynamics of a giving up smoking model, Elsevier, Applied mathematical modelling , 37.7: 5326-5334
[21] Adu , I.K. ,Mojeeb Al-Rahman El-Nur, O. , Yang, C. (2017). mathematical model of drinking epidemic, British journal of mathematics & Computer science, 22(5):1-10, ISSN:2231-0851
[22] Balatif, O., Khajii, B., Labzai,A., Rachik ,M. (2020).Mathematical modeling and analysis of an alcohol drinking model with the ınfluence of alcohol treatment centers, International journal of Mathematics and Matematical sciences, Volume 2020,ID:4903168, pp.12
[23] Röst, G., Tekeli, T. (2020). Stability and oscillations of multistage SIS models depend on the number of stages,Elsevıer :Applied mathematics and computation,380, DOI: 10.1016 / j.amc.2020.12525
[24] Alzzahrani, E., Beleanu, D., El-Desoky, M.M. (2021).Mathematical modeling and analysis of the novel coronavirus using atangana-Baleanu derivative, Elsevıer: Results in Physics, 25, 104240
[25] https://dosyasb.saglik.gov.tr/Eklenti/3613 4,siy2018trpdf.pdf (Access Date:01.02.2021)
[26] https://tuikweb.tuik.gov.tr/UstMenu.do.( Access Date:01.02.2021)
[27] Demirci, M., Eker, E. (2017). Üniversite öğrencilerinin madde bağımlılığı sıklığı ve madde kullanım özellikleri, Anadolu bil Meslek Yüksekokulu Dergisi, Cilt:12,s.10
[28] https://www.milliyet.com.tr/gundem/iste-turkiyenin-alkol-haritasi-1714739 (Access Date:15.01.2021)