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Influence of relapse in a giving up smoking model. (English) Zbl 1402.92241

Summary: Smoking subject is an interesting area to study. The aim of this paper is to derive and analyze a model taking into account light smokers compartment, recovery compartment, and two relapses in the giving up smoking model. Stability of the model is obtained. Some numerical simulations are also provided to illustrate our analytical results and to show the effect of controlling the rate of relapse on the giving up smoking model.

MSC:

92C50 Medical applications (general)
92D30 Epidemiology

Keywords:

smoking
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[1] Ruan, S. G.; Wang, W. D., Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188, 1, 135-163, (2003) · Zbl 1028.34046
[2] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bulletin of Mathematical Biology, 69, 6, 1871-1886, (2007) · Zbl 1298.92101
[3] de la Sen, M.; Alonso-Quesada, S., Vaccination strategies based on feedback control techniques for a general SEIR epidemic model, Applied Mathematics and Computation, 218, 7, 3888-3904, (2011) · Zbl 1238.92030
[4] de La Sen, M.; Agarwal, R. P.; Ibeas, A.; Alonso-Quesada, S., On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls, Advances in Difference Equations, 2010, (2010) · Zbl 1219.34104
[5] Lyapunov, A. M., The General Problem of the Stability of Motion, (1992), London, UK: Taylor & Francis, London, UK · Zbl 0786.70001
[6] Korobeinikov, A., Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bulletin of Mathematical Biology, 71, 1, 75-83, (2009) · Zbl 1169.92041
[7] O’Regan, S. M.; Kelly, T. C.; Korobeinikov, A.; O’Callaghan, M. J. A.; Pokrovskii, A. V., Lyapunov function for SIR and SIRS epidmic models, Applied Mathematics Letters, 23, 446-448, (2010) · Zbl 1193.34102
[8] Buonomo, B.; Rionero, S., On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate, Applied Mathematics and Computation, 217, 8, 4010-4016, (2010) · Zbl 1203.92049
[9] Hu, Z. H.; Bi, P.; Ma, W. B.; Ruan, S. G., Bifurcations of an sirs epidemic model with nonlinear incidence rate, Discrete and Continuous Dynamical Systems B, 15, 1, 93-112, (2011) · Zbl 1207.92040
[10] Li, G. H.; Wang, W. D.; Jin, Z., Global stability of an SEIR epidemic model with constant immigration, Chaos, Solitons and Fractals, 30, 4, 1012-1019, (2006) · Zbl 1142.34352
[11] Korobeinikov, A., Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66, 4, 879-883, (2004) · Zbl 1334.92409
[12] Souza, M. O.; Zubelli, J. P., Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bulletin of Mathematical Biology, 73, 3, 609-625, (2011) · Zbl 1225.92022
[13] Huo, H. F.; Dang, S. J.; Li, Y. N., Stability of a two-strain tuberculosis model with general contact rate, Abstract and Applied Analysis, 2010, (2010) · Zbl 1217.34082
[14] Castillo-Chavez, C.; Song, B. J., Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1, 361-404, (2004) · Zbl 1060.92041
[15] Huo, H. F.; Feng, L. X., Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Applied Mathematical Modelling, 37, 3, 1480-1489, (2013) · Zbl 1351.34044
[16] Xu, R., Global stability of an HIV-1 infection model with saturation infection and intracellular delay, Journal of Mathematical Analysis and Applications, 375, 1, 75-81, (2011) · Zbl 1222.34101
[17] Castillo-Garsow, C.; Jordan-Salivia, G.; Rodriguez-Herrera, A., Mathematical models for the dynamics of tobacco use, recovery, and replase, BU-1505-M, (2000), Cornell University
[18] Zaman, G., Qualitative behavior of giving up smoking models, Bulletin of the Malaysian Mathematical Sciences Society, 34, 2, 403-415, (2011) · Zbl 1221.92067
[19] Zaman, G., Optimal campaign in the smoking dynamics, computation and mathematical methods in medicine, 2011, (2011) · Zbl 1207.92051
[20] Qesmi, R.; Wu, J.; Wu, J. H.; Heffernan, J. M., Influence of backward bifurcation in a model of hepatitis B and C viruses, Mathematical Biosciences, 224, 2, 118-125, (2010) · Zbl 1188.92017
[21] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180, 29-48, (2002) · Zbl 1015.92036
[22] LaSalle, J. P., The Stability of Dynamical Systems. The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, (1976), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 0364.93002
[23] World Health Statistics
[24] WHO Report of the Global Tobacco Epidemic
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