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Deterministic and stochastic stability of a mathematical model of smoking. (English) Zbl 1219.92043
Summary: Our aim in this paper is first constructing a Lyapunov function to prove the global stability of the unique smoking-present equilibrium state of a mathematical model of smoking. Next we incorporate random noise into the deterministic model. We show that the stochastic model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. Then a stochastic Lyapunov method is performed to obtain the sufficient conditions for mean square and asymptotic stability in probability of the stochastic model. Our analysis reveals that the stochastic stability of the smoking-present equilibrium state, depends on the magnitude of the intensities of noise as well as the parameters involved within the model system.

MSC:
92C50 Medical applications (general)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
37N25 Dynamical systems in biology
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[1] Afanas’ev, V.N.; Kolmanowskii, V.B.; Nosov, V.R., Mathematical theory of control systems design, (1996), Kluwer Academic Dordrecht · Zbl 0845.93001
[2] Bandyopadhyaya, M.; Saha, T.; Pal, R., Deterministic and stochastic analysis of a delayed allelopathic phytoplankton model within fluctuating environment, Nonlinear anal. hybrid syst., 2, 958-970, (2008) · Zbl 1218.34098
[3] Beretta, E.; Kolmanovskii, V.; Shaikhet, L., Stability of epidemic model with time delay influenced by stochastic perturbations, Math. comput. simulation, 45, 269-277, (1998) · Zbl 1017.92504
[4] Castillo, G.C., Jordan, S.G., Rodriguez, A.H., 2000. Mathematical models for the dynamics of tobacco use, recovery and relapse. Technical Report Series. BU-1505-M, Department of Biometrics, Cornell University.
[5] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model for internal HIV dynamics, J. math. anal. appl., 341, 1084-1101, (2008) · Zbl 1132.92015
[6] Harrison, G.W., Global stability of predator – prey interactions, J. math. biol., 8, 159-171, (1979) · Zbl 0425.92009
[7] Hasminskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoof & Noordhoof Alphen aan den Rijn, The Netherlands · Zbl 0419.62037
[8] Korobeinikov, A., Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. math. biol., 68, 615-626, (2006) · Zbl 1334.92410
[9] Korobeinikov, A.; Maini, P.K., Nonlinear incidence and stability of infectious disease models, Math. med. biol., 22, 113-128, (2005) · Zbl 1076.92048
[10] Lahrouz, A.; Omari, L.; Kiouach, D., Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear anal. model. control, 16, 59-76, (2011) · Zbl 1271.93015
[11] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J. math. anal. appl., 334, 69-84, (2007) · Zbl 1113.92052
[12] Mackey, M.C.; Nechaeva, I.G., Noise and stability in differential delay equations, J. differential equations, 6, 395-426, (1994) · Zbl 0807.34092
[13] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publishing Limited Chichester · Zbl 0874.60050
[14] Sarkar, R.R.; Banerjee, S., Cancer self remission and tumor stability—a stochastic approach, Math. biosci., 196, 65-81, (2005) · Zbl 1071.92017
[15] Shoromi, O.; Gumel, A.B., Curtailing smoking dynamics: a mathematical modeling approach, Appl. math. comput., 195, 475-499, (2008) · Zbl 1261.92023
[16] Takeuchi, Y.; Adachi, N., The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. math. biol., 401-415, (1980) · Zbl 0458.92019
[17] WHO, 2010a. http://www.emro.who.int/tfi/facts.htm#fact2.
[18] WHO, 2010b. http://www.emro.who.int/tfi/facts.htm#fact9.
[19] World Health Organization report on the global tobacco epidemic, 2009. http://whqlibdoc.who.int/publications/2009/9789241563918_eng_full.pdf.
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