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International Journal Of Chemistry, Mathematics And Physics(IJCMP)

Lyapunov Functions and Global Properties of SEIR Epidemic Model

M. M Ojo , F. O Akinpelu

International Journal of Chemistry, Mathematics And Physics(IJCMP), Vol-1,Issue-1, May - June 2017, Pages 11-16 ,

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The aim of this paper is to analyze an SEIR epidemic model in which prophylactic for the exposed individuals is included. We are interested in finding the basic reproductive number of the model which determines whether the disease dies out or persist in the population. The global attractivity of the disease-free periodic solution is obtained when the basic reproductive number is less than unity and the disease persist in the population whenever the basic reproductive number is greater than unity, i.e. the epidemic will turn out to endemic. The linear and non–linear Lyapunov function of Goh–Volterra type was used to establish the sufficient condition for the global stability of the model.

Epidemic Model; Lyapunov function; Global stability; Basic Reproduction number.

[1] Adebimpe, O., Bashiru, K.A and Ojurongbe, T.A. (2015) stability Analysis of an SIR Epidemic with Non- Linear Incidence Rate and Treatment. Open Journal of Modelling and Simulation, 3, 104-110. http://dx.doi.org/10.4236/ojmsi.2015.33011
[2] Anderson, R.M., May, R.M., (1978). Regulation and stability of host–parasite population interactious II: Destabilizing process. J. Anim. Ecol. 47, 219–267.
[3] Anderson, R.M., May, R.M., (1992). Infectious Disease of Humans, Dynamical and Control. Oxford University Press, Oxford.
[4] Cooke K, Van den Driessche P (1996). Analysis of an SEIRS epidemic model with two delay. J Math Biol; 35:240.
[5] Diekmann, O., Heesterbeek, J.A.P., (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York, Chichester.
[6] Greenhalgh D (1992). Some results for a SEIR epidemic model with density dependence in the death rate. IMA J Math Appl Med Biol; 9:67.
[7] Greenhalgh D (1997). Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity. Math Comput Model; 25:85.
[8] Hethcote H.W. (2000). The mathematics of infectious diseases. SIAM Rev., 42(4): 599-653
[9] Kermark, M.D., Mckendrick, A.G., (1927). Contributions to the mathematical theory of epidemics. Part I Proc. Roy. Soc. A 115, 700–721.
[10] Korobeinikov A (2004). Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math Med Biol; 21: 75-83.
[11] La Salle, J., Lefschetz, S. (1976): The stability of dynamical systems. SIAM, Philadelphia.
[12] Li MY, Smith HL, Wang L. (2001) Global dynamics of an SEIR epidemic model with vertical transmission. SIAM J Appl Math; 62:58.
[13] Li G, Jin Z. (2013) Global dynamic of SEIR epidemic model with saturating contact rate. Math Biosci; 185:15-32.
[14] Li G, Jin Z. (2005) Global dynamic of SEIR epidemic model with infectious force in latent infected and immune period. Chaos, Solutions &Fractals; 25:77-84.
[15] Liu W-M, Hethcote H.W. (1987) Levin SA. Dynamical behaviour of epidemiological models with nonlinear incidence rate. J Math Biol; 25:59-80.
[16] Liu, W.M., Levin, S.A., Iwasa, Y., (1986). Influence of nonlinear incidence transmission rates upon the behavior of SIRS epidemic models. J. Math. Biol. 23, 187–204.
[17] Lu, Z., Chi, X., Chen, L., (2002). The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math.Comput.Model. 36, 1039–1057.
[18] P. van den Driessche, J. Watmough. (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 29–48.
[19] S. Lakshmikantham, S. Leela, A.A. Martynyuk. (1989) Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, Basel.
[20] Tailei Zhang, Zhidong Teng (2008). An SIRVS epidemic model with pulse vaccination strategy .Journal of Theoretical Biology, 250 375–381.
[21] Van den Driessche, P.,Watmough, J. (2002): Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math.Biosci.180, 29–48.