×

zbMATH — the first resource for mathematics

The stability of an SEIRS model with nonlinear incidence, vertical transmission and time delay. (English) Zbl 1329.92139
Summary: In this paper, nonlinear incidence with a more general form and vertical transmission and the immunity period are considered in an SEIRS epidemic model. The basic reproductive number is obtained. If the basic reproductive number is smaller than one, the disease free equilibrium is asymptotically stable. When the basic reproductive number is bigger than one, regardless of the time delay length there exists a unique endemic equilibrium which is locally asymptotically stable under some conditions. By mathematical analysis and numerical simulations, the result shows that the immunity period and vertical transmission can influence the dynamic behaviors of the SEIRS system. To prolong the immunity period of the recovered and to reduce the part of vertical transmission by some measures are both favorable for controlling the disease.

MSC:
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bellenir, K.; Dresser, P., Contagious and Non-Contagious Infectious Diseases Sourcebook, Health Science Series, vol. 8, (1996), Omnigraphics Inc. Detroit
[2] Busenberg, S. N.; Cooke, K. L.; Pozio, M. A., Analysis of a model of a vertically transmitted disease, J. Math. Biol., 17, 305-329, (1983) · Zbl 0518.92024
[3] Busenberg, S. N.; Cooke, K. L., The population dynamics of two vertically transmitted infections, Theor. Popul. Biol., 33, 181-198, (1988) · Zbl 0638.92009
[4] Busenberg, S. N.; Cooke, K. L., Vertical transmitted diseases: models and dynamics, Biomathematics, vol. 23, (1993), Springer-Verlag Berlin · Zbl 0837.92021
[5] Cooke, K. L.; Busenberg, S. N., Vertical transmitted diseases, (Lakshmicantham, V., Nonlinear Phenomena in Mathematical Sciences, (1982), Academic Press New York), 189-197
[6] Cooke, K. L.; Driessche, Van den P., Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35, 240-260, (1990) · Zbl 0865.92019
[7] Cui, J. A.; Sun, Y. H.; Zhu, H. P., The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20, 31-53, (2008) · Zbl 1160.34045
[8] Cui, J. A.; Mu, X. X.; Wan, H., Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. Theor. Biol., 254, 275-283, (2008)
[9] Cui, J. A.; Tao, X.; Zhu, H. P., An SIS infection model incorporating media coverage, Rocky Mountain J. Math., 38, 1323-1334, (2008) · Zbl 1170.92024
[10] Fine, P. M., Vectors and vertical transmission, an epidemiological perspective, Ann. New York Acad. Sci., 266, 173-194, (1975)
[11] Grenhalgh, D., Some results for an SEIR epidemic model with density dependence in the death rate, Med. Biol. IMA J. Math. Appl., 9, 67-85, (1992)
[12] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. Comput. Model., 25, 85-93, (1997) · Zbl 0877.92023
[13] Hethcote, H. W.; Driessche, Van den P., Some epidemiological models with nonlinear incidence, J. Math. Biol., 29, 271-287, (1991) · Zbl 0722.92015
[14] Li, G. H.; Jin, Z., Global stability of a SEIR epedemic model with infectious force in latent, infected and immune period, Chaos Solitons Fractals, 25, 1177-1184, (2005) · Zbl 1065.92046
[15] Li, M. Y.; Muldoweney, J. S., Global stability for SEIR model in epidemiology, Math. Biosci., 125, 155-167, (1995)
[16] Li, M. Y.; Muldoweney, J. S.; Wang, L. C.; Karsai, J., Global dynamics of an SEIR epidemic model with a varying total population size, Math. Biosci., 160, 191-213, (1999) · Zbl 0974.92029
[17] Li, M. Y.; Smith, H. L.; Wang, L. C., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62, 58-69, (2001) · Zbl 0991.92029
[18] Li, X. Z.; Zhou, L. L., Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate, Chaos Solitons Fractals, 40, 874-884, (2009) · Zbl 1197.34077
[19] Liu, W. M.; Levin, S. A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23, 187-204, (1986) · Zbl 0582.92023
[20] Michael, Y.; Smith, H.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62, 58-69, (2001) · Zbl 0991.92029
[21] Sun, C. J.; Lin, Y. P.; Tang, S. P., Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos Solitons Fractals, 33, 290-297, (2007) · Zbl 1152.34357
[22] Wang, W. D., Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15, 423-428, (2002) · Zbl 1015.92033
[23] Zhang, J.; Ma, Z. E., Global stability of SEIR model with saturating contact rate, Math. Biosci., 185, 15-32, (2003) · Zbl 1021.92040
[24] Zhang, T. L.; Teng, Z. D., Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence, Chaos Solitons Fractals, 37, 1456-1468, (2008) · Zbl 1142.34384
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.