Global dynamics of a class of SEIRS epidemic models in a periodic environment. (English) Zbl 1184.34056

Summary: We study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number \(R_{0}\) which is defined through the spectral radius of a linear integral operator. If \(R_{0}<1\), then the disease free periodic solution is globally asymptotically stable and if \(R_{0}>1\), then the disease persists. Our results improve the results in [T. Zhang and Z. Teng, Bull. Math. Biol. 69, No. 8, 2537–2559 (2007; Zbl 1245.34040)] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.


34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations


Zbl 1245.34040
Full Text: DOI


[1] Bacaër, Nicolas, Approximation of the basic reproduction number \(R^0\) for vector-borne diseases with a periodic vector population, Bull. math. biol., 69, 3, 1067-1091, (2007) · Zbl 1298.92093
[2] Bacaër, Nicolas; Guernaoui, Souad, The epidemic threshold of vector-borne diseases with seasonality. the case of cutaneous leishmaniasis in chichaoua, morocco, J. math. biol., 53, 3, 421-436, (2006) · Zbl 1098.92056
[3] Bacaër, Nicolas; Ouifki, Rachid, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. biosci., 210, 2, 647-658, (2007) · Zbl 1133.92023
[4] Hirsch, Morris W.; Smith, Hal L.; Zhao, Xiao-Qiang, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. dynam. differential equations, 13, 1, 107-131, (2001) · Zbl 1129.37306
[5] Li, Michael Y.; Graef, John R.; Wang, Liancheng; Karsai, János, Global dynamics of a SEIR model with varying total population size, Math. biosci., 160, 2, 191-213, (1999) · Zbl 0974.92029
[6] Li, Michael Y.; Muldowney, James S., Global stability for the SEIR model in epidemiology, Math. biosci., 125, 2, 155-164, (1995) · Zbl 0821.92022
[7] Liu, Wei Min; Hethcote, Herbert W.; Levin, Simon A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 4, 359-380, (1987) · Zbl 0621.92014
[8] Ma, Junling; Ma, Zhien, Epidemic threshold conditions for seasonally forced SEIR models, Math. biosci. eng., 3, 1, 161-172, (2006), (electronic) · Zbl 1089.92048
[9] Song, Mei; Ma, Wanbiao; Takeuchi, Yasuhiro, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. comput. appl. math., 201, 2, 389-394, (2007) · Zbl 1117.34310
[10] Sun, Chengjun; Lin, Yiping; Tang, Shoupeng, Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos solitons fractals, 33, 1, 290-297, (2007) · Zbl 1152.34357
[11] Thieme, Horst R., Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. amer. math. soc., 127, 8, 2395-2403, (1999) · Zbl 0918.34053
[12] Thieme, Horst R., Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. biosci., 166, 2, 173-201, (2000) · Zbl 0970.37061
[13] van den Driessche, P.; Watmough, James, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002), John A. Jacquez memorial volume · Zbl 1015.92036
[14] Wang, Wendi; Zhao, Xiao-Qiang, An epidemic model in a patchy environment, Math. biosci., 190, 1, 97-112, (2004) · Zbl 1048.92030
[15] Wang, Wendi; Zhao, Xiao-Qiang, Threshold dynamics for compartmental epidemic models in periodic environments, J. dynam. differential equations, 20, 3, 699-717, (2008) · Zbl 1157.34041
[16] Zhang, Fang; Zhao, Xiao-Qiang, A periodic epidemic model in a patchy environment, J. math. anal. appl., 325, 1, 496-516, (2007) · Zbl 1101.92046
[17] Zhang, Tailei; Teng, Zhidong, On a nonautonomous SEIRS model in epidemiology, Bull. math. biol., 69, 8, 2537-2559, (2007) · Zbl 1245.34040
[18] Zhang, Tailei; Teng, Zhidong; Gao, Shujing, Threshold conditions for a non-autonomous epidemic model with vaccination, Appl. anal., 87, 2, 181-199, (2008) · Zbl 1144.34032
[19] Zhao, Xiao-Qiang, Dynamical systems in population biology, CMS books math./ouvrages math. SMC, vol. 16, (2003), Springer-Verlag New York · Zbl 1023.37047
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.