Global behavior of an SEIRS epidemic model with time delays. (English) Zbl 1015.92033

Wang, Wendi

Appl. Math. Lett. 15, No. 4, 423-428 (2002).

Summary: This is a study of the dynamic behavior of an SEIRS epidemic model with time delays. It is shown that a disease-free equilibrium is globally stable if the reproduction number is not greater than one. When the reproduction number is greater than 1, it is proved that the disease is uniformly persistent in the population, and explicit formulae are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Local stability of an endemic equilibrium is also discussed.

MSC:

92D30	Epidemiology
34K20	Stability theory of functional-differential equations
34K60	Qualitative investigation and simulation of models involving functional-differential equations

Keywords:

global stability; persistence; delay

References:

[1]	Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33, 250-260 (1995) · Zbl 0811.92019
[2]	Beretta, E.; Takeuchi, Y., Convergence results in SIR epidemic model with varying population sizes, Nonlinear Analysis, 28, 1909-1921 (1997) · Zbl 0879.34054
[3]	Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Analysis, 42, 931-947 (2000) · Zbl 0967.34070
[4]	Cooke, K. L.; Van Den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35, 240-260 (1996) · Zbl 0865.92019
[5]	Hale, J. K.; Lunel, S. M.V., Introduction to Functional Differential Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002
[6]	Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston, MA · Zbl 0777.34002
[7]	Busenberg, S.; Cooke, K. L., The effect of integral conditions in certain equations modelling epidemics and population growth, J. Math. Biol., 10, 13-32 (1980) · Zbl 0464.92022
[8]	Hethcote, H. W.; Van Den Driessche, P., Two SIS epidemiological models with delays, J. Math. Biol., 40, 2-26 (2000) · Zbl 0959.92025
[9]	Hethcote, H. W.; Van Den Driessche, P., An SIS epidemic model with variable population size and a delay, J. Math. Biol., 34, 177-194 (1995) · Zbl 0836.92022
[10]	Gao, L. Q.; Hethcote, H. W., Disease transmission models with density-dependent demographics, J. Math. Biol., 30, 717-731 (1992) · Zbl 0774.92018
[11]	Hyman, J. M.; Li, J., Behavior changes in SIS STD models with selective mixing, SIAM. J. Appl. Math., 57, 1082-1094 (1997) · Zbl 0886.34046
[12]	Zhou, J.; Hethcote, H. W., Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32, 809-834 (1994) · Zbl 0823.92027
[13]	Gao, L. Q.; Mena-Lorca, J.; Hethcote, H. W., Four SEI endemic models with periodicity and separatrices, Math. Biosci., 128, 157-184 (1995) · Zbl 0834.92021
[14]	Mena-Lorca, J.; Hethcote, H. W., Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30, 693-716 (1992) · Zbl 0748.92012
[15]	Busenberg, S.; Van Den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28, 257-270 (1990) · Zbl 0725.92021
[16]	Beretta, E.; Takeuchi, Y., Convergence in SIR epidemic models with varying population sizes, Nonlinear Analysis, 28, 1909-1921 (1997) · Zbl 0879.34054