# zbMATH — the first resource for mathematics

Global dynamics of an SEIR epidemic model with saturating contact rate. (English) Zbl 1021.92040
Summary: J. A. P. Heesterbeek and J. A. P. Metz [J. Math. Biol. 31, 529-539 (1993; Zbl 0770.92021)] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number $$R_0$$ is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If $$R_0\leq 1$$, the disease-free equilibrium is globally stable and the disease always dies out. If $$R_0>1$$, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium are also analyzed.

##### MSC:
 92D30 Epidemiology 34D23 Global stability of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text:
##### References:
 [1] Heesterbeek, J.A.P.; Metz, J.A.J., The saturating contact rate in marriage and epidemic models, J. math. biol., 31, 529, (1993) · Zbl 0770.92021 [2] Thieme, H.R.; Castillo-Chavez, C., On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, (), 157 · Zbl 0687.92009 [3] Thieme, H.R., Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. biosci., 111, 99, (1992) · Zbl 0782.92018 [4] Brauer, F.; van den Driessche, P., Models for transmission of disease with immigration of infectives, Math. biosci., 171, 143, (2001) · Zbl 0995.92041 [5] Mena-Lorca, J.; Hethcote, H.W., Dynamic models of infectious diseases as regulators of population sizes, J. math. biol., 30, 693, (1992) · Zbl 0748.92012 [6] Brauer, F., Epidemic models in populations of varying size, () · Zbl 0684.92016 [7] Gao, L.Q.; Hethcote, H.W., Disease transmission models with density-dependent demographics, J. math. biol., 30, 717, (1992) · Zbl 0774.92018 [8] Greenhalgh, D., Some threshold and stability results for epidemic models with a density dependent death rate, Theor. pop. biol., 42, 130, (1992) · Zbl 0759.92009 [9] Bremermann, H.J.; Thieme, H.R., A competitive exclusion principle for pathogen virulence, J. math. biol., 27, 179, (1989) · Zbl 0715.92027 [10] Hethcote, H.W.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. math. biol., 29, 271, (1991) · Zbl 0722.92015 [11] Derrick, W.R.; van den Driessche, P., A disease transmission model in a non-constant population, J. math. biol., 31, 495, (1993) · Zbl 0772.92015 [12] Busenberg, S.; van den Driessche, P., Analysis of a disease transmission model with varying population size, J. math. biol., 29, 257, (1990) · Zbl 0725.92021 [13] Greenhalgh, D., Some results for a SEIR epidemic model with density dependence in the death rate, IMA J. math. appl. med. biol., 9, 67, (1992) · Zbl 0805.92025 [14] Cooke, K.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. math. biol., 35, 240, (1996) · Zbl 0865.92019 [15] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. comput. model., 25, 85, (1997) · Zbl 0877.92023 [16] Li, M.Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math. biosci., 125, 155, (1995) · Zbl 0821.92022 [17] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. biosci., 160, 191, (1999) · Zbl 0974.92029 [18] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. appl. math., 62, 58, (2001) · Zbl 0991.92029 [19] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mount. J. math., 20, 857, (1990) · Zbl 0725.34049 [20] Hale, J.K., Ordinary differential equations, (1969), John Wiley New York · Zbl 0186.40901 [21] Thieme, H.R., Convergence results and a poincaré-bendixson trichotomy for asymptotically automous differential equations, J. math. biol., 30, 755, (1992) · Zbl 0761.34039 [22] Butler, G.; Waltman, P., Persistence in dynamical systems, Proc. am. math. soc., 96, 425, (1986) [23] Freedman, H.I.; Ruan, S.G.; Tang, M.X., Uniform persistence and flows near a closed positively invariant set, J. dynam. diff. equat., 6, 583, (1994) · Zbl 0811.34033 [24] London, D., On derivations arising in differential equations, Linear multilin. algeb., 4, 179, (1976) · Zbl 0358.15011 [25] Li, Y.; Muldowney, J.S., Evolution of surface functionals and differential equations, (), 144 · Zbl 0792.34050 [26] Li, Y.; Muldowney, J.S., On bendixson’s criterion, J. diff. equat., 106, 27, (1993) · Zbl 0786.34033 [27] Smith, H.L., Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM rev., 30, 87, (1988) · Zbl 0674.34012 [28] Hirsch, M.W., Systems of differential equations which are competitive or cooperative. IV: structural stabilities in three dimensional systems, SIAM J. math. anal., 21, 1225, (1990) · Zbl 0734.34042
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.