Global stability for an special SEIR epidemic model with nonlinear incidence rates. (English) Zbl 1152.34357
Summary: A SEIR epidemic model with nonlinear incidence rates, constant recruitment and disease-caused death in epidemiology is considered. It is shown that the global dynamics is completely determined by the contact number \(R_{0}\). If \(R_{0}\leqslant 1\), the disease-free equilibrium is globally stable and the disease dies out. If \(R_{0} > 1\), the unique endemic equilibrium is globally stable in the interior of the feasible region by using the methods established in Butler GJ, Freedman HI, Waltman P. Uniformly persistent systems, Proc Am Math Soc 1986;96:425-30, and the disease persists at the endemic equilibrium.
References:
| [1] | Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math Comput Model, 25, 85-107 (1997) · Zbl 0877.92023 |
| [2] | Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev, 42, 599-653 (2000) · Zbl 0993.92033 |
| [3] | Li, G.; Jin, Z., Global stability of an SEI epidemic model, Chaos, Solitons & Fractals, 21, 925-931 (2004) · Zbl 1045.34025 |
| [4] | Li, G.; Jin, Z., Global stability of an SEI epidemic model with general contact rate, Chaos, Solitons & Fractals, 23, 997-1004 (2005) · Zbl 1062.92062 |
| [5] | Genik, L.; van den Driessche, P., A model for diseases without immunity in a variable size population, Can Appl Math Quart, 6, 5-16 (1998) · Zbl 0940.92024 |
| [6] | Hethcote, H. W.; Stech, H. W.; van den Driessche, P., Periodicity and stability in epidemic models. A survey, (Cook, K. L., Differential equations and applications in ecology, epidemics, and population problems (1981), Academic Press: Academic Press New York), 65-85 · Zbl 0477.92014 |
| [7] | Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J Math Biol, 25, 359-380 (1987) · Zbl 0621.92014 |
| [8] | Li, M. Y.; Muldowney, J. S., Global stability for the SEIR model in epidemiology, Math Biosci, 125, 155-164 (1995) · Zbl 0821.92022 |
| [9] | LaSalle, J. P., The stability of dynamics systems (1976), PA: PA SIAM Philadelphia · Zbl 0355.34037 |
| [10] | Usmani, R. A., Applied linear algebra (1987), Marcel Dekker: Marcel Dekker New York · Zbl 0602.15001 |
| [11] | Thieme, H., Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math Biosci, 111, 99-130 (1992) · Zbl 0782.92018 |
| [12] | Butler, G. J.; Freedman, H. I.; Waltman, P., Uniformly persistent systems, Proc Am Math Soc, 96, 425-430 (1986) · Zbl 0603.34043 |
| [13] | Freedman, H. I.; Ruan, S. G.; Tang, M. X., Uniform persistence and flows near a closed positively invariant set, J Dynam Differen Equat, 6, 583-600 (1994) · Zbl 0811.34033 |
| [14] | Butler, G. J.; Waltman, P., Persistence in dynamics systems, J Differen Equat, 63, 255-263 (1986) · Zbl 0603.58033 |
| [15] | Li, M. Y.; Muldowney, J. S., A geometric approach to global-stability problems, SIAM J Math Anal, 27, 4, 1070-1083 (1996) · Zbl 0873.34041 |
| [16] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems and bifurcations of vector fields (1983), Springer: Springer Berlin · Zbl 0515.34001 |
| [17] | Li, M. Y.; Muldowney, J. S., On Bendixson’s criterion, J Differen Equat, 106, 27-39 (1993) · Zbl 0786.34033 |
| [18] | Smith, R. A., An index theorem and Bendixon’s negative criterion for certain differential equations of higher dimensions, Proc Roy Soc Edinburgh, 91A, 63-77 (1981) · Zbl 0499.34026 |
| [19] | Martin, R. H., Logarithmic norms and projections applied to linear differential systems, J Math Anal Appl, 45, 432-454 (1974) · Zbl 0293.34018 |
| [20] | Hethcote, H. W., Qualitative analysis of communicable disease models, Math Biosci, 28, 335-356 (1976) · Zbl 0326.92017 |
| [21] | Muldowney, J. S., Compound matrices and ordinary differential equations, Rocky Mountain J Math, 20, 857-872 (1990) · Zbl 0725.34049 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
