Zhang, Xin-An; Chen, Lansun The periodic solution of a class of epidemic models. (English) Zbl 0939.92031 Comput. Math. Appl. 38, No. 3-4, 61-71 (1999). Summary: This paper has completely studied the dynamical properties of a class of epidemiological models, obtained necessary and sufficient conditions for a unique periodic solution which is asymptotically orbitally stable, and necessary and sufficient conditions for the global asymptotic stability of the positive equilibrium. Cited in 29 Documents MSC: 92D30 Epidemiology 34C25 Periodic solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations Keywords:nonlinear incidence rate; global asymptotic stability PDFBibTeX XMLCite \textit{X.-A. Zhang} and \textit{L. Chen}, Comput. Math. Appl. 38, No. 3--4, 61--71 (1999; Zbl 0939.92031) Full Text: DOI References: [1] Hethcote, H. W., A thousand and one epidemic models, (Levin, S. A., Lecture Notes in Biology, Vol. 100 (1995), Springer-Verlag), 504-515 · Zbl 0819.92020 [2] Bailey, N. T.J., The Mathematical Theory of Infections Disease (1975), Hafner · Zbl 0115.37202 [3] Chen, L.; Chen, J., Nonlinear Dynamical Systems in Biology (1993), Science Press: Science Press Beijing, (in Chinese) [4] Liu, W.; Levin, S. A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biology, 23, 187-204 (1986) · Zbl 0582.92023 [5] Liu, W.; Hethcote, H. W.; Levin, S. A., Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biology, 25, 359-380 (1987) · Zbl 0621.92014 [6] Zhang, Z., Qualitative theory of differential equations, (Translations of Mathematical Monographs, 101 (1992), Amer. Math. Soc: Amer. Math. Soc Providence, RI) [7] Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. L., Qualitative Theory of Second Order Dynamical Systems (1973), Wiley: Wiley New York · Zbl 0282.34022 [8] Coppel, W. A., Quadratic systems with a degenetate critical point, Bull. Austral. Math. Soc., 38, 1-10 (1988) · Zbl 0634.34013 [9] Coppel, W. A., A new class of quadratic system, J. Diff. Eq., 92, 360-372 (1991) · Zbl 0733.58037 [10] Ye, Y., Theory of limit cycles, Translations of Mathematical Monographs, 66 (1986), Providence, RI [11] Guckenheimer, J.; Holmes, P., Nonlinear Oscillation, Dynamical Systems and Bifurcation of Vector Fields (1983), Springer-Verlag [12] Coppel, W. A., Stability of Asymptotic Behaviour of Differential Equations (1965), D.C. Heath · Zbl 0154.09301 [13] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602 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.