Stability and Hopf Bifurcation for an SEIR Epidemic Model with Delay

Liancheng Wang; Xiaoqin Wu

doi:10.31197/atnaa.380970

Research Article

Stability and Hopf Bifurcation for an SEIR Epidemic Model with Delay

Year 2018, Volume: 2 Issue: 3, 113 - 127, 30.09.2018

Liancheng Wang , Xiaoqin Wu

https://doi.org/10.31197/atnaa.380970

Cited By: 3

Abstract

In this paper, first a third degree transcendental polynomial is studied and the distribution of its zeros is established. Then the results are applied to study an SEIR model with a time delay. We show that, under some conditions, as the time delay increases, a stable endemic equilibrium will become unstable and periodic solution emerges by Hopf bifurcation. By finding the normal form of the system, the direction and the stability of the periodic solution are established. Numerical simulations are performed to demonstrate the theoretical results.

Keywords

Transcendental polynomial, SEIR model, Hopf bifurcation

References

\bibitem{ander92} R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, Oxford, 1992.
\bibitem{green92} D. Greenhalgn, Some results for an SEIR epidemic model with density dependence in the death rate, IMA J. Math. Appl. Med. Biol., 9 (1992), 67-106.
\bibitem{green97} D. Greenhalgn, Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. Comput. Modelling, 25 (1997), 85-107.
\bibitem{hethcote00} H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
\bibitem{li99}M. Y. Li, J. G. Graef, L. Wang, and J. Karsai, Global dynamics of an SEIR model with a varying total population size, Math. Biosci., 160 ( 1999), 191-213.
\bibitem{li95}M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 ( 1995), 155-164.
\bibitem{li01} M. Y. Li, Hal L. Smith and L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69.
\bibitem{liu87} W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rate, J. Math. Biol., 25 (1987), 359-380.
\bibitem{kor04} A. Korobeinikov and P. K. Maini, A Lyaponov function and global properties for AIR and SEIR epidemiological models with nonlinear incidence, Math. Biocsi. Engineering, 1 (2004), 57-60.
\bibitem{hethcote80} H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47.
\bibitem{hethcote81a} H. W. Hethcote, H. W. Stech, and P. van den Driessche, Periodicity and stability in epidemic models: A survey, in Differential Equations and Applications in Ecology, Epidemics, and Population Problems, K. L. Cooke, ed., Academic Press, New York, 1981, pp. 65-85.
\bibitem{hethcote81b} H. W. Hethcote, H. W. Stech, and P. van den Driessche, Nonlinear oscillations in epidemic models, SIAM J. Appl. Math., 40 (1981), 1-9.
\bibitem{hethcote81c} H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity, J. Math. Biol., 13 (1981), 185-198.
\bibitem{hethcote81d} H. W. Hethcote, M. A. Lewis, and P. van den Driessche, An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biol., 27 (1989), 49-64.
\bibitem{hethcote89} H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, in Applied Mathematical Ecology, L. Gross and S. A. Levin, eds., Springer, New York, 1989, pp. 193-211.
\bibitem{hethcote95} H. W. Hethcote and P. van den Driessche, An SIS epidemic model with variable population size and a delay, J. Math. Biol., 34 (1995), 177-194.
\bibitem{cooke96} K. L. Cooke and van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260.
\bibitem{thieme83} H. R. Thieme, Global asmptotical stability for epidemic models, Equdiff 82 Lecture Notes in Mathematics, H. W. Knobloch and K. Schmitt, eds., Spring-Verlag, Berlin, 1983, pp. 608-615.
\bibitem{thieme85} H. R. Thieme, Local stability in epidemic models for heterogeneous populations, in Mathematics in Biology and Medicine, V. Capasso, E. Grosso, and S. L. Paveri-Fontana, eds., Lecture Notes in Biomathematics, Spring-Verelag, Berlin, 1985, pp. 185-189.
\bibitem{khan99} Q. J. A. Khan and D. Greenhalgh, Hopf bifurcation in epidemic models with a time delay in vaccination, IMA J. Math. Appl. Med. Biol., 16 (1999), 113-142.
\bibitem{tch07} J. M. Tchuenche, A. Nwagwo and R. Levins, Global behavior of an SIR epidemic model with time delay, Math. Meth. Appl. Sci., 30 (2007), 733-749.
\bibitem{rost08} G. Rost and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Engineering, 5 (2008), 389-402.
\bibitem{ruan1} S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Mathematics Applied in Medicine and Biology 18 (2001), 41-52.
\bibitem{ruan2} S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems 10 (2003), 863-874.
\bibitem{ch1994} S. N. Chow, C. Li \& S. Wang, {\it Normal forms and bifurcation of planar vector fields,} Cambridge, 2004.
\bibitem{ku2004} Y. Kuznetsov, {\it Elements of applied bifurcation theory}, 3rd Ed., Springer, 2004.