The effect of time delay on the dynamics of an SEIR model with nonlinear incidence |
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| Institution: | 1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum, P.O. Box 321, Khartoum, Sudan;2. Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, 7535 Bellville, South Africa;1. Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan;2. Department of Mathematics, University of Malakand, Chakdara, Dir(Lower),Khyber Pakhtunkhwa, Pakistan;3. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut''s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand;4. Department of Mechanical Engineering, University of Wisconsin, Milwaukee, WI, USA;5. Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia |
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| Abstract: | In this paper, a SEIR epidemic model with nonlinear incidence rate and time delay is investigated in three cases. The local stability of an endemic equilibrium and a disease-free equilibrium are discussed using stability theory of delay differential equations. The conditions that guarantee the asymptotic stability of corresponding steady-states are investigated. The results show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation when using the time delay as a bifurcation parameter. Applying the normal form theory and center manifold argument, the explicit formulas determining the properties of the bifurcating periodic solution are derived. In addition, the effect of the inhibitory effect on the properties of the bifurcating periodic solutions is studied. Numerical simulations are provided in order to illustrate the theoretical results and to gain further insight into the behaviors of delayed systems. |
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