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International Journal of Mathematics and Mathematical Sciences
Volume 2018, Article ID 1467235, 13 pages
Research Article

Hopf Bifurcation Analysis of a New SEIRS Epidemic Model with Nonlinear Incidence Rate and Nonpermanent Immunity

Department of Electrical & Computer Engineering, University of Patras, 26504 Patras, Greece

Correspondence should be addressed to M. P. Markakis; rg.sartapu@sikakram

Received 8 June 2017; Accepted 28 November 2017; Published 17 January 2018

Academic Editor: Hans Engler

Copyright © 2018 M. P. Markakis and P. S. Douris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A new SEIRS epidemic model with nonlinear incidence rate and nonpermanent immunity is presented in the present paper. The fact that the incidence rate per infective individual is given by a nonlinear function and product of rational powers of two state variables, as well as the introduction of an epidemic-induced death rate, leads to a more realistic modeling of the physical problem itself. A stability analysis is performed and the features of Hopf bifurcation are investigated. Both the corresponding critical regions in the parameter space and their stability characteristics are presented. Furthermore, by using algorithms based on a new symbolic form as regards the restriction of an -dimensional nonlinear parametric system to the center manifold and the normal forms of the corresponding Hopf bifurcation, as well, the associated bifurcation diagram is derived, and finally various emerging limit cycles are numerically obtained by appropriate implemented methods.