International Journal of Mathematics and Mathematical Sciences

Volume 2018 (2018), Article ID 1467235, 13 pages

https://doi.org/10.1155/2018/1467235

## 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.

#### Abstract

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.

#### 1. Introduction

The realistic modeling of epidemic models constitutes an important issue of modern research as it can contribute to both a better understanding and more accurate modeling of the actual dynamics and the interrelation of the populations involved. Nonpermanent immunity leads to SEIRS or SIRS models which have been studied with respect to the effects of the epidemiological parameters, with bilinear (see, e.g., [1]) or nonlinear incidence rate (see [2, 3] and the references therein). In particular, in [3] the writers investigate the stability of both the disease-free equilibrium and the endemic one. Also, they determine conditions regarding the existence and stability of Hopf-bifurcated limit cycles with respect to the latter, concerning both SEIRS and SIRS models. The nonlinear incidence rate offers a deeper insight into the actual relation between the populations of susceptible and infective individuals. Furthermore, we introduce an additional death rate solely due to the disease, and hence the SEIRS model is enriched with new nonlinear terms.

Let us now present the specifics of the aforementioned model, described by the following 4D differential system:where , , , , denote the number of susceptible, exposed (incubating), infective, and recovered individuals and the total population, respectively, while represents the incidence rate per infective individual and , , , , , stand for the system parameters. As regards their physical meaning, denotes the birth rate, denotes the physical death rate, denotes the rate of loss of immunity, denotes the rate of incubation, is the additional death rate due to the epidemic, and denotes the recovery rate. Then by normalizing with respect to the total population which is considered constant and taking into account (2), the system becomeswithNow, by eliminating , system (3) is reduced to the following three-dimensional one:By settingwhere , are positive constants and , (5) takes the final formThe analysis is multiparametric in that the parameter space of the system is structured by three varying parameters. Thus in Section 2 a stability analysis of the system is performed, where the active parameters are determined and various graphical representations are obtained concerning the critical (with respect to Hopf bifurcations) values of the varying parameters, as well as the critical, noncritical, and stability regions in the parameter space considered. Then, based on a new proper symbolic form as regards the center manifold analysis (see [4]), the basic features and steps of which are generally presented in Section 3, effective algorithms are implemented, by using symbolic computational software, which result in the associated bifurcation portraits throughout the regions of the parameter space under consideration. These portraits are presented in Section 4, together with limit cycles corresponding to the cases resulting from the respective analysis and obtained by use of a custom orthogonal collocation method on finite elements. Finally, Appendices A and B include algebraic manipulations and formulae related to the analysis carried out throughout this work.

#### 2. Stability Analysis-Hopf Bifurcation

Final reduced system (7) possesses two types of equilibria: a disease-free one, namely,and an endemic one of the form with obtained after some tedious algebraic manipulations (see Appendix A), aswhere , , , , and . We focus on the endemic equilibrium with given in (9)–(11), since it corresponds to persistence of the disease.

As regards the local stability of , taking into account (7), (10), and (11), the Jacobian matrix evaluated at becomeswhereThe associated characteristic equation isNow, by considering the well-known Routh-Hurwitz necessary and sufficient stability conditions, namely,related to the equilibrium , we conclude with the formulaewhere , , , are polynomials with respect to .

Moreover, by solving the equation (resulting from (14) for ) with respect to (after substitution of the right-hand side of (9) for , we finally evaluate the real roots of a 9th-degree polynomial numerically), we further evaluate and by substituting the obtained root of into (9) and (11), respectively. Then, taking into account the fact that , we arrive at by means of (11) in the case where . Thus ifthen represent the critical values .

Then, by solving (10) with respect to and considering varying parameters of the problem we obtain the critical valueprovided that , with being the aforementioned critical equilibrium of the system (7). Thus a* critical surface * is generated in the parameter space (we have and due to (9) we also have , with being fixed), defined over the area of the parameter plane , where the critical values of are obtained via (17) and (18); we call this area* critical region*. Moreover, we differentiate , given by (16), with respect to the active parameters , namely, where and , and also and denote the partial derivatives of and with respect to , provided in Appendix A (A.2). Then by introducing the critical equilibrium and taking into account the fact that is a numerically obtained root of a high degree polynomial, we conclude that by setting the right-hand side of (19) equal to zero, no explicit relation can be extracted involving the parameters of the system. Furthermore, for any fixed values of , we numerically computeeverywhere on the* critical surface*. Thus, considering (17) and (20), according to Liu criterion [5] has a pair of purely imaginary eigenvalues together with a negative real one on this surface; the* transversality condition* (see Appendix B, ) holds, as well. Hence, a Hopf bifurcation occurs at the critical equilibrium. Graphical representations of (evaluated by using (18)) versus (for different values of , with being fixed), versus (for different values of , with being fixed), and versus (for different values of , with being fixed) are presented in Figures 1(a), 1(b), and 1(c), respectively, while* critical regions* are obtained in the parameter plane for fixed values of in Figures 2(a) and 2(b).