Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 2340549, 9 pages
https://doi.org/10.1155/2017/2340549

Research Article

Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

Juan Liu

Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China

Correspondence should be addressed to Juan Liu; liujuan7216@163.com

Received 16 February 2017; Accepted 28 March 2017; Published 9 May 2017

Academic Editor: Lu-Xing Yang

Copyright © 2017 Juan Liu. 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.

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results.

1. Introduction

The outbreak of infectious diseases had not only caused the loss of billions of lives but also badly damaged the social economy in a short time, which brought much pain to human society [1]. Thus, it has been an increasingly urgent issue to understand how to prevent or slow down the transmission of infectious diseases. To this end, many mathematical models have been proposed for describing the spread process of infectious diseases [2–10]. However, all the epidemic models above do not consider the change of delitescence of the infectious diseases. Considering that the diversity of the delitescence period in each infected individual who is infected with disease virus is mainly due to the variation of the virus and the distinct constitution of different people for some disease, such as H1N1 disease, Wang proposed the following SEIS epidemic model with a changing delitescence and a nonlinear incidence rate [11]:

where , , and denote the numbers of the susceptible, exposed, and infectious populations at time , respectively. is the recruitment rate of the susceptible population; is the natural death rate of the population; is the death rate due to the disease of the infected population; is the rate at which the exposed population becomes infectious; is the rate at which the infected population returns to the susceptible population because of the treatment; is the rate at which the infected population becomes the exposed one; and is the rate at which the infected population becomes infectious directly. is the nonlinear incidence rate, where measures the infection force of the disease and measures the inhibition effect from the behavioral change of the susceptible population. Wang investigated global stability of system (1).

In fact, many infectious diseases have different kinds of delays during their spreading process in the population, such as latent period delay [9, 12–16], immunity period delay [17, 18], and infection period delay [19]. The time delay may induce Hopf bifurcation and periodic solutions. The occurrence of a Hopf bifurcation means that the state of the epidemic disease prevalence changes from an equilibrium to a limit cycle. Therefore, the time delay can influence the dynamics of infectious diseases. So it is necessary and useful to investigate system (1) with time delay. Based on this fact and taking the period used to cure the infectious population, we consider the following delayed epidemic system:

where is the time delay due to the period that is used to cure the infectious population. That is, we assume that all the infectious populations will survive after time . The initial conditions for system (2) are

where .

The outline of this paper is as follows. In the next section, stability of the endemic equilibrium is analyzed and the critical value of the time delay at which a Hopf bifurcation occurs is obtained. In Section 3, direction and stability of the Hopf bifurcation are investigated. In Section 4, the obtained theoretical results are verified by some numerical simulations. Finally, this work is summarized in Section 5.

2. Stability of the Endemic Equilibrium and Existence of Hopf Bifurcation

By a direct computation, we know that if (I) and , (II) and , (III) , and , or (IV) and , then system (2) has a unique endemic equilibrium , where

and is the unique positive root of the following equation:

where

Let , , . We can rewrite system (2) as the following form:

where

Then we obtain the linearized system of system (2)

The characteristic equation is

where

When , (10) reduces to

Routh-Hurwitz criterion implies that is locally asymptotically stable without delay if condition holds.

() , .

For , substituting () into (10), we obtain

Then

where

Let ; then

where . According to the analysis about the distribution of roots of (16) in Song et al. [20], we have the following result.

Lemma 1. For the polynomial equation (16),

(1)if , then (16) has at least one positive root;(2)if and , then (16) has no positive roots;(3)if and , then (16) has positive roots if and only if and .

Next, we assume that the coefficients in (16) satisfy the following condition.
(i) or (ii) , , , and .
Thus, (14) has at least one positive root such that (10) has a pair of purely imaginary roots . The corresponding critical value can be obtained from (13)

Taking derivative with respect to on both sides of (10), we obtain

Further, we have

Thus, if the condition : holds, then . Then, based on the Hopf bifurcation theorem in [21], we have the following.

Theorem 2. For system (2), if the conditions hold, then the endemic equilibrium of system (2) is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the endemic equilibrium when , where is defined in (17).

3. Direction and Stability of the Hopf Bifurcation

Let , ; then is the Hopf bifurcation value of system (2). Rescaling the time delay , then system (2) can be transformed into an FDE in as follows:

where

where , , and are defined by Appendix A.

By the Riesz representation theorem, there exists a matrix function , , whose components are of bounded variation, such that

In fact, we choose

For , we define

Then system (20) is equivalent to

For , the adjoint operator of is defined as

and a bilinear inner product is defined by

where .

Let be the eigenvector of belonging to and be the eigenvector of belonging to . By a direct computation, we can get

From (27), we can get

Then we choose

such that .

Next, we can obtain the coefficients , , , and by using the method introduced in [21] and a computation process similar to that in [22–24]. The expressions of , , , and are defined by Appendix B.

Then, we can get the following coefficients which determine the properties of the Hopf bifurcation:

In conclusion, we have the following results.

Theorem 3. For system (2), if   , then the Hopf bifurcation is supercritical (subcritical). If   , then the bifurcating periodic solutions are stable (unstable). If   , then the bifurcating periodic solutions increase (decrease).

4. Numerical Simulations

In order to verify the efficiency of the obtained results in the paper, we carry out some numerical simulations in this section. By extracting some values from [11] and considering the conditions for the existence of the Hopf bifurcation, we consider the special case of system (2) with the parameters , , , , , , , and , Then, system (2) becomes the following form:

from which we can obtain the unique positive root and then we get the unique endemic equilibrium . Then, we can obtain , , and . Thus, based on Theorem 2, we know that the endemic equilibrium is locally asymptotically stable when , which can be illustrated by Figures 1 and 2. In this case, the disease can be controlled easily. Once the value of the delay passes through the critical value , then the endemic equilibrium loses its stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the endemic equilibrium . This property can be shown as in Figures 3 and 4. In this case, the disease will be out of control.

Figure 1: is locally asymptotically stable for with initial values “27.5, 260.5, 65.”

Figure 2: is locally asymptotically stable for with initial values “22, 200, 75.”

Figure 3: becomes unstable and a Hopf bifurcation occurs when with initial values “27.5, 260.5, 65.”

Figure 4: becomes unstable and a Hopf bifurcation occurs when with initial values “22, 200, 75.”

In addition, according to (31), we get , , , and . Therefore, we can conclude that the Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable and increase. Since the bifurcating periodic solutions are stable, it can be concluded that the populations in system (32) can coexist from the view of ecology. Based on this fact, we can conclude that the time delay is harmful for system (32).

5. Conclusions

We generalize a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate in this paper by introducing the time delay due to the period that is used to cure the infectious population into the SEIS model considered in the literature [11]. Compared with the literature [11], we mainly consider the effect of the time delay on the model.

The main results are given in terms of local stability and Hopf bifurcation. Stability of the endemic equilibrium is investigated by analyzing the corresponding characteristic equation. By choosing the time delay as a bifurcation parameter, sufficient conditions have been established for local existence of Hopf bifurcation at the endemic equilibrium. Then, with the help of the normal form theory and the center manifold theorem due to Hassard et al. [21], direction and stability of the Hopf bifurcation are determined. Finally, through numerical simulations, it can be concluded that the period used to cure the infectious population plays an important role in the disease spreading and the disease may be controlled by shortening the period used to cure the infectious population.

Appendix

A. The Expressions of , , and

B. The Expressions of , , , and

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2015A144).

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