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

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.


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][3][4][5][6][7][8][9][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 1− is the rate at which the infected population becomes infectious directly./(1 + ) 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][13][14][15][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  3 + = (, , ) ∈  3 + .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.
( 1 ) For  > 0, substituting  =  ( > 0) into (10), we obtain Then where Let  2 = V; then where According to the analysis about the distribution of roots of ( 16) in Song et al. [20], we have the following result.
Next, we assume that the coefficients in (16) satisfy the following condition.
Next, we can obtain the coefficients  20 ,  11 ,  02 , and  21 by using the method introduced in [21] and a computation process similar to that in [22][23][24].The expressions of  20 ,  11 ,  02 , and  21 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.

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  = 5,  = 0.01,  = 0.5,  = 0.4,  = 0.5,  = 0.65,  = 0.1, and  = 0.02,Then, system (2) becomes the following form: from which we can obtain the unique positive root  * = 73.8562and then we get the unique endemic equilibrium  * (34.3750, 245.5930, 73.8562).Then, we can obtain  0 = 0.3950,  0 = 5.1686, and   ( 0 ) = 0.0012 − 0.0759.3 and 4. In this case, the disease will be out of control.
In addition, according to (31), we get  1 (0) = −1.0027− 0.9244,  2 = 835.5833> 0,  2 = −2.0054< 0, and  2 = 31.5171> 0. 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).

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.
= ( 0 + ) (  11 0  13  21  22  23  31  32  33 Thus, based on Theorem 2, we know that the endemic equilibrium  * (34.3750, 245.5930, 73.8562) is locally asymptotically stable when  <  0 = 5.1686, which can be illustrated by Figures1 and 2. In this case, the disease can be controlled easily.Once the value of the delay passes through the critical value  0 = 5.1686, then the endemic equilibrium  * (34.3750, 245.5930, 73.8562) loses its stability and a Hopf bifurcation occurs, and a family of periodic solutions bifurcate from the endemic equilibrium  * (34.3750, 245.5930, 73.8562).This property can be shown as in Figures