Abstract
A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.
1. Introduction
Mathematical models describing the transmission of infectious diseases have played an important role in understanding the mechanism of disease transmission and controlling the spread of infectious diseases. In the literatures, many classical epidemic models have been proposed and studied, and many authors also attempt to develop more realistic mathematical models. In 1927, by using “compartment modeling,” Kermack and Mckendrick [1] described an epidemic model and computed the theoretical number of infectious individuals. From then on, “compartment modeling” is used until now. Compartmental models are also called models or models, where , and denote the number of susceptible, infectious, and recovered individuals. Some and models have been analyzed in detail [2–7]. However, many diseases incubate inside the hosts for a period of time before the hosts become infectious. Therefore, the epidemic models, where denotes the number of individuals who are infected but not yet infectious, are developed to investigate the role of the incubation period in disease transmission. Some authors have studied the epidemic models [8–12]. For some diseases, it is necessary that some infectious individuals are quarantined. Then, the and models, where denotes the number of infectious individuals that have been quarantined and controlled, are studied [13–16].
Incidence rate plays an important role in the transmission of disease. In earlier models [17], based on the law of mass action, the bilinear incidence rate , where is the average number of contacts per infectious individual per day is used. Capasso and Serio [18] introduced a saturated incidence rate into epidemic models, where , where measures the infection force of the disease and describes the psychological effect or inhibition effect from the behavioral change of the susceptible individuals with the increase of the infectious individuals.
Based on the work of Capasso and Serio [18], in this paper, we investigate an epidemic model with a saturated incidence rate , where depends on the ratio of and . Therefore, we have Then, the saturated incidence rate is . Considering the effects of time delay, we obtain the following epidemic model: where , , , and denote the number of susceptible, infectious, quarantined, and recovered individuals at time . is the recruitment rate of the population. is the natural death rate of the population. and are the extra disease-related death rate in classes and , respectively. is the rate constant for individuals leaving infectious compartment for quarantine compartment . and are the removal rate constants from classes and , respectively. In this model, we assume that the susceptible individuals were infected before time delay which is the latent period. Then, the infected individuals become infectious individuals and some of infectious individuals are quarantined. In classes and , some individuals are cured and removed.
The outline of this paper is as follows. In Section 2, the equilibria and their stability are analyzed. The critical value of time delay at which Hopf bifurcation occurs is obtained. In Section 3, by using an extended method of multiple scales, we obtained the normal form of the Hopf bifurcation. In Section 4, The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. In Section 5, conclusions are given.
2. Stability of Equilibria and Existence of Hopf Bifurcation
In this paper, we will study system (2) which is modeled by delayed differential equations. Noticing that and are uncoupled with and , we will study the system consisting of and :
System (3) always has a disease-free equilibrium . If , system (3) has an unique endemic equilibrium , where
In order to investigate the stability of the two equilibria and simultaneously, we assume that is one of the two equilibria. Then, the Jacobian matrix of system (3) at point is given by that is,where and . Equation (6) can be rewritten as where .
For , the characteristic equation becomes
Remark 1. For , if and , then the roots of (8) are real and negative, and then the equilibrium is asymptotically stable.
For , if is a root of (7), then we have Separating the real and imaginary parts, we have which leads to the following fourth-degree polynomial equation:
Then, by discussing the distribution of the solutions of (11), we analyze the stability of the equilibrium and the existence of Hopf bifurcation. The distribution of the solutions of (11) can be divided into the following three cases.
Case 1. If or then (7) has no positive root.
Case 2. If then (7) has only one positive root .
Case 3. If then (7) has two positive roots: and no such solutions otherwise.
Noticing that in Case 2 is the same as in Case 3, we suppose that (7) has two positive roots . Then, from (10), we can determine thatat which (7) has a pair of purely imaginary roots . Supposing that then
Substituting into (7) and taking the derivative with respect to , we have which, together with (10) and (16), leads to Thus, if , we have Therefore, we can obtain the following theorem.
Theorem 2. Let , be defined by (16) and (17), respectively. For system (3),
(i) if and or , then equilibrium E of system (3) is asymptotically stable for all ;
(ii) if , then equilibrium E of system (3) is asymptotically stable when and unstable when ; system (3) undergoes a Hopf bifurcation at E when ;
(iii) if , , and , then there is positive integer , such that equilibrium E switches times from stability to instability to stability; that is, E is asymptotically stable when and unstable when System (3) undergoes a Hopf bifurcation at E when .
3. The Normal Form of Hopf Bifurcation
In this section, we suppose that system (3) undergoes a Hopf bifurcation from equilibrium . Time delay is chosen as the bifurcation parameter and its critical value is . By using the method of multiple scales [19], we will derive the normal form of the Hopf bifurcation. By translating the equilibrium to the origin, system (3) is rewritten as the following equation: where is the state variable vector and .
According to the MMS, a monoparametric family of solution of the type is as follows: where and is a nondimensional parameter. The solution does not depend on slow scale because secular terms first appear at . Therefore, we assume a two-scale expansion of the solution of (25).
Bifurcation parameter is ordered as
The delay term in (25) can be further expanded as where .
By substituting (26), (27), and (28) into (25), expanding , and balancing the coefficients of , a set of perturbative equations are obtained.
First, for -order terms, we have where
Equation (29) has the following general solution: where is complex constant and is the right eigenvector given by then Next, for -order terms, we have where are the second-order partial derivatives of with respect to and .
Substituting (31) into (34), we obtain Solving (35), it yields where vectors and are obtained by solving the following equations: that is, where , , , − . Then, we can obtain Finally, for -order terms, we get where are the third-order partial derivatives of with respect to and .
Substituting (31) and (36) into (40) and eliminating the secular terms, we can obtain the equations including . Eliminating the coefficients of by using the left eigenvectors and reabsorbing parameter [20], the normal form is determined bywhere the expressions of coefficients and are given bywhere and is the left eigenvector obtained by solving the following equations: where denotes the transpose conjugate and denotes the transpose.
To express the normal form in real form, a polar form representation for the complex amplitude is introduced:
Substituting (44) into (41) and separating the real and imaginary parts in (41), the generalized amplitude and phase modulation equations are drawn: where , and .
From (45), we can get that the amplitude of the periodic solution is The stability of the periodic solution is determined by the sign of the eigenvalue of the Jacobian matrix of . The eigenvalue is Then, we can obtain the following theorem.
Theorem 3. The amplitude of the bifurcating periodic solution of system (3) is ; the stability of the bifurcating periodic solution is determined by : the bifurcating periodic solution is stable (unstable), if .
4. An Example
In this section, we perform some numerical simulations to verify our analysis. For the following parameter values , we can see that endemic equilibria of system (3) exist. Then, a periodic solution with frequency bifurcates from the endemic equilibria at critical value . By Theorem 2, endemic equilibria are asymptotically stable when and unstable when .
By using the dynamical software WinPP, some numerical simulations are given. First, by Theorem 2, endemic equilibria are asymptotically stable when as shown in Figure 1 and the time histories of the state variables and are also given in Figure 1.
Endemic equilibria are unstable when . When crosses critical value , a stable periodic solution yields as shown in Figure 2.
By using the method developed in this paper, the amplitude of the periodic solution can be obtained. The relationship between the amplitude and bifurcation parameter is given in (45). Then, the analysis results are compared with that of numerical simulations as shown in Figure 3.
5. Conclusions
In this paper, the stability and Hopf bifurcation of an epidemic model with time delay are studied. The stability of disease-free equilibrium and endemic equilibrium is studied and the conditions of the stability and unstability of the two equilibria are obtained. Then, the critical value of at which a Hopf bifurcation occurs is obtained. By using an extended method of multiple scales, the normal form of Hopf bifurcation is obtained. In order to verify the theoretical results, the Runge-Kutta scheme is adopted to produce the numerical results. The comparison between analytical predictions and numerical results of the amplitude of Hopf bifurcation reveals a qualitatively and quantificationally excellent agreement.
In order to obtain the normal form of Hopf bifurcation occurring in delayed differential equations, we develop an extended method of multiple scales. Choosing time delay as a bifurcation parameter and using the method, one can get the normal form of Hopf bifurcation without the tedious computation encountered in the center manifold reduction. Moreover, when we develop the method, we choose -dimensional delayed differential equation. Therefore, the method can be used for studying the Hopf bifurcation occurring in a general -dimensional delayed differential equation.
Through observing the results, there are two contributions in this paper. First, we improve an epidemic model and introduce a nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals. Second, we extend the method of multiple scales. By using this method, the normal form of Hopf bifurcation occurring in -dimensional delayed differential equations can be obtained.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this article.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11302072) and the Doctoral Research Fund of Henan Institute of Engineering (D2012021).