Abstract and Applied Analysis

Volume 2013 (2013), Article ID 916130, 11 pages

http://dx.doi.org/10.1155/2013/916130

Research Article

Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

Yakui Xue and Tiantian Li

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received 28 August 2013; Revised 23 September 2013; Accepted 23 September 2013

Academic Editor: Massimiliano Ferrara

Copyright © 2013 Yakui Xue and Tiantian Li. 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.

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any , we show that the disease-free equilibrium is globally asymptotically stable; when , the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any ; when , the disease will persist. However, for any , the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

1. Introduction

Many scholars have been studying the change law of things by analyzing dynamic behavior of the corresponding system [1–28]. Recently, people established some epidemic models according to many realistic factors such as delay factor, isolation, and population change. People prevent many epidemics from spreading by establishing reasonable mathematics models. In fact, many diseases have different kinds of delays when they spread, such as immunity period delay [2–4], infection period delay [5], and incubation period delay [3, 6–13, 16–18, 25]. The time delay may influence the dynamics of infectious diseases. So it is necessary and useful to discuss delayed epidemic model.

Many scholars have discussed how delay affected the spread of infectious diseases. And some delays do not have an influence on transmission of epidemics [2–12]. Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate were discussed by Kyrychko and Blyuss in [2]; they considered time delay representing temporary immunity period and showed the endemic equilibrium was globally asymptotically stable, which indicated that the delay had no effect on the system with nonlinear incidence rate. And McCluskey studied the global stability for an SIR epidemic model with delay and nonlinear incidence rate [7]; he showed that the endemic equilibrium was globally asymptotically stable by using a Lyapunov function. And global stability of a delayed SIRS epidemic model with a nonmonotonic incidence rate was discussed by Muroya et al. [8], who proved that the endemic equilibrium was globally asymptotically stable. This illustrated that the delay could not affect the dynamics of the system with nonmonotonic incidence rate. At the same time, Xu and Ma studied stability of a delayed SIRS epidemic model with a nonlinear incidence rate [10]; the global asymptotic stability of system at the endemic equilibrium was proved. Besides, Yoshida and Hara discussed global stability of a delayed SIR epidemic model with density dependent birth and death rates in [11]. They investigated the stability of the endemic equilibrium, which was showed to be globally asymptotically stable. These researches above indicate delay has no effect on transmission of epidemics under certain conditions.

However, some delays impact the dynamics of system, which cause Hopf bifurcation to occur [13–20, 24, 27]. And Akkocaoğlu et al. discussed Hopf bifurcation analysis of a general nonlinear differential equation with delay in [13]. They get necessary conditions for the linear stability and Hopf bifurcation. There are also other scholars who analyze the existence of Hopf bifurcation in system. Hopf bifurcation analysis for a model of genetic regulatory system with delay was investigated by Wan and Zou in [14]. They analyzed the direction of the bifurcation and the stability of bifurcated periodic solutions. At the same time, density Hopf bifurcation in two SIRS dependent epidemic models was studied by Greenhalgh et al. [15], who found that Hopf bifurcation was theoretically possible but appeared not to occur for realistic parameter values. Meanwhile, Song et al. studied properties of stability and Hopf bifurcation for a HIV infection model with time delay [18] and showed that Hopf bifurcation would occur under certain conditions. However, Some professors researched how population change affected the dynamic behavior of system [21–25, 28]. Ferrara studied an Ak Solow model with a nonpositive rate of population growth [21] and analyzed how a nonpositive population growth rate hypothesis affected the dynamics of the standard Ak Solow model. At the same time, a note on the Ak Solow-Swan model with bounded population growth rate was researched by him [22]; he examined the effects of assuming a negative lower bound for the population growth rate within the Solow-Swan model with Ak technology and bounded population growth rate introduced by Guerrini. Besides, he researched the Ak Solow model with logistic law technology [23] and analyzed how a logistic law technology hypothesis affected the dynamics of the standard Ak Solow model. Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays was studied by Fang and Jiang [24]; they proved that the system was locally asymptotically stable in a range of the delay and Hopf bifurcation occurred as crossed a critical value. Enatsu et al. analyzed stability of delayed SIR epidemic models with a class of nonlinear incidence rates in [25]. In this paper, on the basis of [23–25, 27], we consider the incidence rate like and establish the following models with logistic growth, obtaining conditions for Hopf bifurcation. At the same time, the death rate due to disease and the same natural death rate of , , and are considered.

The organization of this paper is as follows. In Section 2, SIR epidemic models, their basic reproduction number, and existence of equilibrium are given. In addition, the stability and existence of Hopf bifurcation of system (2) and system (6) are discussed in detail. In Section 3, we present the numerical simulations to verify the conclusions. In Section 4, a brief discussion is given to conclude this work.

2. The SIR Epidemic Model, the Basic Reproduction Number, and Equilibrium

We establish the delayed SIR epidemic model. Here, represents the number of individuals who are susceptible to the disease, that is, who are not yet infected at time . represents the number of infected individuals who are infectious and are able to spread the disease by contacting with susceptible individuals at time . represents the number of recovered individuals at time .

The transfer diagram is depicted in Figure 5.

SIR epidemic model with nonlinear incidence rate is as follows:

In system (1), it is assumed that the population growth in susceptible host individuals is governed by the logistic growth with a carrying capacity as well as intrinsic birth rate constant , where the parameters , , , , , and are positive constants in which is the natural death rate of the population; is the disease transmission coefficient; is the parameter which measures the psychological or inhibitory effect; is the death rate due to disease; is the recovery rate and is the latent period of the epidemic.

For simplicity, define , , . , , , , , , , . When dropping the , system (1) becomes the following system:

The initial conditions for system (2) are , , . , , .         ,  . And the feasible region of the model with the initial condition above is

It is easy to show that is positively invariant with respect to system (1). According to the calculation and the practical significance of the epidemic model, system (2) always has a disease-free equilibrium . Define the basic reproduction number as follows:

Let . If the basic reproduction number , there exists a unique endemic equilibrium , , .

SIR epidemic model with bilinear incidence rate is as follows: when , system (1) becomes

After the same transformation, system (5) becomes

System (6) always has a disease-free equilibrium . Define the basic reproduction number as follows:

Let . If the basic reproduction number , there exists a unique endemic equilibrium .

First of all, system (2) is studied. We prove that the disease-free equilibrium is globally asymptotically stable. And the endemic equilibrium is locally asymptotically stable when and is unstable when . Meanwhile, system (2) undergoes Hopf bifurcation at when . And it is proved that the same conclusions appear in system (6).

2.1. SIR Epidemic Model with Nonlinear Incidence Rate

In this section, we discuss the local stability of the disease-free equilibrium in system (2) by analyzing its corresponding characteristic equation. By defining a reasonable Lyapunov function, we resolve the global dynamics of it without requiring any extra conditions. At the same time, we study the dynamics of system (2) at the endemic equilibrium . And the conditions for Hopf bifurcation are get.

Theorem 1. If , the disease-free equilibrium of system (2) is locally asymptotically stable for any . If , it is unstable for any .

Proof. When , the characteristic equation at the disease-free equilibrium of system (2) takes the form

Clearly, system (2) always has two negative real roots: and . All other roots are given by the roots of equation

Assume ,

Because ,

Therefore the disease-free equilibrium of system (2) is locally asymptotically stable.

If , let , because , , there is one positive real root at least. As a result, the disease-free equilibrium of system (2) is unstable.

When , it is easy for us to prove that the disease-free equilibrium of system (2) is locally asymptotically stable.

Theorem 2. If , the disease-free equilibrium of system (2) is globally asymptotically stable for any .

Proof. For , define a differentiable Lyapunov function

Obviously ,

Calculating the derivative of along positive solutions of system (2), it follows that

Because , . And when , ,

While , if and only if, , , . For all , it is easy to show that is the largest invariant subset of the set . Because of LaSalle’s invariance principle, the disease-free equilibrium of system (2) is globally asymptotically stable. This completes the proof.

Theorem 3. If , the endemic equilibrium is asymptotically stable when ; Hopf bifurcation arises at when passes through a sequence of critical values in system (2).

Proof. The characteristic equation at the endemic equilibrium is

Here , .

When , the characteristic equation at the endemic equilibrium is

Here let

Because , , and , order . One has

According to Hurwitz criterion, the endemic equilibrium of system (2) is locally asymptotically stable.

When , let

The characteristic equation at the endemic equilibrium of system (2) takes the form

Suppose that is a root of (21). On substituting , we derive that

Separating real and imaginary parts, it follows that

Squaring and adding equations the both, then we have

Letting , then (24) becomes

Here . It is easy to show that . One has

So when , . Then there is a unique positive satisfying (25). That is, there is a single pair of purely imaginary roots of (21).

From (23), we get the corresponding such that the characteristic equation (21) has a pair of purely imaginary roots:

Next we show

This will signify that there exists at least one eigenvalue with positive real part for . Differentiating (21) with respect to , we will obtain

Therefore

Here, we will show . Let .

One has

According to the formula above, if , , that is, when

We can derive when . Therefore, the transversality condition holds and the conditions for Hopf bifurcation are satisfied at in system (2).

2.2. SIR Epidemic Model with Bilinear Incidence Rate

Using the same methods in system (2), we can prove that the disease-free equilibrium in system (6) is globally asymptotically stable for any . Here the proof is ignored. And we only study the dynamics of system (6) at the endemic equilibrium .

Theorem 4. If , the endemic equilibrium is asymptotically stable when ; system (6) undergoes Hopf bifurcation at when .

Proof. The characteristic equation at the endemic equilibrium of system (6) is

When , the characteristic equation at the endemic equilibrium is

Because

the endemic equilibrium of system (6) is locally asymptotically stable. When , let

The characteristic equation at the endemic equilibrium of system (6) will become

Let ; using the same methods in system (2), we can obtain the equation with , as follows:

Order ; (38) becomes

We know . When , . That is, there is a single pair of purely imaginary roots of (39). At the same time, can be gotten:

And then

Let ,       +    +  . Order

Obviously, . When ,  . At the same time, if , and in theory. In addition, the conclusion is also illustrated by the graphics which show the trends of the functions. See Figures 1(a) and 1(b).

In a word, when , . Therefor, . That is, the transversality condition holds and the conditions for Hopf bifurcation are satisfied at in system (6). Here, we can find that conditions for Hopf bifurcation become weaker in system (6) than that in system (2). It means that system (6) undergoes Hopf bifurcation at when easily.

(a)

(b)

Figure 1: The graphics of and .

3. The Numerical Simulations

In this section, we study system (1) and system (5) numerically. According to the different dates that can reflect the actual situation, we get the different simulations to prove our conclusions obviously.

3.1. The Numerical Simulations of System (1)

The disease-free equilibrium of system (1) is globally asymptotically stable.

Let , , , , , , , ; here ; see Figure 2(a).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2: The numerical simulation graphics of system (1).

When , the dynamics of system (1) are shown by the following simulations.

Let , , , , ,, and; here . When , the endemic equilibrium is asymptotically stable; see Figure 2(b).

When , the endemic equilibrium is also asymptotically stable; see Figures 2(c) and 2(d).

When , the endemic equilibrium is unstable, and the Hopf bifurcation arises at ; see Figures 2(e)–2(h).

3.2. The Numerical Simulations of System (5)

When , the disease-free equilibrium of system (5) is globally asymptotically stable.

Let , , , , , and ; here ; see Figure 3(a).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: The numerical simulation graphics of system (5).

When , the dynamics of system (5) are showed by the following simulations.

Let , , , ,, and ; here .

When , the endemic equilibrium is asymptotically stable; see Figure 3(b).

When , the endemic equilibrium is also asymptotically stable; see Figures 3(c) and 3(d).

When , the endemic equilibrium is unstable, and the Hopf bifurcation arises at ; see Figures 3(e)–3(h).

Besides, when increases, will decrease. Let , , , , , , . We can get the changing trends of as changes in system (1). Obviously, there exists such that . That is, if , the disease will die out.

At the same time, let ,, , , , . We can get the changing trends of as changes in system (5). Obviously, there exists such that . That is, if , the disease will die out.

See Figures 4(a) and 4(b).

(a)

(b)

Figure 4: The numerical simulation graphics of .

Figure 5

4. Discussions

In this paper, we investigate the dynamics of SIR epidemic models with the logistic growth of population of susceptible individuals. The incidence rate like is considered here. In addition, the death rate due to disease and the same natural death rate of , , and are considered in two systems. We prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity for any delay and the endemic equilibrium is locally asymptotically stable when the basic reproduction number is greater than unity for in system (2) and system (6), respectively. At the same time, when , the sufficient conditions for the Hopf bifurcations are obtained, and the periodic solution will occur in the two systems. From the biological point of view, the delay here has influence on the transmission of the infectious diseases. That is, the delay is harmful. Besides, the conditions for Hopf bifurcations in system (2) are different from those in system (6). The conditions for Hopf bifurcation become weaker in system (6) than those in system (2). Therefore, according to the different situations, people should take different measures to control diseases.

Acknowledgments

This work is supported by the National Science Foundation of China (10471040) and the National Sciences Foundation of Shanxi Province (2009011005-1).

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