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Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 636153, 16 pages
http://dx.doi.org/10.1155/2008/636153
Research Article

Analysis of a Delayed SIR Model with Nonlinear Incidence Rate

1School of Mechantronic Engineering, North University of China, Taiyuan 030051, China
2Department of Basic Science, Taiyuan Institute of Technology, Taiyuan 030008, China
3Department of Mathematics, North University of China, Taiyuan 030051, China

Received 8 September 2008; Accepted 6 October 2008

Academic Editor: Manuel de La Sen

Copyright © 2008 Jin-Zhu Zhang et al. 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

An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value and time delay (i.e., incubation time length). If is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one there will be an endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.

1. Introduction

Epidemic models with nonlinear incidence have been studied by many authors, and related literature of SIR disease transmission model is quite large, where S denotes the number of individuals that are susceptible to infection, I denotes the number of individuals that are infectious, and R denotes the number of individuals that have been removed with immunity. For example, a detailed dynamical analysis of the nonlinear incidence rate (where is the average number of contacts per infective per unit time) is given by Liu et al. [1, 2], Hethcote and van den Driessche [3], Moghadas and Alexander [4], Korobeinikov and Maini [5], and others. After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio [6] introduced a saturated incidence rate into epidemic models, the incidence rate seems more reasonable than because the number of effective contacts between infective individuals and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the protection measures by the susceptible individuals [7], such as the incidence rate (see [1, 8] and references therein).

Zhang and Chen [9] investigated a class of SIR epidemiological models under assumption that the susceptible satisfies the logistic equation and the incidence rate is of the form . More recently, Zhang et al. [10], serve as an extended version to [9], have carried out a long-term qualitative analysis incorporating incubation time delay into incidence rate in the case of , that is, with the force of infection , which was proposed by Cooke [11]. The incubation period is a time, during which the infectious agents develop in the vector, and only after that time the infected vector becomes itself infectious. The detailed biological meanings and transmission mechanisms were given in [11]. The results obtained in [10] represent that, if the epidemic is persistent, introducing time delay changes the dynamical behaviors of the epidemic state. If the threshold value determining whether the disease dies out is larger than one and less than three, the endemic equilibrium is absolutely stable (in the sense that it is asymptotically stable for all values of the delays [12]); when it exceeds three, the endemic equilibrium is conditionally stable (i.e., it is asymptotically stable for the delays in some intervals), and limit cycles arise by Hopf-type bifurcation with increasing time delay.

In 1978, May and Anderson [13] proposed the saturated incidence rate of the form , and used by some authors [1417], recently. The effect of saturation factor (refer to ) stems from epidemical control. In the absence of effective therapeutic treatment and vaccine, the epidemical control strategies are based on taking appropriate preventive measures. For example, if transmission vector is mosquito, these measures include mosquito reduction mechanisms and personal protection against exposure to mosquitos. Mosquito reduction mechanisms entail the elimination of mosquito breeding sites (such as clearing culverts, roadside ditches, eliminating standing water, etc.), larvaciding (killing of larvae before they become adults) and adulticiding (killing of adult mosquitoes by spraying). On the other hand, personal protection is based on preventing vector mosquitoes from biting humans (by using mosquito repellents, avoiding locations where mosquitoes are biting, and using barrier methods such as window screens and long-sleeved clothing) [1820].

From a practical point of view, instead of the bilinear incidence rate in [10], we consider saturation incidence rate in this paper and assume the force of infection is in this version which is saturated with the susceptible. The susceptible host population is also assumed to have the logistic growth with carrying capacity with a specific growth rate constant . We can get a generalized SIR epidemiological model as follows:where and are positive constants. is the parameter that measures the inhibitory effect, is the natural recovery rate of the infective individuals, and represent the per capita death rates of infectives and recovered, respectively. Notice that when , the system (1.1) becomes the system of bilinear incidence rate in [10], throughout this paper, we assume .

By mathematical analysis, we derive a threshold value and prove that the values of and incubation time length completely determine the global dynamics of system (1.1), that is, this two factors determine whether the disease approaches an endemic value or whether solutions oscillate. If , the disease-free equilibrium is globally asymptotically stable and the disease always dies out, whereas if , the disease persists if it is initially present. By taking the incubation time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical simulations show that the system with time delay admits rich complex dynamic, and a sequence of periodic solutions will emanate with increasing time delay, which exhibits quite complex periodic and chaotic patterns.

We arrange our paper as follows. In Section 2, results on positivity and boundedness of solutions are presented. In addition, we also consider the equilibria of system (1.1) and give the threshold for the existence of endemic equilibrium. In Section 3, we consider the global stability of the disease-free equilibrium and obtain the necessary and sufficient conditions for the permanence of endemic equilibrium. The local stability analysis of system (1.1) is considered in Section 4. Some numerical results will be given as applications in Section 5.

2. Preliminary Results

The initial conditions of (1.1) are defined in the Banach spacewhere . By a biological meaning, we assume that .

It can be verified that the positive cone is positively invariant with respect to (1.1) from [21, Lemma 2.1].

Lemma 2.1. All feasible solutions of the system (1.1) are bounded and enter the regionwhere

Proof. From the first equation of (1.1), we getby comparison, we have The total host population size can be determined by , andThus, we have as . Therefore, all feasible solutions of the system (1.1) are bounded and enter the region . This completes the proof of Lemma 2.1.

Lemma 2.1 shows that the solutions of system (1.1) are bounded and, hence, lie in a compact set and are continuable for all positive time.

Let . For system (1.1), there always exists the equilibria , , if , there also exists an endemic equilibrium , where

3. Permanence

Before starting our theorem, we give the following lemma.

Lemma 3.1 (see[22]). Consider the following equation:where ; and for . One has (i)if , then (ii)if , then

Theorem 3.2. If , then the solutions of (1.1), with respect to for any , satisfy as

Proof. We consider first the case when .
From the first equation of (1.1), then there exists a such that for some when Since is increasing function with respect to , then from the second equation of (1.1), we have
Since we haveBy Lemma 3.1, we have in terms of is the global attractivor of
Next, we shall consider the case when Noticing that is equal to Since , is always decreasing when above . If should ever get below then must stay strictly below for all subsequent time. This implies there are two possible cases, either (i) from above as , or(ii)there exists such that for all
In the first of these cases, we have only to show that Integrating the first equation for from to in (1.1), we getThen,Letting , we conclude that and, therefore, .
In the second of these cases, consider the functionalThen, for all ,
A direct application of Liapunov-LaSalle type theorem [22] shows that . By the third equation of (1.1), we get that implies This proves is the sufficient condition for .

According to (1.1) and the definitions for permanence in [23], we have the following lemma.

Lemma 3.3. Permanence of in system (1.1) implies that of .

Next, we represent our main results in this section.

Theorem 3.4. System (1.1) is permanent if it satisfies

In order to prove Theorem 3.4, we present uniform persistence theory for infinite dimensional systems from [24]. Let be a complete metric space. Suppose that is open and dense in and . Assume that is a semigroup on satisfyingLet and let be the global attractor for .

Lemma 3.5 (see[24]). Suppose that satisfies (3.8) and one has the following: (i)there is a such that is compact for (ii) is point dissipative in (iii) is isolated and thus has an acyclic covering , where (iv) for

Then is a uniform repeller with respect to , that is, there is an such that for any , , where is the distance of from .

Proof. We first prove that leads to the permanence of system (1.1).
By Lemma 3.3, we only need to consider the following subsystem of (1.1) and prove that in system (3.10) are permanent if and only if holdswhere are continuous on , and
We begin by showing that the boundary planes of repel the positive solutions to system (3.10) uniformly. Let denote the space of continuous functions mapping into . We choose Denote and Int
Next, we verify that the conditions of Lemma 3.5 are satisfied. By the definition of and and system (3.10), it is easy to see that and are positively invariant. Moreover, conditions (i) and (ii) of Lemma 3.5 are clearly satisfied. Thus, we only need to verify conditions (iii) and (iv). Since system (3.10) possesses two constant solutions in withand we have , then we get for all , according to the second equation of (3.10), we have , hence all points in approach , Similarly, we have all points in approach , that is, . This shows that invariant sets and are isolated invariant, then is isolated and is an acyclic covering, satisfying condition (iii) of Lemma 3.5.
Now, we show that We only need to prove , since the proof for is simple.
Assume the contrary, that is, , then there exists a positive solution to system (3.10) with Since , then for a sufficiently small with , there exists a positive constant such that By the second equation of (3.10) and noting that is an increasing function with respect to , then we haveAccording to the comparison principle, when , contradicting . Then we have . At this time, we are able to conclude from Lemma 3.5 that repels the positive solutions of (3.10) uniformly. Incorporating the above results into Lemmas 3.3 and 3.5, we know that system (1.1) is permanent.
Next, we verify that permanence of system (1.1) indicates . Assume that the contrary holds, that is, , then by Theorem 3.2, as contradicting permanence of system (1.1). This proves Theorem 3.4.

4. Linearized Analysis

The dynamics of model (1.1) are determined by the first two equations. Therefore, throughout the remainder of this paper, we consider the subsystem (3.10), and rewrite it as follows:

Let be any equilibrium of (4.1), linearized system of (4.1) at , we getThen the characteristic equation of (4.1) at is given byAt the equilibrium , characteristic equation (4.3) reduces to Obviously, (4.4) has a positive root independent of any parameters. Hence, is always a unstable saddle point.

Theorem 4.1. For the system (4.1), the equilibrium is(i)asymptotic stable if ;(ii)linearly neutrally stable if ;(iii)unstable if .

Proof. The characteristic equation at is
Equation (4.5) has a negative real part characteristic root and roots of
(i) Assume that , (4.6) has characteristic root when . If is a root of (4.6), it must satisfyWhen , there are no positive real roots . This shows that all roots of must have negative real parts, therefore, is an asymptotically stable equilibrium.
(ii) Assume that , then is a root of (4.6). It is easy to verify that is a simple characteristic root. If the other roots are , then they must satisfyand we must have . Therefore is linearly neutrally stable.
(iii) Assume that , then , and . Hence, has at least one positive root and is unstable.

By the arguments to Theorems 3.2 and 4.1, we directly have the following corollary.

Corollary 4.2. The equilibrium of system (1.1) is global asymptotically stable if holds true in the feasible region for any

In the following, we will study the linear stability of the positive equilibrium of (4.1). We can see that the characteristic roots of (4.3) at positive equilibrium are the roots of Since and at , we havewhere When , the DDE (4.1) becomes ODE which has the same equilibria as follows:and (4.10) becomes

Define

If , that is, , the system (4.12) is locally asymptotically stable. If the unique positive equilibrium of (4.12) is unstable, system (4.12) becomes oscillatory in a stable limit cycle [25], and this limit cycle is unique [26, 27].

If is a root of (4.10), then by separating the real and imaginary parts, we get Squaring and adding both equations, then we have

Define .

If , that is, , there is no positive real satisfying (4.15), thus eigenvalues of (4.10) do not approach the imaginary axis for any . This shows that is absolutely stable when .

If , there is a unique positive satisfying (4.15). That is, (4.10) has a unique pair of purely imaginary roots .

From (4.14), corresponding to can be obtained as follows: where denotes and denotes

Further, Under the condition , that is, , then we have

If , there exists a critical value when is stable; when is unstable.

Summarizing the discussion above, we have the following conclusion.

Theorem 4.3. For system (4.1), one has(i)if holds true, then is absolutely stable;(ii)if holds true, then is conditionally stable, that is, there is a critical delay value such that is asymptotically stable when and unstable when . Furthermore, system (4.1) undergoes Hopf bifurcation at when ;(iii)if , then there is also a critical delay value such that the periodic solution is still stable when , however, there are a sequence of periodic solutions emanate when

These results are summarized in Table 1.

tab1
Table 1: Compare DDE (4.1) with ODE (4.12).

Remark 4.4. In fact, in the case of , we have known that ODE (4.12) has a stable periodic solution [25]. If we consider the impact of the incubation time on ODE (4.12), that is, DDE (4.1), from above discussion, we can see that there are a sequence of periodic solutions bifurcate from the positive equilibrium when the time delay takes the critical delay such that previous stable periodic solution losses stability, which will lead to complex dynamic phenomena. This can be seen from the simulation results Section 5.

In addition, we want to mention that Theorem 4.3(ii) cannot determine the direction and stability of bifurcation periodic solutions, this can be done by analyzing the high-order terms in terms of [28]. The method is very complex and trivial, here we omit it.

From the point of biology, in comparison with the results of [10], we see that if there is the inhibition effect from the behavioral change of the susceptible individuals when the infective increases (i.e., we take use of the saturation incidence rate), then the threshold decline and less than 3. It is crucial for the government to take the corresponding control measures and policies against the disease when the epidemic outbreaks.

5. Numerical Results

The main goal of the previous section was to qualitatively characterize the dynamic behaviors of system (1.1) at long term. In this section, we confirm our previous theoretical analysis in Section 4 and demonstrate that the local behaviors in the regions of the parameters space correspond to complex population dynamics to system (1.1). The objective is to explore the possibility of chaotic behavior in system (1.1). It is difficult to test whether there exists chaos in a time-delayed system, but numerical simulation analysis is a valid method for such a system. Extensive numerical simulations are carried out for different values of saturation parameter and recover rate . The quality results are as follows.

First, we consider the property of system (1.1) in the regions of the parameter space corresponds to complex population dynamics in the case of . To illustrate the transition from the periodic pattern to chaotic pattern, we concentrate on the regions of small and large time delay as an example. We consider the set of parameter values as , , , , , and . By calculating, there exists the relation of (, ) for system (1.1). Then, from Theorem 4.3(iii), the system (1.1) has a stable period solution if the time delay is less than the critical delay (see Figure 1), and the periodic solution will lost stability when the time delay is greater than the critical delay , and then a typical chaos was observed with increasing the time delay (see Figure 2). This phenomenon has been verified by the bifurcation diagram via delay , as shown in Figure 3.

fig1
Figure 1: Temporal behavior of the infected and corresponding three-dimensional phase are plotted for the system (1.1) subject to (, ). The parameters are , , , , , , and with initial value (28, 3, 7).
fig2
Figure 2: The parameters like in Figure 1 but .
636153.fig.003
Figure 3: Bifurcation diagram of system (1.1): successive maxima of the infected are plotted for increasing values of the time delay , with parameters , , , , , and .

We increase the recovery rate , let , and fix the other parameters as above. By calculating, the system (1.1) satisfies (, , ). In this context, by Theorem 4.3(ii), system (1.1) is conditionally stable at unique positive equilibrium , that is, there exists a critical delay such that is stable if (see Figure 4), will lose stability by an Hopf bifurcation if , as shown in Figure 5. We find that the periodic solution, quasiperiod, and chaos patterns emerge with increasing time delay. This may be clear from the bifurcation diagram (see Figure 7).

fig4
Figure 4: Temporal behavior of the infected and corresponding three-dimensional phase are plotted for the system (1.1) subject to (, , ). The parameters are , , , , , , and with initial value (35, 10, 15).
fig5
Figure 5: Temporal behavior of the infected and corresponding three-dimensional phase are plotted for the system (1.1) subject to (, , ). The parameters are , , , , , , , and with initial value (40, 3, 7).
fig6
Figure 6: The parameters like in Figure 5 but .
636153.fig.007
Figure 7: Bifurcation diagram of system (1.1): successive maxima of the infected are plotted for increasing time delay , with parameters , , , , , and .

Now, we fix the parameter , change and let (i.e., we take some measures to protect on susceptibles), and the other parameters are also as above. By calculating, system (1.1) always exists a relationship (, ). By Theorem 4.3(i), we know that system (1.1) is absolutely stable at unique positive equilibrium for any value of the time delay, as shown in Figure 8. Our numerical simulations have demonstrated the validity of our theoretical analysis, that is, the values of threshold value and incubation time length completely determine the dynamics of system (1.1).

fig8
Figure 8: Temporal behavior of the infected and corresponding three-dimensional phase are plotted for the system (1.1) subject to . The parameters like in Figure 5 but , , and with initial value (25, 4, 20).

It is necessary to indicate that the system (1.1) is realistic at the initial phase of the disease emergence since the number of the infected is rare. However, when the number of the infected is large, it is more reasonable that one should replace the term in the first equation of (1.1) by . Hence, a profound understanding for this case is still desirable and could motivate further investigations.

Acknowledgments

The authors are grateful to Associate Editor Manuel de la Sen and anonymous referee for their valuable comments and suggestions that greatly improved the original version of this paper. This work is supported by the National Sciences Foundation of China (60771026), Science Foundations of Shanxi Province (2007011019), the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005), and supported by Program for New Century Excellent Talents in University (NCET050271).

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