Abstract and Applied Analysis

Volume 2014 (2014), Article ID 542154, 15 pages

http://dx.doi.org/10.1155/2014/542154

Research Article

SVEIRS: A New Epidemic Disease Model with Time Delays and Impulsive Effects

Tongqian Zhang,¹ Xinzhu Meng,² and Tonghua Zhang³

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

²Shandong University of Science and Technology, Qingdao 266590, China

³Department of Mathematics, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia

Received 5 January 2014; Revised 12 April 2014; Accepted 25 April 2014; Published 26 May 2014

Academic Editor: Zhiming Guo

Copyright © 2014 Tongqian 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.

We first propose a new epidemic disease model governed by system of impulsive delay differential equations. Then, based on theories for impulsive delay differential equations, we skillfully solve the difficulty in analyzing the global dynamical behavior of the model with pulse vaccination and impulsive population input effects at two different periodic moments. We prove the existence and global attractivity of the “infection-free” periodic solution and also the permanence of the model. We then carry out numerical simulations to illustrate our theoretical results, showing us that time delay, pulse vaccination, and pulse population input can exert a significant influence on the dynamics of the system which confirms the availability of pulse vaccination strategy for the practical epidemic prevention. Moreover, it is worth pointing out that we obtained an epidemic control strategy for controlling the number of population input.

1. Introduction

In epidemic modeling, susceptible-infectious-recovered type of models is well known [1–18] although such models very often ignore the incubation period in the development of mathematical models for some diseases. However, recent research shows for certain diseases, such as smallpox, rabies, BSE, and some skin diseases, the incubation period has significant effect on the epidemic dynamics so that it is nonnegligible. The incubation period varies greatly from a couple of days (e.g., H1N1 outbreaking worldwide has generally an incubation period of one to seven days) to several years (e.g., AIDS virus sometimes can be several years). When taking the incubation period into account in the development of models, we reach SEIR model, which is short for susceptible, exposed, infectious, and recovered [19–30]. And some researchers used time delay to describe the incubation period; for example, Cooke [31], Beretta and Takeuchi [4], Takeuchi et al. [32], and Ma et al. [5] studied a SIR model with time delay and nonlinear incidence rate . Liu et al. [33, 34] used a nonlinear incidence rate , and Meng et al. [35] and Jiang et al. [30], respectively, studied an impulsively vaccinating SIR model with nonlinear incidences and , which are better to describe the spread process of diseases than linear one.

In order to prevent infectious diseases, [36, 37] suggested that vaccination to the susceptible population is an important strategy. The traditional vaccinations are applied to each individual, while impulsive ones are to periodically vaccinate people within certain age groups [7–10, 38]. Some diseases may have a vaccination period after being cured but may cause losing immunity gradually. In this case, people might be infected again. So it is of great significance to investigate epidemic models with time delay and impulsive effects due to the incubation period and vaccination period [26–29]. For some certain regional systems, the immigrations can be periodic impulsive population input because the immigratory population might be susceptible. Certainly two different impulsive effects for periodic vaccination and population input do not usually happen simultaneously. Therefore, motivated by Jiang et al. [30] and Song et al. [19], we built a new mathematical model: susceptible, vaccinated, exposed, infectious, recovered, and susceptible epidemic model with two time delays and two nonlinear incidences with pulse vaccination and a constant periodic population input at two different moments as follows:

Here all parameters of system (1) are nonnegative constants. For the significance of parameters in (1), please see literatures Jiang et al. [30] and Song et al. [19]. Terms and are the nonlinear incidence rates, and in our paper we only discuss the case

2. Preliminaries

Let , and then it is easy to see that satisfies the following:

Hence, for time which is large, we obtain . Let and ; here is bounded function on interval . Since variable only appears in the fifth equation, system (1) can be further reduced as

with the initial conditions

Lemma 1 (see [39, 40]). For the following impulse differential inequalities

where , , , and are constants.

Assume the following:

the sequence satisfies , with ; and is left-continuous at .

Then

Lemma 2 (see [41]). For the following delay differential equation

where , , and are all positive constants and for , then we have

Lemma 3 (see [42]). The following system,

has a unique positive -periodic solution:

and we further have and as .

3. The Existence and Global Attractivity of “Infection-Free” Periodic Solution

3.1. Existence

In this section, we are committed to investigate the existence of “infection-free” periodic solution. In this case, we have

From systems (4) and (12), we obtain

By Lemma 3, system (13) has a unique positive -periodic solution:

Furthermore, we can prove that it is the unique globally asymptotically stable positive periodic solution of system (4). We summarize this conclusion in the following lemma.

Lemma 4. The system (4) has an “infection-free” periodic solution , for and ; for any solution of it, the following holds true:

as .

This lemma indicates that in between the vaccination the susceptible and vaccinated populations oscillate with period in synchronization with the periodic pulse vaccination. Next we prove the global attractivity of such solution.

3.2. Global Attractivity

In this section, we will prove our main result on the global attractivity of the infection-free solution. It is stated in the following theorem.

Theorem 5. The system (4) has a unique infection-free periodic solution , and when it exists, it is globally attractive if

where

with

Proof. Let be a solution of (4) satisfied initial condition (5). Since , one can choose an small enough such that

where

For , we have

By impulsive differential inequality Lemma 1, we have

where

Thus

and then we have

Thus there exists a positive integer and constant small enough such that, for all ,

For , system (4) yields

We obtain the following comparison impulsive differential system:

By Lemma 3, the system has a periodic solution given by

which is globally asymptotically stable.

Now, assume that is the solution of system (28) with initial value . Then by Lemma 1, we know there exists a positive integer such that

Hence,

From (27), (31), and the third equation in (4), for we have

Consider the comparison equation:

From (19), we have

According to Lemma 2, we then obtain

Notice the fact that for all and , and the comparison theorem implies as . Without loss of generality, we may assume that for all . By using the first and second equations in (4), we reach

For , we have

considering the following system:

We obtain

Now by using comparison theorem of impulsive equations, for any there exists a such that

for . On the other side, from the first and second equations of (4), we have

Then we have , and , , as , where is a unique positive periodic solution of

from which we have that, for ,

Applying the comparison theorem again, for any , there exists a such that

for . Let , and then from (40) and (44) we have

for large enough, which implies , as . This completes the proof.

Corollary 6. If or , then the infection-free periodic solution is globally attractive, where the critical values are given below:

4. Permanence

In this section, we discuss the permanence of the infectious population. First, we introduce the following definition.

Definition 7. System (4) is said to be permanent if there exist positive constants , , (independent of initial value), and a finite time , which may depend on the initial condition, such that every positive solution with initial condition (5) satisfies , , for all .

Let

Then we have our main result of this section.

Theorem 8. Let , if , and then there exists a positive constant small enough such that

with large enough.

Proof. As before, we suppose that is a positive solution of system (4) with initial condition (5). Then for , we construct a function as follows:

And then differentiating along the trajectory of (4) yields

for . Let

Since , we get , . Then we have . And from , we can get

Thus, we have

Form , we have

that is,

We can take small enough such that

Thus we can choose to be small enough such that

Then we claim that there exists an such that for is large enough. We next prove this claim in two steps.

Step I. For any positive constant , that for all is not true.

Otherwise, there is a positive constant , such that for all . First, if for all , it follows from the first, fourth, and fifth equations of (4) that, for ,

By Lemma 1, there exists so that for

Similarly, from the second and the fourth equations of (4), we have

and for ,

Then, by (50), for ,

Let

We can prove that for all . Otherwise, there exists a nonnegative constant such that for , and . Then from the second equation of (4) and (37), we easily see that

which is a contradiction. Hence for all . Equation (62) implies

It then follows that as . This is a contradiction to . Therefore, for any positive constant , the inequality cannot hold for all .

Step II. From Step I, we only need to consider the followng: (i) for all large enough and (ii) oscillates about for all large . However, Case (i) is obvious in the result of this theorem, so we only need to consider Case (ii), in which we will show that for all large where

First, we notice there exist two positive constants , such that

Second, because is bounded continuous function and has no pulse, we can get that is uniformly continuous. Therefore there exists a constant (with and is independent of the choice of ) such that for all .

If , our aim is obtained.

If , from the second equation of (4) we have that for . Then we have for since . It is clear that for .

If , then we have for . We then can easily prove for . Since the interval is arbitrarily chosen, we know that holds for large enough. Finally, noticing the choice of is independent of the positive solution of (4), we completed our proof.

Theorem 9. Let , if , and then system (4) is permanent.

Proof. Suppose that is a positive solution of system (4) with initial conditions (5). Then from system (4), we have

As what we did in the proof of Theorem 5, we can prove that there exist large enough and small enough such that

Then for , by Theorem 8, we have

for large enough. Thus the system (4) is uniformly permanent.

5. Numerical Simulations and Discussions

Next, we carry out numerical simulations to illustrate the theoretical results obtained in the previous sections. We first set the parameters as follows: , , , , , , , , , , , , and . Straightforward calculation shows . Then by Theorem 8, the disease will be permanent (please see Figures 1(a), 1(b), 1(c), and 1(d)). In order to show the effect of , we decrease to , and other parameters are the same with those in Figure 1, and the infection-free periodic solution of system (4) is globally attractive. This phenomenon is also seen from our theoretical analysis as in this case and then according to Theorem 5, the disease will be eradicated; please see Figure 2(a).

(a) Time series of , , and . The disease will be permanent

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 1: The results of numerical simulation on the threshold values , where , .

(a) Time series of , , and . The disease dies out

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 2: The results of numerical simulation on the threshold values , where , .

If we keep and , as the same with those in Figure 1, but increase vaccination proportion of susceptible persons to , then the disease will be eradicated; see Figure 3(a). If we keep and and decrease to , then the disease also will be eradicated; see Figure 4(a).

(a) Time series of , , and . The disease dies out

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 3: The results of numerical simulation on the threshold values , where , .

(a) Time series of , , and . The disease dies out

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 4: The results of numerical simulation on the threshold values , where , .

And if we keep , but decrease to , then the disease will be permanent; see Figure 5. If we keep and and increase to , then the disease also will be permanent; see Figure 6. For details please see Table 1.

Table 1

Status of the disease

1
1
0.4
1.4

Permanence

4
4
0.4
1.4

Eradication

1
1
0.9
1.4

Eradication

1
1
0.4
0.2

Eradication

4
4
0.1
1.4

Permanence

4
4
0.4
2

Permanence

(a) Time series of , , and . The disease will be permanent

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 5: The results of numerical simulation on the threshold values , where , .

(a) Time series of , , and . The disease will be permanent

(b) Phase portrait of , of the system  (4)

(c) Phase portrait of , of the system  (4)

(d) Phase portrait of , , and of the system  (4)

Figure 6: The results of numerical simulation on the threshold values , where , .

Lastly, we conclude our paper as follows. In this paper, we proposed an SVEIRS model, which is a new epidemic model with periodic pulse vaccination and pulse population input at two different fixed moments. Our primary result is to investigate the effect of impulsive vaccination, pulse population input, and time delays to the dynamics of population model. With the help of comparison theorems, we proved the existence of the “infection-free” periodic solution and obtained the conditions for global attractivity of the “infection-free” periodic solution and the conditions for the permanence of the system. All the theoretical results show that we believe it might be helpful in disease control: people can select appropriate vaccination rate and population input rate according to our theoretical results to control diseases.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. T. Q. Zhang and X. Z. Meng are financially supported by the National Natural Science Foundation of China (no. 11371230), the Shandong Provincial Natural Science Foundation of China (no. ZR2012AM012), and the Project of Shandong Province Higher Educational Science and Technology Program of China (no. J13LI05).

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