Abstract
We consider a nonautonomous SEIR endemic model with saturation incidence concerning pulse vaccination. By applying Floquet theory and the comparison theorem of impulsive differential equations, a threshold parameter which determines the extinction or persistence of the disease is presented. Finally, numerical simulations are given to illustrate the main theoretical results and it shows that pulse vaccination plays a key role in the disease control.
1. Introduction
Millions of human beings suffer or die of various infectious diseases every year. Infectious diseases such as measles, polio, diphtheria, tetanus, and pertussis have tremendous influence on human life. Controlling infectious diseases has been an increasingly complex issue worldwide in recent years. Pulse vaccination is an effective method to use in controlling the transmission of diseases which has gained prominence. Epidemiological models with pulse vaccination have been set up and investigated in many literatures [1–8]. This kind of vaccination strategy is usually called pulse vaccination strategy (PVS). This vaccination is called impulsive when all the vaccine doses are applied in a very short span of time.
As we all known, some diseases may be governed directly or indirectly by environmental factors such as temperature, humidity, and barometric pressure; see [9–13], for example. It is realistic to investigate this kind of epidemic models with periodic (seasonal) fluctuations. Bai and Zhou [14] studied the existence and the number of periodic solutions in an SIR epidemic model with seasonal contact rate. One of the most important subjects in this field is to obtain a threshold that determines the persistence or extinction of a diseases. According to [15, 16], we know that the basic reproduction number for the periodic epidemic models is generally different from the basic reproduction number of the tim-averaged autonomous system. They defined the basic reproduction number which is a threshold between the extinction and the uniform persistence of the disease. In [17], Nakata and Kuniya considered the following SEIRS epidemic model: where , , , and denote susceptible, exposed (infected but not infectious), infectious, and recovered population at time , respectively. , , , , , and are continuous, positive -periodic functions.
Many epidemic models with constant coefficients and pulse vaccination have been proposed and studied to understand mechanism of disease transmission. However, there have been less results on the pulse vaccination models with variable coefficients. Zhang and Teng [18] formulated a nonautonomous SEIRS model in epidemiology. Under the quite weak assumptions, they establish some sufficient conditions to prove the permanence and extinction of disease. The nonautonomous models are more realistic. On the basis of the above discussion, we formulate the following SEIR epidemic model which takes into account both the saturation incidence and the effects of pulse vaccination: where , , , , and are continuous, positive -periodic functions. is the saturation incidence. The function is the recruitment rate of the susceptible population and is the death rate of the population. is the transmission coefficient and and are the instantaneous per capita rates of leaving the latent stage and infected stage, respectively. is the constant period between two pulse vaccinations, is the fraction of susceptible to whom the vaccination is inoculated at times , is a positive integer, and , .
This paper is organized as follows. In Section 2, sufficient conditions for the global attractivity of disease-free periodic solution of system (2) are obtained. In Section 3, the permanence of disease is discussed. Numerical analysis and briefly discuss are given in the last Section.
2. Stability of the Disease-Free Periodic Solution
In the section, we will study the global asymptotical stability of the disease-free periodic solution of system (2). Before staring our theorem, we give the following lemma.
Lemma 1. Consider the following system: Then system (3) has a unique positive periodic solution which is globally asymptotically stable, where
Proof. By calculating, we obtain the solution of system (3) with respect to and , Set . From (6) and , we have where is the stroboscopic map. It is easy to know that the map has the unique positive fix point which implies that the unique periodic solution of system (3) is globally asymptotically stable. Similarly, we can obtain that the periodic solution of system (3) is unique and globally asymptotically stable. The proof is completed.
We will present the Floquet theory for the following linear -periodic impulsive equation:
Introduce the following conditions:: and ,:, , ,:there exists a , such that , .
Note. means the set of functions which are piecewise continuous.
Let be a fundamental matrix of (9); then there exists a unique nonsingular matrix such that By equality (10) there corresponds to the fundamental matrix the constant matrix which we call the monodromy matrix of (9). All monodromy matrices of (9) are similar and have the same eigenvalues. The eigenvalues of the monodromy matrices are called the Floquet multipliers of (9).
Lemma 2 (see [19, Chapter 2, Theorem 3.5] (Floquet theory)). Let conditions hold. Then the linear -periodic impulsive equation (9) is(1)stable if and only if all multipliers of (9) satisfy the inequality and, moreover, to those for which correspond simple elementary divisors;(2)asymptotically stable if and only if all multipliers of (9) satisfy the inequality ;(3)unstable if for some .
Let be the standard ordered -dimensional Euclidean space with a norm . For , we write provided , provided , respectively.
Let on the interval be a bounded, continuous, cooperative, and irreducible matrix function and . We consider the following system: Let be the fundamental solution matrix of (11), and let be the spectral radius of . By the Perron-Frobenius theorem, is the principle eigenvalue of in the sense that it is simple and admits an eigenvector . By the similar proof of Lemma 2.1 in [17], we have the following result.
Lemma 3. Denote . Then there exists a positive, -periodic function such that is a solution of (11).
Define
Theorem 4. If , then the disease-free period solution of system (2) is globally asymptotically stable.
Proof. Firstly, we demonstrate the local stability of the disease-free period solution . Define , , , and . Then (2) can be expanded in a Taylor series. After neglecting higher order terms, the linearized equations can be written as
where and
Thus the monodromy matrix of system (13) is
There is no need to calculate the exact form of as it is not required in the following analysis. The stability of the period solution is determined by the eigenvalues of . In view of Lemma 2, the disease-free period solution of system (2) is local stable if .
Secondly, we will prove the global attractivity of the disease-free period solution. According to the first equation of system (2) and the positivity of the solutions, we have
Consider the following auxiliary system:
Lemma 1 implies that system (17) admits a positive periodic solution which is globally asymptotically stable. From the comparison theorem in impulsive differential equations, we have . Hence, for any , there exits such that for . Then it holds that
for . Put be matrix function such that
Since and is continuous for small , we can choose sufficiently small so that .
Consider the following system:
By Lemma 3 and the comparison principle, there exists a positive -periodic function such that where and . Then we get that and . From the first and third equations of system (2), Lemma 1 and the limiting system theory, we have and . This completes the proof.
3. Permanent
Next, we give the conditions which assure that the disease is endemic (i.e., the disease is permanent) if the infectious population persists above a certain positive level for large enough time. Let , , , , , , , , , , and . Using the same discussions as those in the proof of Lemma 2.2 in [17], we have With the similarly method of Theorem 6 see also in [20], we have the following lemma.
Lemma 5. The susceptible population is bounded away from zero so that for sufficiently large , where
Define
We consider following system:
Define . By Lemma 3, there exist a positive, -periodic function such that is a solution of (24), where .
Theorem 6. Assume that holds; then there exists a positive constant , such that, for all , the solution of system (2) satisfies
Proof. We first prove the following claim:
where is sufficiently small. Suppose that (26) is not true; then there exists a such that for all . Then according to the first equation of system (2), we get
Consider an auxiliary system
By the comparison theorem, it following that holds. From Lemma 1, we see that system (28) admits a positive periodic solution which is globally asymptotically stable and . Hence, for any , there exits such that for . Then for , we obtain that
If , choosing , such that . By the comparison principle, there exists a positive constant such that where and , which implies that which is a contradiction. Thus (26) holds.
In view of (26), we know that there are two possible cases.
Case I. for all large . The conclusion is evident in this case.
Case II. oscillates about for all large .
Set , and satisfy
for .
It follows from system (2) that
for .
Denote . Since is sufficiently large, by Lemma 5 and , we have
Then, by the comparison principle, there exists a positive such that
where is independent of the choice of the interval . Thus we have that
Since the kind of interval is chosen in an arbitrarily way (we only need to be sufficiently large) and is independent of the positive solution of system (2), we have proved that any solution of system of (2) satisfies , for sufficiently large . The proof is completed.
The following are two immediate corollaries of Theorem 6.
Corollary 7. The diseases-free periodic solution of (2) is unstable when .
It follows from Lemma 5 and Theorem 6 that the following result holds.
Theorem 8. If , then the system (2) is permanent.
4. Discussion and Numerical Results
In this paper, we have established and studied a nonautonomous SEIR endemic model with saturation incidence and pulse vaccination. From the theoretically analysis, we obtain a threshold , and there exists a globally asymptotically stable disease-free periodic solution if (see Theorem 4). However, when , the stability of disease-free periodic solution is lost, and system (2) is permanent (see Theorem 6). According to [16] we can define the spectral radius of the as the basic reproduction ratio for system (2). It is worthwhile for us to study how to define the basic reproduction ratio for a general system with pulse interval interruption.
To illustrate the mathematical results, we investigate future by using numerical simulations. Now consider the following two examples which Theorems 4 and 6 are satisfied.
Consider the following set of system (2): , , , , , and .
We have By a approximation method, we obtain . Hence, by Theorem 4, the disease-free periodic solution is globally asymptotically stable. The disease will die out. Numerical simulation illustrates this fact (see Figure 1).
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The following choice of parametric values: , , , , , , and .
We have By a approximation method, we obtain . Clearly, in view of Theorem 6, the disease will become endemic (see Figure 2).
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It is interesting to examine how the pulse vaccination affects the transmission of the disease. Fix the parameters of system (2) as follows: , , , , , , and .
Numerical results shown in Figure 3 imply that the disease will be eradicated if we set . In contrast, the disease will become endemic shown in Figure 4 if set . Obviously, the pulse vaccination plays an important role in the system.
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Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province (PKLHB1302), the Soft Science Research Project of Hubei Province (2012GDA01309), Natural Science Foundation of Education Committee of Hubei Province (B2013073), and Key Discipline of Hubei Province-Forestry.