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The Scientific World Journal
Volume 2013 (2013), Article ID 871393, 5 pages
http://dx.doi.org/10.1155/2013/871393
Research Article

Dynamic Analysis of an SEIR Model with Distinct Incidence for Exposed and Infectives

Department of Mathematics and Sciences, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou, Hebei 075000, China

Received 17 April 2013; Accepted 9 May 2013

Academic Editors: J. Banaś and M. M. Cavalcanti

Copyright © 2013 Junhong Li and Ning Cui. 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 SEIR model with vaccination strategy that incorporates distinct incidence rates for the exposed and the infected populations is studied. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The sufficient conditions for the global stability of the endemic equilibrium are obtained using the compound matrix theory. Furthermore, the method of direct numerical simulation of the system shows that there is a periodic solution, when the system has three equilibrium points.

1. Introduction

Mathematical models have become important tools in analyzing the spread and the control of infectious diseases. Many infectious diseases in nature, such as measles, HIV/AIDS, SARS, and tuberculosis (see [16]), incubate inside the hosts for a period of time before the hosts become infectious. Li and Fang (see [7]) studied the global stability of an age-structured SEIR model with infectivity in latent period. Yi et al. (see [8]) discussed the dynamical behaviors of an SEIR epidemic system with nonlinear transmission rate. Li and Zhou (see [9]) considered the global stability of an SEIR model with vertical transmission and saturating contact rate.

In this paper, we will consider an SEIR model that the diseases can be infected in the latent period and the infected period. The population size is divided into four homogeneous classes: the susceptible , the exposed (in the latent period) , the infective , and the recovered . It is assumed that all the offsprings at birth are susceptible to the disease. The inflow rate (including birth and immigration) and outflow rate (including natural death and emigration) are denoted by and , respectively. The rate of disease-caused death is taken as . We assume that susceptible individuals are vaccinated at a constant per capita rate . Due to the partial efficiency of the vaccine, only fraction of the vaccinated susceptibles goes to the recovered class. The remained fraction of the vaccinated susceptibles has no immunity at all and goes to the exposed class after infected by contact with the infectives. If , it means that the vaccine has no effect at all, and if , the vaccine is perfectly effective. The positive parameter is the rate at which the exposed individuals become infectious. is the constant rate, at which the infectious individuals recover with acquiring permanent immunity. The transfer mechanism from the class to the class is guided by the function , where is the force of infection. denotes the relative measure of infectiousness for the asymptomatic class .

Based on these considerations, and with reference to [1012], the SEIR model is given by the following system of differential equations: where the derivative is denoted by and .

Thus, the total population size implies . Let , , , and .

Because the variable does not appear in the equations of , , and , we only need to consider the following subsystem: The system (2) is equivalent to (1). From biological considerations, we study (2) in the following closed set: where denotes the nonnegative cone of including its lower dimensional faces.

2. Equilibria and Global Stability

It is easy to visualize that (2) always has a disease-free equilibrium . The Jacobian matrix of (2) at an arbitrary point takes the following form: where

Theorem 1. If , the disease-free equilibrium is locally asymptotically stable, where

Proof. Let
We calculate the characteristic equation of as follows:
The stability of is equivalent to all eigenvalues of (8) being with negative real parts, which can be guaranteed by . Consequently, the disease-free equilibrium is local asymptotical stability. This proves the theorem.

Theorem 2. If , the disease-free equilibrium is globally asymptotically stable, where

Proof. Consider the following function: Its derivative along the solutions to the system (2) is as follows:
Furthermore, only if . The maximum invariant set in is the singleton . When , the global stability of follows from LaSalle’s invariance principle (see [13]). This completes the proof.

Theorem 3. Equation (2) has a unique endemic equilibrium if and , where

Proof. Let the right side of each of the first three differential equations equal to zero in (2); we obtain the following: with , and . So we get When the three equations of (14) are multiplied together, we obtain the following: Define the following: where and the roots of are , , and the other two are , which satisfy . is the root of . Direct calculations show the following: Since , the linear function has exactly one intersection with the function where lies in the interval . Furthermore, and can be uniquely determined from by the following:
From this, we can easily see that (2) has a unique endemic equilibrium. This completes the proof.

Denote the interior of by . In this paper, we obtain sufficient conditions that the equilibrium is globally asymptotically stable using the geometrical approach of Li and Muldowney in [14].

Theorem 4. The unique endemic equilibrium is globally asymptotically stable in , when

Proof. Since , namely, and is unstable, we can easy see that (2) satisfies the assumptions and (see [14]) in the interior of its feasible region . The unique equilibrium is locally asymptotically stable using simple calculation.
Let and denotes the vector field of (2) and where Set the the following function: Then the matrix can be written in block form as follow: where
Let denote the vectors in , we select a norm in as and let denote the Lozinskii measure with respect to this norm. Using the method of estimating in [15], we have , where , are matrix norms with respect to the vector norm, and denotes the Lozinskii measure with respect to norm. More specifically, , , and .
Rewriting the system (2), we have the following: Therefore, Since and , there is and Along each solution to (2) such that , where is the compact absorbing set, we thus have the following: which implies This completes the proof.

3. Conclusion

In this paper, we discuss an SEIR model that the diseases can be infected in the latent period and the infected period. The vaccine effectiveness is also taken into account. We investigate the global dynamics of the reduced proportional system. If , the disease-free equilibrium is globally asymptotically stable. The unique equilibrium of the system (2) is globally asymptotically stable in , when , , , and . When , (2) becomes the SEIR model without infectivity in latent and disease-caused death (see [8]). When , (2) becomes the SEIR model without infectious in latent (see [12]).

The parameters are considered in the following cases: (see [16]), and At this case, there are three fixed points: and , are unstable, stable. The method of direct numerical simulation of (2) shows that there is a periodic solution. The phase portraits of the system (2) in Figure 1(a), and time series of are given in Figures 1(b), 1(c), and 1(d).

fig1
Figure 1: (a) Phase portraits of (2). (b) Time series of . (c) Time series of . (d) Time series of .

Acknowledgments

The authors are very grateful to the reviewers for their valuable comments and suggestions. This work was supported by the Youth Science Foundations of Education Department of Hebei Province (no. 2010233, 2011236).

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