Discrete Dynamics in Nature and Society

Volume 2010, Article ID 727168, 7 pages

http://dx.doi.org/10.1155/2010/727168

## Persistence of an SEIR Model with Immigration Dependent on the Prevalence of Infection

Department of Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, China

Received 31 May 2010; Accepted 7 October 2010

Academic Editor: Guang Zhang

Copyright © 2010 Wenjuan Wang 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 incorporate the immigration of susceptible individuals into an SEIR epidemic model, assuming that the immigration rate decreases as the spread of infection increases. For this model, the basic reproduction number, , is found, which determines that the disease is either extinct or persistent ultimately. The obtained results show that the disease becomes extinct as and persists in the population as .

#### 1. Introduction

Mathematical models have been used to predict the spread of infectious diseases of humans and animals since the pioneering work of Anderson and May [1]. Many diseases such as tuberculosis and chronic hepatitis have the longer exposed period; thus, in some common researches, a population is divided into four classes: susceptible, exposed, infective, and recovered. In many studies on epidemic models, the goal is to understand the key factors affecting disease transmission [2–5], and this often includes determining a threshold condition for the persistence and extinction of the disease.

Many diseases such as influenza, measles, and sexually transmitted diseases are easily spread between regions (such as countries and cities) due to travel. This population dispersal is an important aspect to consider when studying the spread of a disease [6–8]. We will investigate a disease transmission model with population immigration from other regions to the one considered.

In many models, it is assumed that, in the absence of infection, the growth rate of population is given by , where is thought to be the input rate of population. Here, we consider as the sum of two parts, and , where is the birth rate of the population and is the immigration rate from other regions. Since the spread of the infection usually affects the immigration to the region, then we will introduce the effect into an SEIR epidemic model and consider this persistence and extinction of the disease in this paper.

#### 2. Model

In this paper, we consider an SEIR epidemic model with immigration: Here, , , , and represent the numbers of susceptible, exposed, infectious, and recovery individuals at time , respectively. is the input rate; is the immigration rate from other regions (such as countries or cities); it depends on the number of infectious individuals in the region considered, where is the immigration rate in the absence of disease and reflects the effect of infection on immigration from other regions; is the percapita natural death rate; is the transmission coefficient of infection; is the transfer rate from the exposed compartment to the infectious one; is the percapita recovery rate; is the percapita disease-induced death rate.

From model (2.1) we have It follows that then system (2.1) is bounded.

Since the variable does not appear explicitly in the first three equations in system (2.1), then we need only to consider the dynamics of a subsystem consisting of the first three equations in system (2.1). For this subsystem, making the following variable transformations: and removing the bar in , , , and , then we obtain the simplified system where , and

From the first equation in system (2.4), we have it implies that therefore, the set is positively invariant to system (2.4). Thus, we only consider the dynamical behavior of system (2.4) on the set .

#### 3. The Existence and Local Stability of Equilibria

It is obvious that system (2.4) always has the disease-free equilibrium . Its endemic equilibrium is determined by the following equations:

From the last two equations in (3.1), we have and for . Substituting into the first equation in (3.1) gives then is the positive root of (3.2).

According to the monotonicity of functions at the two sides of (3.2), we know that (3.2) has a unique positive root if and no positive roots if . Therefore, with respect to the existence of equilibria of system (2.4), we have the following theorem.

Theorem 3.1. *Denote that . When , system (2.4) has only the disease-free equilibrium on the set ; when , besides the disease-free equilibrium , system (2.4) also has a unique endemic equilibrium , where and is determined by (3.2).*

With respect to the local stability of equilibria and of system (2.4), we have the following theorem.

Theorem 3.2. *The disease-free equilibrium is locally asymptotically stable as and unstable as . The endemic equilibrium is locally asymptotically stable as it exists.*

*Proof. *(i) From the Jacobian matrix of system (2.4) at the disease-free equilibrium , it is easy to know that the disease-free equilibrium is locally asymptotically stable as and unstable as .

(ii) For the Jacobian matrix of system (2.4) at the endemic equilibrium , the characteristic equation is given by , where , , and then

Notice that (3.2) can be rewritten as

Using (3.4) gives

On the other hand, (3.2) can become
then
where , and . It follows from that , that is, Therefore, it follows from Hurwitz criterion that the endemic equilibrium is locally asymptotically stable.

#### 4. The Extinction and Persistence of Infection

In this section, we will consider the ultimate state of infection; that is, the disease will be whether extinct or persistent ultimately.

When , define function , where , then the derivative of with respect to along the solution of (2.4) on the set is given by It follows from that and , then there exists a positive number such that and . Therefore, from (4.1) we have , then where , therefore, for ; that is, as . It implies that the disease will be extinct ultimately when .

In order to discuss the persistence of the disease, we first introduce some definitions and lemmas.

Assume that is a locally compact metric space with metric and let be a closed subset of with the boundary and the interior . Let be a semidynamical system defined on .

We say that is persistent if, for all , and that is uniformly persistent if there is such that, for all ,

In [3], Fonda gives a result about persistence in terms of repellers. A subset of is said to be a uniform repeller if there is an such that, for each , A semiflow on a closed subset of a locally compact metric space is uniformly persistent if the boundary of is repelling in a suitable strong sense [9]. The result by Fonda is as follows.

Lemma 4.1. * Let be a compact subset of such that is positively invariant. A necessary and sufficient condition for to be a uniform repeller is that there exists a neighborhood of and a continuous function satisfying*(1)* if and only if *(2)*for all there is a such that .*

For any , there is a unique solution of system (2.4), which is defined in and satisfies . Since is a positively invariant set of system (2.4), then for and is a semidynamical system in .

In the following, we will prove that, when , is a uniform repeller, which implies that the semidynamic system is uniformly persistent.

Obviously, for if , then is invariant to (2.4). Again the set is a compact subset of .

Let be defined by and let where is small enough so that Since is equivalent to , then there exists a positive number such small that inequality (4.2) holds.

Assume that there is such that for each we have which implies that for . From the first equation in system (2.4) we have then So there is a sufficiently large number such that for

Define the auxiliary function , where is a sufficiently small constant so that . Direct calculation gives the derivative of along with as follows: Then, for , we have where therefore, .

On the other hand, the boundedness of the solution of (2.1) implies that of on the set . It implies that the assumption above is not true. Therefore, the above proof shows that, for each with belonging to a suitably small neighborhood of , there is some such that . Therefore, it follows from Lemma 4.1 that is a uniform repeller when that is, the infection is uniformly persistent. So we have the following theorem.

Theorem 4.2. *For system (2.4), the infection will be extinct when and persistent when .*

#### 5. Conclusion and Discussion

In Sections 3 and 4, for system (2.4) we investigated the qualitative behavior and obtained the threshold determining the persistence of infection. Corresponding to the original model (2.1), the basic reproduction number is . According to the results in Sections 3 and 4, model (2.1) only has the disease-free equilibrium which is globally stable when ; it implies that the disease is extinct ultimately; when , model (2.1) has a unique endemic equilibrium which is locally asymptotically stable and the disease persists in the population. Since the expression of here is independent of the parameter , then this shows that this parameter has no effect on the persistence of disease, but it can affect the strength of spread of disease according to Theorem 3.1.

#### Acknowledgments

This work was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), and the Major Subject Foundation of Shanxi (20091028).

#### References

- R. M. Anderson and R. M. May,
*Infectious Diseases of Humans: Dynamics and Control*, Oxford University Press, New York, NY, USA, 1991. View at Zentralblatt MATH - V. Capasso,
*Mathematical structures of epidemic systems*, vol. 97 of*Lecture Notes in Biomathematics*, Springer, Heidelberg, Germany, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Fonda, “Uniformly persistent semidynamical systems,”
*Proceedings of the American Mathematical Society*, vol. 104, no. 1, pp. 111–116, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. W. Hethcote, “The mathematics of infectious diseases,”
*SIAM Review*, vol. 42, no. 4, pp. 599–653, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. W. Hethcote and P. van den Driessche, “Two SIS epidemiologic models with delays,”
*Journal of Mathematical Biology*, vol. 40, no. 1, pp. 3–26, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Wang, “Population dispersal and disease spread,”
*Discrete and Continuous Dynamical Systems Series B*, vol. 4, no. 3, pp. 797–804, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Wang and X.-Q. Zhao, “An epidemic model with population dispersal and infection period,”
*SIAM Journal on Applied Mathematics*, vol. 66, no. 4, pp. 1454–1472, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-Y. Yang, F.-Q. Zhang, and X.-Y. Wang, “SIV epidemic models with age of infection,”
*International Journal of Biomathematics*, vol. 2, no. 1, pp. 61–67, 2009. View at Publisher · View at Google Scholar - G. Butler and P. Waltman, “Persistence in dynamical systems,”
*Journal of Differential Equations*, vol. 63, no. 2, pp. 255–263, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH