Research Article | Open Access

Jinghai Wang, "Analysis of an SEIS Epidemic Model with a Changing Delitescence", *Abstract and Applied Analysis*, vol. 2012, Article ID 318150, 10 pages, 2012. https://doi.org/10.1155/2012/318150

# Analysis of an SEIS Epidemic Model with a Changing Delitescence

**Academic Editor:**Allan Peterson

#### Abstract

An SEIS epidemic model with a changing delitescence is studied. The disease-free equilibrium and the endemic equilibrium of the model are studied as well. It is shown that the disease-free equilibrium is globally stable under suitable conditions. Moreover, we also show that the unique endemic equilibrium of the system is globally asymptotically stable under certain conditions.

#### 1. Introduction

Infectious diseases have tremendous influence on human life. Every year millions of human beings suffer from or die of various infectious diseases. It has been an increasingly complex issue to control infectious diseases. In order to predict the spreading of infectious diseases among regions, many epidemic models have been proposed and analyzed in recent years (see [1โ14]). Bilinear and standard incidence rates have been frequently used in classical epidemiological models (see [5]). However, it is more effective by using nonlinear incidence. Mathematical models describing the population dynamics of infectious diseases have been playing an important role in disease control for a long time. Many scholars have studied mathematical models which describe the dynamical behavior of the transmission of infectious diseases (see also [1โ14] and the references therein). A variety of nonlinear incidence rates have been used in epidemiological models (see [6โ14]). However, the models with a changing delitescence have seldom been studied.

In recent years, the spread of infectious diseases is diversiform, such as H1N1 disease. The diversity of the delitescence period in each infected body who is infected with H1N1 virus is mainly due to the variation of the virus and the distinct constitution of different people. The study of the models with a changing delitescence plays an important role in controlling infectious diseases. In this paper, we consider an SEIS epidemic model with a changing delitescence and a nonlinear incidence rate, and we study the existence and stability of the equilibriums of the SEIS epidemic model.

By a standard nonlinear incidence rate and a changing delitescence , we consider an SEIS epidemic model which consists of the susceptible individuals (), exposed individuals but not yet infected (), infectious individuals (), and the total population ().

The model is given as follows: where is the recruitment rate of individuals (including newborns and immigrants) into the susceptible population; is the natural death rate; is the rate at which infected individuals are treated or recovered; is disease-related death rate; is the rate at which exposed individuals become infectious; is the rate at which infected individuals become exposed; is the rate at which infected individuals become infectious. The nonlinear incidence rate is assumed to be of the form . , and are all normal numbers with . This, together with , implies Thus, substituting and (1.2) into (1.1), we have Form (1.2), in the absence of the disease, that is, . Since the spread of the disease in the population will lead to the decrease of , it follows that . Note that is a positively invariant region for the original model: and model (1.3) is obviously well-pased in .

#### 2. Existence of Equilibria

Now, we study equilibria of model (1.3), (note that ). Steady states of model (1.3) satisfy the following equations:

If , then and , so (2.1) has the disease-free equilibrium .

If , from the third equation of (2.1), we obtain

From the first and second equations of (2.1), we obtain

By substituting (2.2) and (2.3) into the first equation of (2.1), we obtain the following equation for : Let .

It is easy to see that (2.4) has a positive root if and only if . So (2.1) has a unique endemic equilibrium with Then, we have the following theorem.

Theorem 2.1. *If , model (1.3) only has the disease-free equilibrium ; if , model (1.3) has a unique endemic equilibrium except the disease-free equilibrium . *

#### 3. Stability of Equilibria

Theorem 3.1. *If , then the disease-free equilibrium is locally asymptotically stable; if , is stable; if , is unstable.*

*Proof. *The linearization of model (1.3) about the equilibrium gives
Thus, we have
Assume that , , and are the roots of the above equation. Then, we know that , and are the roots of the following equation:
According to Viete theorem, we have
If , we have
It is easy to see that all the roots of (3.3) have negative real parts if and only if . If , it is obvious that one of eigenvalue of (3.3) is 0. If , one of the roots of (3.3) has positive real part. This completes the proof.

Theorem 3.2. *If , then the disease-free equilibrium is globally asymptotically stable. *

*Proof. * Consider a Liapunov function as , then we have
If , , then if and only if .

Hence, according to Theorem 3.1, if , the disease-free equilibrium is globally asymptotically stable. This completes the proof.

Now, we study the local stability of the endemic equilibrium .

Substituting into the first equation of (2.1), we have

Then, the Jacobi matrix of (1.3) about is

Denote

According to [15], if and , then . Note that , , , , and then is stable.

If and , then . Meanwhile, , that is, hence, , which proves that is stable.

If and , to obtain , , , , there must be , but it is impossible.

From the above discussion, we get the following conclusion.

Theorem 3.3. *If , system (1.3) has a unique endemic equilibrium , which is locally asymptotically stable.*

From the third equation of (1.3), we have . Note that as , then as . The limit system of (1.3) is It is easy to see that system (3.11) has a disease-free equilibrium . If , system (3.11) has a unique endemic equilibrium with Consider the Dulac function . Note that then , there is no limit cycle in the first quadrant of the - plane. System (3.11) has a unique endemic equilibrium , then we prove that is globally asymptotically stable.

From the above discussion, we get the following conclusion.

Theorem 3.4. *There is no limit cycle, and the endemic equilibrium of system (3.11) is globally asymptotically stable.*

When , . It is easy to see that the stability of is equivalent to that of . Since is globally asymptotically stable, the unique endemic equilibrium is also globally asymptotically stable.

#### 4. Numerical Simulation

Figures 1โ5 are drawn by MATLAB. With different parameters, they describe the changes of the number of exposed individuals who are not yet infectious, infectious individuals, and the total population.

With , , , , , , , , we consider (2,3,5), (3,2,7), (2,2,5), (3,4,8), (2,5,9), (5,4,9), (2,4,9) as the initial values. Figure 1 describes system (1.3) with (), Figure 2 describes system (1.3) with ().

For the initial values (20,30,500), (30,20,700), (20,20,500), (30,40,800), (20,50,900), (50,40,900), (20,40,900), Figures 3, 4, and 5 are drawn with , , , , , , , by MATLAB. With (), (), and (), Figures 3โ5 describe the endemic equilibriums of system (1.3).

From Figures 1 and 2, it is easy to see that the value of decides the development trend of the infectious disease, that is, whether the infectious disease dies out or does not exist forever when the other conditions are the same.

From Figures 3โ5, with a unique endemic equilibrium of system (1.3), the value of will affect the numbers of exposed individuals who are not yet infectious, infectious individuals, and the total population.

With , , , , , , , Figure 6 shows that system (3.11) has no limit cycle.

#### 5. Conclusion

Because of distinct constitutions of individuals, some infected individuals who are not yet infectious become exposed individuals, while other infected individuals immediately become infectious. So, we establish an epidemic model with a changing delitescence between SEIS and SIS model by using the proportional number .

This paper mainly considers the existence and stability of equilibriums. We use Jacobi matrix to discuss the local asymptotical stability of the endemic equilibrium, and we obtain sufficient conditions for this. When we discuss its global asymptotical stability, we use the Dulac function in its limit system (3.11), and we get relatively complete conclusions.

After the discussion of the stability of the equilibriums, we have made numerical simulations of the epidemic model with different values of . By discussing the basic reproduction number , we show that the value of is important in the tendency of the infectious diseases with the other conditions being the same. We must pay attention to the changing delitescence, develop effective control strategies, then we can better control the tendency of infectious diseases.

#### References

- E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, โGlobal asymptotic stability of an SIR epidemic model with distributed time delay,โ
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 6, pp. 4107โ4115, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Takeuchi, W. Ma, and E. Beretta, โGlobal asymptotic properties of a delay SIR epidemic model with finite incubation times,โ
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 42, no. 6, pp. 931โ947, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. M. Anderson and R. M. May, โPopulation biology of infectious diseases: part 1,โ
*Nature*, vol. 280, pp. 361โ367, 1979. View at: Google Scholar - L. Q. Gao, J. Mena-Lorca, and H. W. Hethcote, โFour SEI endemic models with periodicity and separatrices,โ
*Mathematical Biosciences*, vol. 128, no. 1-2, pp. 157โ184, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Ma, Y. Zhou, W. Wang, and Z. Jin,
*Mathematical Ecology*, Springer, New York, NY, USA, 1990. - S. Ruan and W. Wang, โDynamical behavior of an epidemic model with a nonlinear incidence rate,โ
*Journal of Differential Equations*, vol. 188, no. 1, pp. 135โ163, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Xiao and S. Ruan, โGlobal analysis of an epidemic model with nonmonotone incidence rate,โ
*Mathematical Biosciences*, vol. 208, no. 2, pp. 419โ429, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. M. Liu, H. W. Hethcote, and S. A. Levin, โDynamical behavior of epidemiological models with nonlinear incidence rates,โ
*Journal of Mathematical Biology*, vol. 25, no. 4, pp. 359โ380, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. d'Onofrio, โStability properties of pulse vaccination strategy in SEIR epidemic model,โ
*Mathematical Biosciences*, vol. 179, no. 1, pp. 57โ72, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Liao, โOn the homotopy analysis method for nonlinear problems,โ
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499โ513, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. R. Shi, X. J. Xu, Z. X. Wu et al., โApplication of the homotopy analysis method to solving nonlinear evolution equations,โ
*Acta Physica Sinica*, vol. 55, no. 4, pp. 1555โ1560, 2006. View at: Google Scholar | Zentralblatt MATH - J. Hui and D. Zhu, โGlobal stability and periodicity on SIS epidemic models with backward bifurcation,โ
*Computers & Mathematics with Applications*, vol. 50, no. 8-9, pp. 1271โ1290, 2005. View at: Publisher Site | Google Scholar - J. Li and Z. Ma, โQualitative analyses of SIS epidemic model with vaccination and varying total population size,โ
*Mathematical and Computer Modelling*, vol. 35, no. 11-12, pp. 1235โ1243, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L.-I. Wu and Z. Feng, โHomoclinic bifurcation in an SIQR model for childhood diseases,โ
*Journal of Differential Equations*, vol. 168, no. 1, pp. 150โ167, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Jeffries, V. Klee, and P. van den Driessche, โWhen is a matrix sign stable?โ
*Canadian Journal of Mathematics*, vol. 29, no. 2, pp. 315โ326, 1977. View at: Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 Jinghai Wang. 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.