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Abstract and Applied Analysis
Volume 2012, Article ID 318150, 10 pages
http://dx.doi.org/10.1155/2012/318150
Research Article

Analysis of an SEIS Epidemic Model with a Changing Delitescence

Institute of Mathematics and Computer Science, Fuzhou University, Fuzhou, FuJian 350002, China

Received 17 May 2012; Accepted 26 July 2012

Academic Editor: Allan Peterson

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.

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 [114]). 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 [114] and the references therein). A variety of nonlinear incidence rates have been used in epidemiological models (see [614]). 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 𝛽𝑆𝐼/(1+𝛼𝐼) and a changing delitescence 𝜇, we consider an SEIS epidemic model which consists of the susceptible individuals (𝑆), exposed individuals but not yet infected (𝐸), infectious individuals (𝐼), and the total population (𝑁).

The model is given as follows: 𝑆=𝐴𝑑𝑆𝛽𝑆𝐼𝐸1+𝛼𝐼+𝛾𝐼,=𝜇𝛽𝑆𝐼𝐼1+𝛼𝐼(𝑑+𝜀)𝐸,=(1𝜇)𝛽𝑆𝐼1+𝛼𝐼+𝜀𝐸(𝑑+𝛾+𝛿)𝐼,(1.1) where 𝐴 is the recruitment rate of individuals (including newborns and immigrants) into the susceptible population; 𝑑 is the natural death rate; 𝛾 is the rate at which infected individuals are treated or recovered; 𝛿 is disease-related death rate; 𝜀 is the rate at which exposed individuals become infectious; 𝜇 is the rate at which infected individuals become exposed; 1𝜇 is the rate at which infected individuals become infectious. The nonlinear incidence rate is assumed to be of the form 𝛽𝑆𝐼/(1+𝛼𝐼). 𝐴,𝑑,𝛾,𝛿,𝜀, and 𝜇 are all normal numbers with 0<𝜇<1. This, together with 𝑁=𝑆+𝐸+𝐼, implies 𝑁=(𝑆+𝐸+𝐼)=𝐴𝑑𝑁𝛿𝐼.(1.2) Thus, substituting 𝑆=𝑁𝐸𝐼 and (1.2) into (1.1), we have 𝐸=𝜇𝛽𝐼𝐼1+𝛼𝐼(𝑁𝐸𝐼)(𝑑+𝜀)𝐸,=(1𝜇)𝛽𝑆𝐼𝑁1+𝛼𝐼(𝑁𝐸𝐼)+𝜀𝐸(𝑑+𝛾+𝛿)𝐼,=𝐴𝑑𝑁𝛿𝐼.(1.3) Form (1.2), in the absence of the disease, that is, 𝐼=0,𝑁𝐴/𝑑. Since the spread of the disease in the population will lead to the decrease of 𝑁, it follows that 𝑁[0,𝐴/𝑑]. Note that 𝐷 is a positively invariant region for the original model: 𝐴𝐷=(𝐸,𝐼,𝑁)𝐸0,𝐼0,𝑁0,𝐸+𝐼𝑁𝑑,(1.4) and model (1.3) is obviously well-pased in 𝐷.

2. Existence of Equilibria

Now, we study equilibria of model (1.3), (note that 𝑚=𝑑+𝜀,𝑛=𝑑+𝛾+𝛿). Steady states of model (1.3) satisfy the following equations: 𝜇𝛽𝐼1+𝛼𝐼(𝑁𝐸𝐼)(𝑑+𝜀)𝐸=0,(1𝜇)𝛽𝑆𝐼1+𝛼𝐼(𝑁𝐸𝐼)+𝜀𝐸(𝑑+𝛾+𝛿)𝐼=0,𝐴𝑑𝑁𝛿𝐼=0.(2.1)

If 𝐼=0, then 𝐸=0 and 𝑁=𝐴/𝑑, so (2.1) has the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑).

If 𝐼0, from the third equation of (2.1), we obtain 𝐴𝑁=𝑑𝛿𝑑𝐼.(2.2)

From the first and second equations of (2.1), we obtain 𝐸=𝜇𝑛𝑚(1𝜇)+𝜇𝜀𝐼.(2.3)

By substituting (2.2) and (2.3) into the first equation of (2.1), we obtain the following equation for 𝐼: 𝛽𝛿𝑑+1+𝜇𝑛+𝑚(1𝜇)+𝜇𝜀𝑚𝑛𝛼𝑚(1𝜇)+𝜇𝜀𝐼=𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝛽𝐴(𝑚(1𝜇)+𝜇𝜀)𝑚𝑛𝑑1.(2.4) Let 𝑅0=𝛽𝐴(𝑚(1𝜇)+𝜇𝜖)/𝑚𝑛𝑑.

It is easy to see that (2.4) has a positive root if and only if 𝑅0>1. So (2.1) has a unique endemic equilibrium 𝑃(𝐸,𝐼,𝑁) with 𝐼=(𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀))((𝛽𝐴(𝑚(1𝜇)+𝜇𝜀)/𝑚𝑛𝑑)1),𝐸𝛽((𝛿/𝑑)+1+𝜇𝑛/(𝑚(1𝜇)+𝜇𝜀))+𝑚𝑛𝛼/(𝑚(1𝜇)+𝜇𝜀)=𝜇𝑛𝐼𝑚(1𝜇)+𝜇𝜀,𝑁=𝐴𝑑𝛿𝑑𝐼.(2.5) Then, we have the following theorem.

Theorem 2.1. If 𝑅01, model (1.3) only has the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑); if 𝑅0>1, model (1.3) has a unique endemic equilibrium 𝑃(𝐸,𝐼,𝑁) except the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑).

3. Stability of Equilibria

Theorem 3.1. If 𝑅0<1, then the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑) is locally asymptotically stable; if 𝑅0=1, 𝑃0(0,0,𝐴/𝑑) is stable; if 𝑅0>1, 𝑃0(0,0,𝐴/𝑑) is unstable.

Proof. The linearization of model (1.3) about the equilibrium 𝑃0(0,0,𝐴/𝑑) gives |||||||||||𝜆+𝑚𝜇𝛽𝐴𝑑0𝜀𝜆(1𝜇)𝛽𝐴𝑑|||||||||||+𝑛00𝛿𝜆+𝑑=0.(3.1) Thus, we have ||𝑃𝜆𝐸𝐽0||𝜆=(𝜆+𝑑)2+𝑚+𝑛(1𝜇)𝛽𝐴𝑑𝜆+𝑚𝑛(1𝜇)𝑚𝛽𝐴𝑑𝜀𝜇𝛽𝐴𝑑=0.(3.2) Assume that 𝜆1, 𝜆2, and 𝜆3 are the roots of the above equation. Then, we know that 𝜆1=𝑑<0, and 𝜆2,𝜆3 are the roots of the following equation: 𝜆2+𝑚+𝑛(1𝜇)𝛽𝐴𝑑𝜆+𝑚𝑛(1𝜇)𝑚𝛽𝐴𝑑𝜀𝜇𝛽𝐴𝑑=0.(3.3) According to Viete theorem, we have 𝜆1+𝜆2=(1𝜇)𝑚𝛽𝐴𝑑𝑚𝑛,𝜆1𝜆2=𝑚𝑛1𝑅0.(3.4) If 𝑅0<1, we have 𝜆1+𝜆2=(1𝜇)𝑚𝛽𝐴𝑑𝑚𝑛<𝑚𝑛(1𝜇)𝑚(1𝜇)+𝜇𝜀𝑚𝑛<0,𝜆1𝜆2=𝑚𝑛1𝑅0>0.(3.5) It is easy to see that all the roots of (3.3) have negative real parts if and only if 𝑅0<1. If 𝑅0=1, it is obvious that one of eigenvalue of (3.3) is 0. If 𝑅0>1, one of the roots of (3.3) has positive real part. This completes the proof.

Theorem 3.2. If 𝑅0<1, then the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑) is globally asymptotically stable.

Proof. Consider a Liapunov function as 𝑉=𝜀𝐸+𝑚𝐼, then we have 𝑉=(𝑚(1𝜇)+𝜇𝜀)𝛽𝐼(𝑁𝐸𝐼)𝐴1+𝛼𝐼𝑚𝑛𝐼(𝑚(1𝜇)+𝜇𝜀)𝛽𝑑𝑅𝐼𝑚𝑛𝐼=𝑚𝑛𝐼0.1(3.6) If 𝑅0<1, 𝑉0, then 𝑉=0 if and only if 𝐸=𝐼=0.
Hence, according to Theorem 3.1, if 𝑅0<1, the disease-free equilibrium 𝑃0(0,0,𝐴/𝑑) is globally asymptotically stable. This completes the proof.

Now, we study the local stability of the endemic equilibrium 𝑃(𝐸,𝐼,𝑁).

Substituting 𝐸=(𝜇𝑛/(𝑚(1𝜇)+𝜇𝜀))𝐼 into the first equation of (2.1), we have 𝛽1+𝛼𝐼𝑁𝐸𝐼=𝑚𝑛𝑚.(1𝜇)+𝜇𝜀(3.7)

Then, the Jacobi matrix of (1.3) about 𝑃(𝐸,𝐼,𝑁) is 𝐽𝑃=|||||||||||||𝜇𝛽𝐼1+𝛼𝐼𝜇𝑚1+𝛼𝐼𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝛽𝐼𝜇𝛽𝐼1+𝛼𝐼(1𝜇)𝛽𝐼1+𝛼𝐼+𝜀1𝜇1+𝛼𝐼𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝛽𝐼𝑛(1𝜇)𝛽𝐼1+𝛼𝐼|||||||||||||0𝛿𝑑.(3.8)

Denote 𝑗11=𝜇𝛽𝐼1+𝛼𝐼𝑚,𝑗12=𝜇1+𝛼𝐼𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝛽𝐼,𝑗13=𝜇𝛽𝐼1+𝛼𝐼,𝑗21=(1𝜇)𝛽𝐼1+𝛼𝐼+𝜀,𝑗22=1𝜇1+𝛼𝐼𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝛽𝐼𝑛,𝑗23=(1𝜇)𝛽𝐼1+𝛼𝐼,𝑗31=0,𝑗32=𝛿,𝑗33=𝑑.(3.9)

According to [15], if (1𝜇)𝛽>𝜀𝛼 and 𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀)𝛽𝐼, then 𝑚𝑛/𝛽(𝑚(1𝜇)+𝜇𝜀)𝐼𝜀/((1𝜇)𝛽𝜀𝛼). Note that 𝑗𝑖𝑖<0(𝑖=1,2,3), 𝑗12𝑗210, 𝑗13𝑗31=0, 𝑗23𝑗32<0, and then 𝐽(𝑃) is stable.

If (1𝜇)𝛽>𝜀𝛼 and 𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀)>𝛽𝐼, then 𝜀/((1𝜇)𝛽𝜀𝛼)𝐼𝑚𝑛/𝛽(𝑚(1𝜇)+𝜇𝜀). Meanwhile, ((1𝜇)/(1+𝛼𝐼))[𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀)𝛽𝐼]𝑛<0, that is, (𝑛𝛼+(1𝜇)𝛽)𝐼>0>(1𝜇)𝑚𝑛𝑚(1𝜇)+𝜇𝜀𝑛=𝑛𝜇𝜀,𝑚(1𝜇)+𝜇𝜀(3.10) hence, 𝑗𝑖𝑖<0(𝑖=1,2,3),𝑗12𝑗210,𝑗13𝑗31=0,𝑗23𝑗32<0, which proves that 𝐽(𝑃) is stable.

If (1𝜇)𝛽𝜀𝛼 and 𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀)>𝛽𝐼, to obtain 𝑗𝑖𝑖<0(𝑖=1,2,3), 𝑗12𝑗210, 𝑗13𝑗31=0, 𝑗23𝑗32<0, there must be 𝜀((1𝜇)𝛽𝜀𝛼)𝐼, but it is impossible.

From the above discussion, we get the following conclusion.

Theorem 3.3. If 𝑅0>1, system (1.3) has a unique endemic equilibrium 𝑃(𝐸,𝐼,𝑁), which is locally asymptotically stable.

From the third equation of (1.3), we have 𝑁=𝐴𝑑𝑁𝛿𝐼𝐴𝑑𝑁. Note that 𝐼0 as 𝑡, then 𝑁(𝑡)𝐴/𝑑 as 𝑡. The limit system of (1.3) is 𝐸=𝜇𝛽𝐼𝐴1+𝛼𝐼𝑑𝐼𝐸𝐼𝑚𝐸,=(1𝜇)𝛽𝐼𝐴1+𝛼𝐼𝑑𝐸𝐼+𝜀𝐸𝑛𝐼.(3.11) It is easy to see that system (3.11) has a disease-free equilibrium 𝑃0(0,0). If 𝑅0>1, system (3.11) has a unique endemic equilibrium 𝑃(𝐸,𝐼) with 𝐼=(𝑚𝑛/(𝑚(1𝜇)+𝜇𝜀))(𝛽𝐴(𝑚(1𝜇)+𝜇𝜀)/𝑚𝑛𝑑1),𝐸𝛽(1+𝜇𝑛/(𝑚(1𝜇)+𝜇𝜀))+𝑚𝑛𝛼/(𝑚(1𝜇)+𝜇𝜀)=𝜇𝑛𝑚𝐼(1𝜇)+𝜇𝜀.(3.12) Consider the Dulac function 𝐵=1/𝐼. Note that 𝑃=𝜇𝛽𝐼𝐴1+𝛼𝐼𝑑𝐸𝐼𝑚𝐸,𝑄=(1𝜇)𝛽𝐼𝐴1+𝛼𝐼𝑑𝐸𝐼+𝜀𝐸𝑛𝐼,(3.13) then 𝜕(𝐵𝑃)/𝜕(𝐸)+𝜕(𝐵𝑄)/𝜕(𝐼)=𝜇(𝛽/(1+𝛼𝐼))𝑚/𝐼((1𝜇)𝛽/(1+𝛼𝐼)2)((1+𝛼𝐼)𝛼(𝐴/𝑑𝐸𝐼))(𝜀/𝐼2)𝐸<0, there is no limit cycle in the first quadrant of the 𝐼-𝐸 plane. System (3.11) has a unique endemic equilibrium 𝑃(𝐸,𝐼), then we prove that 𝑃(𝐸,𝐼) 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 𝑃(𝐸,𝐼,𝑁) of system (3.11) is globally asymptotically stable.

When 𝑡, 𝑃(𝐸,𝐼,𝑁)𝑃(𝐸,𝐼,𝐴/𝑑). It is easy to see that the stability of 𝑃(𝐸,𝐼,𝐴/𝑑) is equivalent to that of 𝑃(𝐸,𝐼). Since 𝑃(𝐸,𝐼) is globally asymptotically stable, the unique endemic equilibrium 𝑃(𝐸,𝐼,𝑁) is also globally asymptotically stable.

4. Numerical Simulation

Figures 15 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.

318150.fig.001
Figure 1: Dynamical behavior of system (1.3) with 𝜇=0.9.
318150.fig.002
Figure 2: Dynamical behavior of system (1.3) with 𝜇=0.001.
318150.fig.003
Figure 3: Dynamical behavior of system (1.3) with 𝜇=0.99.
318150.fig.004
Figure 4: Dynamical behavior of system (1.3) with 𝜇=0.5.
318150.fig.005
Figure 5: Dynamical behavior of system (1.3) with 𝜇=0.01.

With 𝛽=0.02, 𝑑=0.01, 𝐴=0.5, 𝑚=0.4, 𝑛=0.5, 𝜀=0.1, 𝛼=0.4, 𝛿=0.01, 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 𝜇=0.9 (𝑅0=0.65), Figure 2 describes system (1.3) with 𝜇=0.001 (𝑅0=1.99).

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 𝛽=0.5, 𝑑=0.01, 𝐴=5, 𝑚=0.3, 𝑛=0.5, 𝜀=0.1, 𝛼=0.4, 𝛿=0.01 by MATLAB. With 𝜇=0.99 (𝑅0=170), 𝜇=0.5 (𝑅0=333.33), and 𝜇=0.01 (𝑅0=496.67), Figures 35 describe the endemic equilibriums of system (1.3).

From Figures 1 and 2, it is easy to see that the value of 𝜇 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 35, with a unique endemic equilibrium of system (1.3), the value of 𝜇 will affect the numbers of exposed individuals who are not yet infectious, infectious individuals, and the total population.

With 𝛽=0.5, 𝑑=0.01, 𝑚=0.3, 𝑛=0.5, 𝜇=0.5, 𝛼=0.4, 𝜀=0.1, Figure 6 shows that system (3.11) has no limit cycle.

318150.fig.006
Figure 6: Dynamical behavior of system (3.11).

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 𝜇.

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 𝜇. By discussing the basic reproduction number 𝑅0=𝛽𝐴(𝑚(1𝜇)+𝜇𝜀)/𝑚𝑛𝑑, we show that the value of 𝜇 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.

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