AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 318150 10.1155/2012/318150 318150 Research Article Analysis of an SEIS Epidemic Model with a Changing Delitescence Wang Jinghai Peterson Allan Institute of Mathematics and Computer Science Fuzhou University Fuzhou, FuJian 350002 China fzu.edu.cn 2012 30 08 2012 2012 17 05 2012 26 07 2012 2012 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.

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 ). Bilinear and standard incidence rates have been frequently used in classical epidemiological models (see ). 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  and the references therein). A variety of nonlinear incidence rates have been used in epidemiological models (see ). 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: (1.1)S=A-dS-βSI1+αI+γI,E=μβSI1+αI-(d+ε)E,I=(1-μ)βSI1+αI+εE-(d+γ+δ)I, 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; 1-μ 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 (1.2)N=(S+E+I)=A-dN-δI. Thus, substituting S=N-E-I and (1.2) into (1.1), we have (1.3)E=μβI1+αI(N-E-I)-(d+ε)E,I=(1-μ)βSI1+αI(N-E-I)+εE-(d+γ+δ)I,N=A-dN-δI. 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: (1.4)D={(E,I,N)E0,I0,N0,E+INAd}, 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: (2.1)μβI1+αI(N-E-I)-(d+ε)E=0,(1-μ)βSI1+αI(N-E-I)+εE-(d+γ+δ)I=0,A-dN-δI=0.

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 (2.2)N=Ad-δdI.

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

By substituting (2.2) and (2.3) into the first equation of (2.1), we obtain the following equation for I: (2.4)[β(δd+1+μnm(1-μ)+με)+mnαm(1-μ)+με]I=mnm(1-μ)+με[βA(m(1-μ)+με)mnd-1]. Let R0=βA(m(1-μ)+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 (2.5)I*=(mn/(m(1-μ)+με))((βA(m(1-μ)+με)/mnd)-1)β((δ/d)+1+μn/(m(1-μ)+με))+mnα/(m(1-μ)+με),E*=μnm(1-μ)+μεI*,N*=Ad-δdI*. 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.

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 (3.6)V=(m(1-μ)+με)βI(N-E-I)1+αI-mnI(m(1-μ)+με)βAdI-mnI=mnI(R0-1). 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(1-μ)+με))I* into the first equation of (2.1), we have (3.7)β1+αI*(N*-E*-I*)=mnm(1-μ)+με.

Then, the Jacobi matrix of (1.3) about P*(E*,I*,N*) is (3.8)J(P*)=|-μβI*1+αI*-mμ1+αI*[mnm(1-μ)+με-βI*]μβI*1+αI*-(1-μ)βI*1+αI*+ε1-μ1+αI*[mnm(1-μ)+με-βI*]-n(1-μ)βI*1+αI*0-δ-d|.

Denote (3.9)j11=-μβI*1+αI*-m,j12=μ1+αI*[mnm(1-μ)+με-βI*],j13=μβI*1+αI*,j21=-(1-μ)βI*1+αI*+ε,j22=1-μ1+αI*[mnm(1-μ)+με-βI*]-n,  j23=(1-μ)βI*1+αI*,j31=0,  j32=-δ,j33=-d.

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

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

If (1-μ)βεα and mn/(m(1-μ)+με)>βI*, to obtain jii<0  (i=1,2,3), j12j210, j13j31=0, j23j32<0, there must be ε((1-μ)β-εα)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=A-dN-δIA-dN. Note that I0 as t, then N(t)A/d as t. The limit system of (1.3) is (3.11)E=μβI1+αI(Ad-E-I)-mE,I=(1-μ)βI1+αI(Ad-E-I)+εE-nI. It is easy to see that system (3.11) has a disease-free equilibrium P¯0(0,0). If R0>1, system (3.11) has a unique endemic equilibrium P¯*(E¯*,I¯*) with (3.12)I*=(mn/(m(1-μ)+με))(βA(m(1-μ)+με)/mnd-1)β(1+μn/(m(1-μ)+με))+mnα/(m(1-μ)+με),E*=μnm(1-μ)+μεI*. Consider the Dulac function B=1/I. Note that (3.13)P=μβI1+αI(Ad-E-I)-mE,Q=(1-μ)βI1+αI(Ad-E-I)+εE-nI, then (BP)/(E)+(BQ)/(I)=-μ(β/(1+αI))-m/I-((1-μ)β/(1+αI)2)(-(1+αI)-α(A/d-E-I))-(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.

Dynamical behavior of system (1.3) with μ=0.9.

Dynamical behavior of system (1.3) with μ=0.001.

Dynamical behavior of system (1.3) with μ=0.99.

Dynamical behavior of system (1.3) with μ=0.5.

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.

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(1-μ)+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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