# Analysis of an SEIS Epidemic Model with a Changing Delitescence

Authors: Jinghai Wang
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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 [1–14]). 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 [1–14] and the references therein). A variety of nonlinear incidence rates have been used in epidemiological models (see [6–14]). 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 rateβSI/(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,N→A/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)∣E≥0,I≥0,N≥0,E+I≤N≤Ad},
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.IfI=0, then E=0 and N=A/d, so (2.1) has the disease-free equilibrium P0(0,0,A/d).IfI≠0, 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.
IfR0≤1, 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.
IfR0<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
(3.1)|λ+m-μβAd0ελ-(1-μ)βAd+n00δλ+d|=0.
Thus, we have
(3.2)|λE-J(P0)|=(λ+d)[λ2+(m+n-(1-μ)βAd)λ+mn-(1-μ)mβAd-εμβAd]=0.
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:
(3.3)λ2+(m+n-(1-μ)βAd)λ+mn-(1-μ)mβAd-εμβAd=0.
According to Viete theorem, we have
(3.4)λ1+λ2=(1-μ)mβAd-m-n,λ1λ2=mn(1-R0).
If R0<1, we have
(3.5)λ1+λ2=(1-μ)mβAd-m-n<mn(1-μ)m(1-μ)+με-m-n<0,λ1λ2=mn(1-R0)>0.
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.
IfR0<1, then the disease-free equilibrium P0(0,0,A/d) is globally asymptotically stable.Proof.
Consider a Liapunov function asV=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, V′≤0,  then V′=0 if and only if E=I=0.
Hence, according to Theorem3.1, 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 equilibriumP*(E*,I*,N*).SubstitutingE*=(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 [15], 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),  j12j21≤0,  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),j12j21≤0,j13j31=0,j23j32<0, which proves that J(P*) is stable.If(1-μ)β≤εα and mn/(m(1-μ)+με)>βI*, to obtain jii<0(i=1,2,3), j12j21≤0, 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.
IfR0>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-δI≤A-dN. Note that I→0 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 equilibriumP*(E*,I*,N*) of system (3.11) is globally asymptotically stable.Whent→∞, 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

Figures1–5 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.Figure 1
Dynamical behavior of system (1.3) with μ=0.9.Figure 2
Dynamical behavior of system (1.3) with μ=0.001.Figure 3
Dynamical behavior of system (1.3) with μ=0.99.Figure 4
Dynamical behavior of system (1.3) with μ=0.5.Figure 5
Dynamical behavior of system (1.3) with μ=0.01.Withβ=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), Figures3, 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 3–5 describe the endemic equilibriums of system (1.3).From Figures1 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 Figures3–5, 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.Withβ=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.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 numberμ.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 ofμ. 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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