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

Authors: Wenjuan Wang; Jingqi Xin; Fengqin Zhang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## 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,R0, is found, which determines that the disease is either extinct or persistent ultimately. The obtained results show that the disease becomes extinct as R0<1 and persists in the population as R0>1.


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## Body

## 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 byN'=A-μN, where A is thought to be the input rate of population. Here, we consider A as the sum of two parts, A1 and A2, where A1 is the birth rate of the population and A2 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:(2.1)S′=μ1B1+μ1B21+mI-μ1S-βSI,E′=βSI-(μ1+ɛ)E,I′=ɛE-(μ1+α+γ)I,R′=γI-μ1R.
Here, S=S(t), E=E(t), I=I(t), and R=R(t) represent the numbers of susceptible, exposed, infectious, and recovery individuals at time t, respectively. μ1B1 is the input rate; μ1B2/(1+mI) 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 μ1B2 is the immigration rate in the absence of disease and m reflects the effect of infection on immigration from other regions; μ1 is the percapita natural death rate; β is the transmission coefficient of infection; ɛ is the transfer rate from the exposed compartment to the infectious one; γ is the percapita recovery rate; α is the percapita disease-induced death rate.From model (2.1) we have(2.2)(S+E+I+R)′=μ1B1+μ1B21+mI-μ1(S+E+I+R)-αI≤μ1[(B1+B2)-(S+E+I+R)].
It follows that lim supt→∞(S+E+I+R)≤B1+B2, then system (2.1) is bounded.Since the variableR 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:(2.3)S=β(ɛm2)·S¯,E=β(ɛm2)·E¯,I=1m·I¯,t=mβ·t¯,
and removing the bar in S¯, E¯, I¯, and t¯, then we obtain the simplified system(2.4)S′=μA1+μA21+I-μS-SI,E′=SI-b1E,I′=E-b2I,
where μ=μ1m/β,A1=ɛm2B1/β,A2=ɛm2B2/β,b1=(μ1+ɛ)m/β, and b2=(μ1+α+γ)m/β.From the first equation in system (2.4), we have S′≤μ[(A1+A2)-S]; it implies that lim supt→∞S(t)≤A1+A2; therefore, the set Ω={(S,E,I)∈R+3:S≤A1+A2} is positively invariant to system (2.4). Thus, we only consider the dynamical behavior of system (2.4) on the set Ω.


## 3. The Existence and Local Stability of Equilibria

It is obvious that system (2.4) always has the disease-free equilibrium E0(A1+A2,0,0). Its endemic equilibrium E*(S*,E*,I*) is determined by the following equations:(3.1)μA1+μA21+I-μS-SI=0,SI-b1E=0,E-b2I=0.From the last two equations in (3.1), we have S=b1b2 and E=b2I for I≠0. Substituting S=b1b2 into the first equation in (3.1) gives(3.2)μ(A1+A21+I)=b1b2(μ+I),
then I* 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 (A1+A2)/(b1b2)>1 and no positive roots if (A1+A2)/(b1b2)≤1. Therefore, with respect to the existence of equilibria of system (2.4), we have the following theorem.Theorem 3.1.
Denote thatR0=(A1+A2)/(b1b2). When R0≤1, system (2.4) has only the disease-free equilibrium E0(A1+A2,0,0) on the set Ω; when R0>1, besides the disease-free equilibrium E0, system (2.4) also has a unique endemic equilibrium E*(S*,E*,I*), where S*=b1b2,E*=b2I*, and I* is determined by (3.2).With respect to the local stability of equilibriaE0 and E* of system (2.4), we have the following theorem.Theorem 3.2.
The disease-free equilibriumE0 is locally asymptotically stable as R0<1 and unstable as R0>1. The endemic equilibrium E* is locally asymptotically stable as it exists.Proof.
(i) From the Jacobian matrix of system (2.4) at the disease-free equilibrium E0, it is easy to know that the disease-free equilibrium E0 is locally asymptotically stable as R0<1 and unstable as R0>1.
(ii) For the Jacobian matrix of system (2.4) at the endemic equilibrium E*, the characteristic equation is given by λ3+a1λ2+a2λ+a3=0, where a1=μ+I*+b1+b2, a2=(μ+I*)(b1+b2), and a3=I*[μA2/(1+I*)2+b1b2], then
(3.3)a1a2-a3=(μ+I*+b1+b2)(μ+I*)(b1+b2)-I*[μA2(1+I*)2+b1b2].
Notice that (3.2) can be rewritten as
(3.4)(μ+I)(1+I)=μ[(A1+A2)+A1I](b1b2),μA2(1+I)=b1b2I+μ(b1b2-A1).
Using (3.4) gives
(3.5)(1+I*)(a1a2-a3)=[μA1(b1+b2)b1b2-2b1b2]I*2+μ(A1+A2)(b1+b2)(μ+b1+b2)b1b2+{μ(b1+b2)b1b2[(A1+A2)+A1(μ+b1+b2)]-(1+μ)b1b2+μA1}I*=Δf1(I*).
On the other hand, (3.2) can become
(3.6)f2(I)=Δb1b2I2+[b1b2(1+μ)-μA1]I+μ[b1b2-(A1+A2)]=0,
then
(3.7)f1(I*)+2f2(I*)=c1I*2+c2I*+c3μ>0,
where c1=μA1(b1+b2)/(b1b2), c2=b1b2(1+μ)+μ[(b1+b2)(A1+A2+μA1)+A1(b12+b22+b1b2)]/(b1b2), and c3=(A1+A2)[μ(b1+b2)+b12+b22]/(b1b2)+2b1b2. It follows from f2(I*)=0 that f1(I*)>0, that is, a1a2-a3>0. Therefore, it follows from Hurwitz criterion that the endemic equilibrium E* 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.WhenR0<1, define function V1=ρE+I, where ρ∈(1/b1,b2/(A1+A2)), then the derivative of V1 with respect to t along the solution of (2.4) on the set Ω is given by(4.1)V1′=(1-ρb1)E+[ρ(A1+A2)-b2]I.
It follows from ρ∈(1/b1,b2/(A1+A2)) that 1-ρb1<0 and ρ(A1+A2)-b2<0, then there exists a positive number σ such that 1-ρb1<σρ(R0-1) and ρ(A1+A2)-b2<σ(R0-1). Therefore, from (4.1) we have V1′≤σ(R0-1)V1, then V1(t)≤V1(0)exp[σ(R0-1)t], where V1(0)=ρE(0)+I(0), therefore, limt→∞V1(t)=0 for R0<1; that is, limt→∞E(t)=limt→∞I(t)=0 as R0<1. It implies that the disease will be extinct ultimately when R0<1.In order to discuss the persistence of the disease, we first introduce some definitions and lemmas.Assume thatX is a locally compact metric space with metric d, and let F be a closed subset of X with the boundary ∂F and the interior intF. Let π be a semidynamical system defined on F.We say thatπ is persistent if, for all u∈intF, lim inft→+∞d(π(u,t),∂F)>0 and that π is uniformly persistent if there is ξ>0 such that, for all u∈intF, lim inft→+∞d(π(u,t),∂F)>ξ.In [3], Fonda gives a result about persistence in terms of repellers. A subset Σ of F is said to be a uniform repeller if there is an η>0 such that, for each u∈F∖Σ, lim inft→+∞d(π(u,t),Σ)>η. A semiflow on a closed subset F of a locally compact metric space is uniformly persistent if the boundary of F is repelling in a suitable strong sense [9]. The result by Fonda is as follows.Lemma 4.1.
LetΣ be a compact subset of X such that X∖Σ is positively invariant. A necessary and sufficient condition for Σ to be a uniform repeller is that there exists a neighborhood U of Σ and a continuous function P:X→R+ satisfying(1)
P(u)=0 if and only if u∈Σ,(2)
for allu∈U∖Σ there is a Tu>0 such that P(π(u,Tu))>P(u).For anyu0=(S0,E0,I0)∈Ω, there is a unique solution π(u0,t)=(S,E,I)(t;u0) of system (2.4), which is defined in R+ and satisfies π(u0,0)=(S0,E0,I0). Since Ω is a positively invariant set of system (2.4), then π(u0,t)∈Ω for t∈R+ and is a semidynamical system in Ω.In the following, we will prove that, whenR0>1, Σ={(S,E,I)∈Σ:I=0} is a uniform repeller, which implies that the semidynamic system π is uniformly persistent.Obviously,I(t)>0 for t>0 if I(0)>0, then Ω∖Σ is invariant to (2.4). Again the set Σ is a compact subset of Ω.LetP:Ω→R+ be defined by P(S,E,I)=I, and let U={(S,E,I)∈Ω:P(S,E,I)<η1}, where η1>0 is small enough so that(4.2)μμ+2βη1(A1+A21+η1)>b1b2.
Since R0>1 is equivalent to A1+A2>b1b2, then there exists a positive number η1 such small that inequality (4.2) holds.Assume that there isu¯∈U(u¯=(S¯,E¯,I¯)) such that for each t>0 we have P(π(u¯,t))<P(u¯)<η1, which implies that I(t;u¯)<η1 for t>0. From the first equation in system (2.4) we have (4.3)S'≥μ(A1+A21+η1)-(μ+η1)S,
then (4.4)lim inft→∞S(t;u¯)≥μμ+η1(A1+A21+η1).
So there is a sufficiently large number T>0 such that S(t;u¯)>μ[A1+A2/(1+η1)]/(μ+2η1) for t≥T.Define the auxiliary functionV2(t)=(1-η2)E(t)+b1I(t), where η2(0<η2<1) is a sufficiently small constant so that μ(1-η2)/(μ+2η1)·[A1+A2/(1+η1)]>b1b2. Direct calculation gives the derivative of V2(t) along with π(u¯,t) as follows:(4.5)V2′=b1η2E+[(1-η2)S-b1b2]I.
Then, for t≥T, we have (4.6)V2′≥b1η2E+[μ(1-η2)μ+2η1(A1+A21+η1)-b1b2]I>σV2,
where (4.7)σ=min{b1η21-η2,1b1[μ(1-η2)μ+2η1(A1+A21+η1)-b1b2]}>0,
therefore, limt→∞V2(t)=+∞.On the other hand, the boundedness of the solution of (2.1) implies that of V2(t) on the set Ω. It implies that the assumption above is not true. Therefore, the above proof shows that, for each u∈Ω∖Σ with u belonging to a suitably small neighborhood of Σ, there is some Tu such that P(π(u,Tu))>P(u). Therefore, it follows from Lemma 4.1 that Σ={(S,E,I)∈Σ:I=0} is a uniform repeller when R0>1; 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 R0<1 and persistent when R0>1.


## 5. Conclusion and Discussion

In Sections3 and 4, for system (2.4) we investigated the qualitative behavior and obtained the threshold R0 determining the persistence of infection. Corresponding to the original model (2.1), the basic reproduction number is R0=(β1+β2)ɛ/[(μ1+ɛ)(μ1+α+γ)]. According to the results in Sections 3 and 4, model (2.1) only has the disease-free equilibrium which is globally stable when R0<1; it implies that the disease is extinct ultimately; when R0>1, model (2.1) has a unique endemic equilibrium which is locally asymptotically stable and the disease persists in the population. Since the expression of R0 here is independent of the parameter m, 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.



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