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

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.


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 by N A − μN, where A is thought to be the input rate of population.Here, we consider A as the sum of two parts, A 1 and A 2 , where A 1 is the birth rate of the population and A 2 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.

Model
In this paper, we consider an SEIR epidemic model with immigration:

2.2
It follows that lim sup t → ∞ S E I R ≤ B 1 B 2 , then system 2.1 is bounded.Since the variable R 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: and removing the bar in S, E, I, and t, then we obtain the simplified system where From the first equation in system 2.4 , we have is positively invariant to system 2.4 .Thus, we only consider the dynamical behavior of system 2.4 on the set Ω.

The Existence and Local Stability of Equilibria
It is obvious that system 2.4 always has the disease-free equilibrium E 0 A 1 A 2 , 0, 0 .Its endemic equilibrium E * S * , E * , I * is determined by the following equations: According to the monotonicity of functions at the two sides of 3.2 , we know that 3.2 has a unique positive root if Therefore, with respect to the existence of equilibria of system 2.4 , we have the following theorem.With respect to the local stability of equilibria E 0 and E * of system 2.4 , we have the following theorem.

has only the diseasefree equilibrium E
Theorem 3.2.The disease-free equilibrium E 0 is locally asymptotically stable as R 0 < 1 and unstable as R 0 > 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 E 0 , it is easy to know that the disease-free equilibrium E 0 is locally asymptotically stable as R 0 < 1 and unstable as R 0 > 1.
ii For the Jacobian matrix of system 2.4 at the endemic equilibrium E * , the characteristic equation is given by λ 3 a 1 λ 2 a 2 λ a 3 0, where Notice that 3.2 can be rewritten as

3.5
On the other hand, 3.2 can become where Therefore, it follows from Hurwitz criterion that the endemic equilibrium E * is locally asymptotically stable.

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.
then the derivative of V 1 with respect to t along the solution of 2.4 on the set Ω is given by It implies that the disease will be extinct ultimately when R 0 < 1.
In order to discuss the persistence of the disease, we first introduce some definitions and lemmas.
Assume that X 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 int F. Let π be a semidynamical system defined on F.
We say that π is persistent if, for all u ∈ int F, lim inf t → ∞ d π u, t , ∂F > 0 and that π is uniformly persistent if there is ξ > 0 such that, for all u ∈ int F, lim inf t → ∞ 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 inf t → ∞ 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.Let Σ 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 2 for all u ∈ U \ Σ there is a T u > 0 such that P π u, T u > P u .
For any u 0 S 0 , E 0 , I 0 ∈ Ω, there is a unique solution π u 0 , t S, E, I t; u 0 of system 2.4 , which is defined in R and satisfies π u 0 , 0 S 0 , E 0 , I 0 .Since Ω is a positively invariant set of system 2.4 , then π u 0 , t ∈ Ω for t ∈ R and is a semidynamical system in Ω.
In the following, we will prove that, when R 0 > 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.
Let P : Ω → R be defined by P S, E, I I, and let U { S, E, I ∈ Ω : P S, E, I < η 1 }, where η 1 > 0 is small enough so that then there exists a positive number η 1 such small that inequality 4.2 holds.
Assume that there is u ∈ U u S, E, I such that for each t > 0 we have P π u, t < P u < η 1 , which implies that I t; u < η 1 for t > 0. From the first equation in system 2.4 we have So there is a sufficiently large number T > 0 such that S t; u > μ A 1 A 2 / 1 η 1 / μ 2η 1 for t ≥ T. Define the auxiliary function Direct calculation gives the derivative of V 2 t along with π u, t as follows: Then, for t ≥ T , we have where On the other hand, the boundedness of the solution of 2.1 implies that of V 2 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 T u such that P π u, T u > P u .Therefore, it follows from Lemma 4.1 that Σ { S, E, I ∈ Σ : I 0} is a uniform repeller when R 0 > 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 R 0 < 1 and persistent when R 0 > 1.

Conclusion and Discussion
In Sections 3 and 4, for system 2.4 we investigated the qualitative behavior and obtained the threshold R 0 determining the persistence of infection.Corresponding to the original model 2.1 , the basic reproduction number is R 0 β 1 β 2 ε/ μ 1 ε μ 1 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 R 0 < 1; it implies that the disease is extinct ultimately; when R 0 > 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 R 0 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.

3 . 1 From
the last two equations in 3.1 , we have S b 1 b 2 and E b 2 I for I / 0. Substituting S b 1 b 2 into the first equation in 3.1 gives μ A 1 A 2 1 I b 1 b 2 μ I , 3.2 then I * is the positive root of 3.2 .
4 also has a unique endemic equilibrium E * S * , E * , I * , where S * b 1 b 2 , E * b 2 I * , and I * is determined by 3.2 .
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 μ 1 B 2 is the immigration rate in the absence of disease and m reflects the effect of infection on immigration from other regions; μ 1 is the percapita natural death rate; β is the transmission coefficient of infection; ε 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 2.1Here, 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.μ 1 B 1 is the input rate; μ 1 B 2 / 1