eu DYNAMIC PROPERTIES FOR AN EPIDEMIC MODEL WITH PARTIAL IMMUNITY

In this paper, an epidemic model with waning preventive vaccine is formulated. The analysis of the model is presented in terms of the basic reproduction number. If the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, the system is permanent and there is a unique endemic equilibrium. In this case, sufficient conditions are obtained for the global attractiveness of the endemic equilibrium. Numerical simulations are carried out to illustrate the main results. AMS Subject Classification: 34K20, 34K60, 92D30


Introduction
Over the past few decades, many epidemic models have been proposed and analyzed to investigate the transmission dynamics of infectious diseases (see, for Received: January 31, 2015 c 2015 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author instance, [1], [2], [3] and the the references therein).In these studies, it was assumed that the infection is acquired following effective contact with infected population.However, some patients with infectious diseases can discharge infectious pathogens at the end of the latent period, such as tuberculosis, measles and chicken pox.Hence, the infection can also be acquired following effective contact with the latened population(see, for instance, [4] and the references therein).
Vaccination has been useful in controlling diseases spread.The mathematical models with vaccination have been studied by many authors(see, for instance, [5], [6], [7] and the the references therein).These articles all assumed that the vaccinees obtained the permanent immunity.However, some clinical studies have shown that the permanent immunity induced by the preventive vaccines may wane over time(see, for instance, [8], [9] and the the references therein).
Motivated by the work of Lianbing Li et al. [4] and Jianwen Jia et al. [9], we study the following differential equations: where S, V, E, I and R denote the susceptible, vaccinated, exposed, infectious and recovered individuals, respectively.Π is the constant recruitment rate of individuals, p is the fraction of recruited individuals who are vaccinated, β is the effective contact rate, α is the rate at which exposed individuals become infectious, δ is the recovery rate, µ is the natural mortality rate, d is the diseaseinduced mortality rate, ω is the rate at which vaccine wanes, (that is 1/ω is the duration of the loss of immunity acquired by preventive vaccine or by infection), 0 < η ≤ 1 is the constant describing the decrease in the relative infectiousness of population in the exposed individuals E in comparison to those in the infectious individuals I, and 0 ≤ τ ≤ 1 is the vaccine efficacy (τ = 1 represents a vaccine that offers 100% protection against infection, τ = 0 represents a vaccine that offers no protection at all).
(1.2) Notice that the recovered population R(t) does not feature in the first four equations of the model, we will only discuss Equations (1.1a) − (1.1d) in the following.The dynamic behaviors of R(t) can be obtained from Equation (1.1e).
The paper is organized as follows.In the next section, some basic properties are presented.In Section 3, the local stability and the global asymptotic stability of a disease-free equilibrium of the model (1.1a) − (1.1d) are discussed.The permanence of the model (1.1a) − (1.1d) is given by means of the persistence theory on infinite dimensional systems in Section 4. In Section 5, sufficient conditions are received for the global attractiveness of the endemic equilibrium by using the theory of compound matrices.Numerical simulations are carried out in Section 6 to illustrate the main theoretical results.A brief conclusion is given in Section 7.

Basic Properties
In this section, we study the basic properties of the model (1.1a) − (1.1d).Proof.Suppose (S(t), V (t), E(t), I(t)) be a positive solution of the model (1.1a) − (1.1d) with initial conditions in R + 4 .Define Calculating the derivative of L(t) along the solutions of the model (1.1a) − (1.1d), it follows that Hence, for ε > 0 sufficiently small, there exists a Hence, for ε > 0 sufficiently small, there is a We therefore derive from Equation (1.1a) that, for t > T 2 , Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that Hence, for ε > 0 sufficiently small, there exists a T 3 > T 2 such that if t > T 3 , That is, the arbitrary positive solution (S(t), V (t), E(t), I(t)) of the model (1.1a) − (1.1d) is ultimately bounded.This completes the proof. Denote Theorem 2.1 implies that the set D is a positively invariant and the attracting region for the disease transmission model given by the model (1.1a) − (1.1d) with initial conditions in R 4 + .
If R 0 < 1, Hence, the characteristic roots of Equation (3.2) have negative real parts.Therefore, P 0 is locally asymptotically stable.Theorem 3.1 implies that the disease can be eliminated when R 0 < 1 if the initial sizes of the sub-populations of the model are in the basin of attraction of P 0 .To ensure the disease eradication is independent of the initial sizes of the sub-populations of the model, we study the global stability of the disease-free equilibrium P 0 in the following.
Proof.Let (S(t), V (t), E(t), I(t)) be any positive solution of the model (1.1a) − (1.1d) with initial conditions in R + 4 .Since R 0 < 1, we can choose ε > 0 small enough such that By (2.1) and (2.2), for ε > 0 satisfying (3.3), there exists a T 4 > 0 such that if t > T 4 , From Equation(1.1c), it is easy to know that if t > T 4 , Consider the following auxiliary system It is easy to prove that the equilibrium (0, 0) of system (3.4) is globally asymptotically stable for (3.3).By comparison, it follows that Hence, for arbitrary ε > 0, there exists a Consider the following auxiliary system It is easy to prove that the equilibrium (( Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that By (3.5) and (3.8), P 0 is globally asymptotically stable when R 0 < 1.The proof is complete.

Permanence
In this section, we study the permanence of the model (1.1a) − (1.1d) by the persistence theory on infinite dimensional systems developed by Hale and Waltman [11].
Let X be a complete metric space.Suppose that X The following is a small variant of Theorem 4.1 in [11].
Lemma 4.1.(see [11]) Suppose that T (t) satisfies (4.1) and we have the following: is isolated and has an acyclic covering M , where Proof.Let (S(t), V (t), E(t), I(t)) be any solution of the model (1.1a) − (1.1d) with initial conditions in R 4 + .By Equation (1.1b), we obtain that Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that By (4.2), for ε > 0 sufficiently small, there is a Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that If there exists m 3 > 0 such that lim inf t→+∞ E(t) ≥ m 3 , then for ε > 0 sufficiently small, there is a Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that lim inf t→+∞ Hence, it suffices to prove that lim inf t→+∞ E(t) ≥ m 3 for the permanence of the model (1.1a) − (1.1d).Let It is easy to show that X 0 and ∂X 0 are positively invariant and the condition (ii) in Lemma 4.
and It is easy to prove that the equilibrium P 1 is globally asymptotically stable.
Hence, Ω = {P 0 }.That is, {P 0 } is a covering of Ω.Therefore, Ω is isolated and has an acyclic covering satisfying the conditions (iii) in Lemma 4.1.
We now show that W s (P 0 ) ∩ X 0 = ∅.Assume W s (P 0 ) ∩ X 0 = ∅.Then there exists a solution (S(t), Since R 0 > 1, we can choose ε > 0 small enough such that and for ε > 0 satisfying (4.6), there exists a T 8 > T 7 such that if t > T 8 , From Equation (1.1c), it is easy to know that if t > T 8 , Consider the following auxiliary system (4.7) The Jacobian matrix of system (4.7) is Since J ε admits positive off-diagonal elements, the Perron-Frobenius theorem implies that there is a positive eigenvector p = (p 1 , p 2 ) for the maximum root γ of J ε .
The characteristic equation of system (4.7) is where Since (4.6) holds, it is shown that the maximum root γ of J ε is positive by a simple computation.Let z(t) = (z 1 (t), z 2 (t)) be a solution of system (4.7) through (lp 1 , lp 2 ) at t = t 0 , where l > 0 satisfies lp 1 < E(t 0 ), lp 2 < I(t 0 ).
We know that Clearly, z i (t) is strictly increasing and z i (t) → +∞ as t → +∞, i = 1, 2.
Consequently, ) is permanent for R 0 > 1.This proof is complete.

Existence and Stability of the Endemic Equilibrium
In this section, we are concerned with the existence of the endemic equilibrium and prove that the endemic equilibrium is globally attractive by the theory of the compound matrices.
In conclusion, we have the following results.
In the following, the method developed in [13], [14] is used to discuss the global attraction of the endemic equilibrium P * (S * ,V * ,E * ,I * ).First, we introduce this method briefly.
Let G ⊂ R n be an open set.Consider the differential equation: where, the function f : where the matrix Q f is the derivative of Q in the direction of the vector field f in system (5.6), and J [2] is the second additive compound matrix of the Jacobian matrix of system (5.6).
Lemma 5.1. [13]If G 1 is a compact absorbing subset in the interior of G, and there exist a γ > 0 and a Lozinski ȋ measure μ(A) ≤ −γ for all x ∈ G 1 , then every omega limit point of system (5.6) in the interior of G is an equilibrium in G 1 .
According to [15], the Lozinski ȋ measure in Lemma 5.1 can be evaluated as: where D + is the right-hand derivative.
Case 1: U 1 (z) > U 2 (z), sgn(z 1 ) = sgn(z 2 ) = sgn(z 3 ) and Taking the right-hand derivative of z , we get Thus, Case 2: Taking the right-hand derivative of z , we get Thus, Case 3: Taking the right-hand derivative of z , we get Taking the right-hand derivative of z , we get Thus, (5.12) Taking the right-hand derivative of z , we get Case 6: Taking the right-hand derivative of z , we get Thus, Thus, D + z ≤ −µ z . (5.17 Thus, Thus, Thus,  Thus, μ(A) < 0 by (5.8).Since R 0 > 1, the model (1.1a) − (1.1d) is permanent by Theorem 4.1.Hence, there exists a compact absorbing set in D. By Lemma 5.1, the endemic equilibrium is globally attractive in the interior of D .The proof is complete.

Numerical Simulations
In this section, we show the feasibility of the conditions of Theorem 5.

Conclusion
In this paper, the dynamics of a SEIR epidemic model with vaccination is investigated.It is shown that if the basic reproductive number R 0 < 1, the disease-free equilibrium is globally asymptotically stable while the endemic equilibrium is not feasible.In this case, the disease dies out.If R 0 > 1, the system is permanent, and the endemic equilibrium is globally attractive provided u > max{α + ω, 2ηβΠ((1 − p)µ + ω)/(µ(µ + ω)) + ω}.To control the disease, a strategy should reduce the basic reproduction number to below unity.Clearly, if τ , p or 1/ω increases, the basic reproduction number decreases.Hence, it is useful to control the disease by increasing the rate τ , p or 1/ω.

Theorem 2 . 1 .
The arbitrary positive solution of the model (1.1a)−(1.1d)with initial conditions in R + 4 is ultimately bounded.