An SEIV Epidemic Model for Childhood Diseases with Partial Permanent Immunity

An SEIV epidemic model for childhood disease with partial permanent immunity is studied. The basic reproduction number R 0 has been worked out. The local and global asymptotical stability analysis of the equilibria are performed, respectively. Furthermore, if we take the treated rate τ as the bifurcation parameter, periodic orbits will bifurcate from endemic equilibrium when τ passes through a critical value. Finally, some numerical simulations are given to support our analytic results.


Introduction
It is primarily important for health administrators to protect children from disease that can be prevented by vaccination. Although preventive vaccines have reduced the incidence of infectious diseases among children, childhood disease is an important public health problem. We often use mathematical models to realize the transmission dynamics of childhood diseases and to estimate control programs [1][2][3][4]. Recently, many scholars study the SEIV epidemic models [5,6]. In those models, let ( ), ( ), and ( ), respectively, represent the number of susceptible individuals at time , infective individuals at time , and vaccinated individuals at time . At the earliest, most researches on these types of models assume that the disease incubation is negligible, so that each susceptible individual, once infected, instantaneously turns into infectious and later recovers obtaining a permanent immunity. Soon afterwards, the models become more general. Researchers assume that a susceptible individual first goes through a latent period after infection before becoming infectious (we called represents exposed individuals but not yet infectious).
In [7], the authors discussed the following model: where all parameters are positive. Parameter represents the number of additional populations of childhood; represents the rate at which vaccine wanes; represents the natural death rate; represents the rate at which susceptible individuals become infected by those who are infectious; represents the fraction of recruited individuals who are vaccinated; represents the rate at which infected individuals are treated; and represents the rate at which exposed individuals become infectious.
In model (1), is called incidence rate which plays an important role in the transmission dynamics. In addition, incidence rate can determine the tendency of epidemics. At the earliest, in the classical epidemic disease model, scholars made much focus on the bilinear incidence [8,9]. In 1945, Wilson and Worcester discussed the nonlinear incidence rate [10,11]. Later, the incidence function grows into more general nonlinear forms. In [12], the authors have considered a SEIV model with nonlinear incidence rate (1 + ). The paper discussed the basic reproduction of the system and bifurcation phenomenon. And this incidence function is more in line with actual situation. One of the strategies to control infectious diseases is vaccination in [13,14].

Computational and Mathematical Methods in Medicine
And under the above circumstance, in [15], the authors have studied the following model: In [15] they have supplied a framework of discussing the transmission dynamics of the epidemic model where the preventive vaccine may lose efficacy over time. And it has showed that if the vaccination coverage level is below the threshold, the disease will persist within the population. In addition, if the vaccination coverage level exceeds a certain threshold value, the disease can be eradicated from the population through constructing a proper Lyapunov function by using global stability analysis of the model.
In the process of treatment, some patients can not be cured; therefore we should consider the disease-caused death on the basic of the above models and make the parameter be the rate at which infectious individuals lose their life due to disease during the process of treatment. Moreover, about some diseases, some cured patients can not obtain a permanent immunity. Thus, this paper also considers the SEIV epidemic models for childhood disease with partial permanent immunity based on above models and denotes as the rate of transforming to . Namely, when = 0, all recoverers obtain permanent immunity. When = 1, all recoverers become susceptible individuals. When 0 < < 1, partial infective individuals become susceptible individuals and the number is . So model (2) is transformed to model (3). Model (3) is described as follows: Assume the initial values are satisfied with the following: System (3) which we present will be analyzed to decide the optimal vaccine coverage level needed to control the disease. The rest of this paper is organized as follows. In Section 2, we calculate the basic reproduction number 0 , which determines the spread of infection. In Section 3, the local stability of equilibria is analyzed. We discuss the bifurcation phenomenon and illustrate that when the treated rate crosses through a critical value, system (3) undergoes Hopf bifurcation at the positive equilibrium in Section 4. By constructing the Lyapunov function and a generalization of the Poincaré-Bendixson criterion, we discuss the global stability of disease-free equilibrium and endemic equilibrium, respectively, in Section 5. Some numerical examples are presented to illustrate theoretical analysis in Section 6. In Section 7 we discuss our findings.

The Basic Reproduction Number
In the following, we will calculate the basic reproduction number of system (3). The basic reproduction number, denoted by 0 , is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [16] is positively invariant for system (3). Next we will discuss the dynamic characteristic of system (3) on Ω. Set = ( , , , ) ; then system (3) can be rewritten as where It is easy to get ] . (11) where = ( + )( + )( + + ).

Local Stability of Equilibria
In the following, we will discuss the local stability of the equilibria 0 and * .

(25)
The characteristic equation is where 1 = ( + ) + (2 + + + ) = 3 + + + + , It is easy to get It is clear that 1 > 0. By the Hurwitz criterion, epidemic equilibrium * is locally asymptotically stable for 3 > 0 and 5 need the results in [17,18]. In order to introduce it, consider the following equation which has a parameter : Without loss of generality, for all values of the parameter , assume 0 is an equilibrium for system (29); that is, Lemma 5 (see [17]). Suppose the following. Define as the th component of , and And and totally decide the local dynamic of system (29) around = 0.
(iv) Consider < 0, > 0. If changes from negative to positive, = 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.
Remark 6. The requirement that is nonnegative is unnecessary by [17].
It seems that a transcritical bifurcation occurs at = 0: more specifically, the bifurcation at = 0 is forward when < 0 and > 0; the bifurcation at = 0 is backward when > 0 and > 0.
Next consider = 0 as the bifurcation parameter, so that 0 < 1 for < 0 and 0 > 1 for > 0 and so that 0 is a disease-free equilibrium for system (29) of all values of .
Take into account the following system: where is continuously differentiable at least twice in both and . The disease-free equilibrium is the line ( 0 ; ). And the disease-free equilibrium changes its local stability at the point ( 0 ; ) [16].
Next we will exhibit that there exist nontrivial equilibria near the bifurcation point ( 0 ; 0). Let = 1 , = 2 , = 3 , and = 4 ; then system (3) becomeṡ1 We will show that system (33)  Calculate the eigenvalues of the following matrix: we can obtain 1 = − , 2 = − − , 3 = −2 − − , and The matrix ( 0 , * ) has a simple eigenvalue of 0; and all others have negative real parts. Thus, we can make use of the center manifold theory. The disease-free equilibrium 0 is a nonhyperbolic equilibrium when = * (i.e., when 0 = 1). This completes the verification with respect to (A1) of Lemma 5.
Expanding (35), we can have Expanding ( and all the other second-order partial derivatives are equal to zero. So, we evaluate and as follows: From system (33), and the terms ( 2 / )( 0 , * ) and ( 2 / )( 0 , * ) which are nonzero, the following are deduced: Obviously is always positive. Therefore the sign of the coefficient determines the local dynamics around the disease-free equilibrium for = * by Lemma 5.  (12), we get that the condition > 0 is equivalent to the condition * 1 < 1 at the bifurcation, that is, when 0 = 1.  Furthermore, taking the treated rate as the bifurcation parameter, we can get the following. Theorem 9. Let 0 > 1. When passes through a critical value, system (3) undergoes Hopf bifurcation at the positive equilibrium * . Proof. If system (3) shows Hopf bifurcation, there must exist = * , which satisfies the following conditions: From (28) and (44), we can calculate the critical value * . For = * , we have From (26) and (46), we have which has three roots: For all , the roots are all in the following general forms: Next, we prove the transversality condition We substitute ( ) = 1 ( ) + 2 ( ) into (47) and calculate the derivative, getting where 1 = 6 1 ( ) 2 ( ) + 2 1 ( ) 2 ( ) , For we obtain Re ( ( )) Hence, the transversality condition is confirmed. This verifies the result.
Consider the system as follows: where and are locally Lipschitz in ∈ and continuous. And for all positive values its solutions exist. If ( , ) → ( ) when → ∞ locally uniformly for ∈ , then system (58) is defined as asymptotically autonomous with limit system (59).

Lemma 11. Set is a locally asymptotically stable equilibrium of (59) and is the -limit set of a forward bounded solution
( ) of (58). If includes a point 0 such that the solution of (59) with (0) = 0 converges to when → ∞, then = ; that is, ( ) → when → ∞.

Corollary 12. If solutions of system (58) are bounded and the equilibrium of the limit system (59) is globally asymptotically stable, then any solution ( ) of system (58) satisfies ( ) → when → ∞.
Next, we obtain sufficient conditions that endemic equilibrium * is globally asymptotically stable for 0 > 1 by the geometrical approach [9]. Firstly, we briefly introduce this geometrical approach.
Let a 1 function → ( ) ∈ be in an open set ∈ . Consider the differential equation Denote (0, 0 ) = 0 by ( , 0 ) which is the solution to (60). We establish the following two assumptions.
If the equilibrium is locally stable, it is globally stable in and all trajectories in converge to . For ≥ 2, we mean a condition satisfied by which rules out the existence of nonconstant periodic solutions of (60) by Bendixson's criterion. The classical Bendixson's condition div ( ) < 0 for = 2 is robust under 1 local perturbations of . About higher-dimensional systems, the 1 robust properties have been discussed.
If there exists a neighborhood of 0 and > 0 such that ∩ ( , ) is empty for all > , then a point 0 ∈ is called wandering for (59). For example, all limit points and equilibria are nonwandering. We will introduce the global stability principle in [19] which is suited for autonomous systems.

And suppose that (60) satisfies Bendixson's criterion that is robust under 1 local perturbations of at all nonequilibrium nonwandering points for (60). Then, is globally stable in provided it is stable.
To have the robustness required by Lemma 13, we show the following Bendixson criterion [19]. Let → ( ) be a matrix-valued function that is 1 for ∈ . Assume that −1 ( ) exists and is continuous for ∈ , which is the compact absorbing set. Define a quantity 2 as where By substituting the derivative in the direction of into each entry of , the matrix is obtained. ( ) is the Lozinski1 measure of in terms of a vector norm | ⋅ | in : If is simply connected, the condition 2 < 0 excludes the existence of any orbit that attracts a simple closed rectifiable curve that is invariant for (62), such as homoclinic orbits, heteroclinic cycles, and periodic orbits in [19]. And it is robust under 1 local perturbations of near any nonequilibrium point that is nonwandering. In particular, the following lemma is proved in [19].

Lemma 14.
Assume that is simply connected and that the hypotheses ( 1) and ( 2) hold. Then, if 2 < 0, the unique equilibrium of (62) is globally stable in .
Next, we will obtain the main result.
From Theorem 3(ii), we obtain that there exists the endemic equilibrium * and it is unique due to 0 > 1. We will analyse the stability of * by the method in [9]. Due to Lemma 14, the global stability of * requires the following sufficient conditions: (i) there must exist a compact absorbing Computational and Mathematical Methods in Medicine 9 set in the interior of Ω (i.e., condition (H1)); (ii) * in the interior of Ω is unique (i.e., condition (H2)); and (iii) the requirement 2 < 0.
As Ω is bounded, the uniform persistence implies that there exist a compact absorbing set in the interion of Ω for system (3) (see [20]). Therefore, (H1) is verified. Also, * is the only equilibrium in the interior of Ω, so that * is unique; that is, (H2) is verified, too.
Set ( , , ) be the vectors in 3 . We choose a standard in 3 as |( , , )| = max{| , + |} and set be the Lozinski1 measure in term of this standard. Applying the technique in [21], the following can be obtained: We can obtain Considering ≤ , ≤ / , where is the constant of uniform persistence; it is obvious that And if + < , or + < , holds, then > 0.

Discussion
In this paper, considering disease-caused death and partial permanent immunity, we modified the SEIV epidemic model  in [15]. Applying the method of [16], we calculated the basic reproduction number 0 and found that when 0 = 1 and * 1 < 1 system (3) shows backward bifurcation. If * 1 < 1, system (3) has a unique endemic equilibrium when 0 ≥ 1 and has two endemic equilibria when 0 < 1. If * 1 > 1, system (3) has a unique endemic equilibrium when 0 ≥ 1 and has no endemic equilibrium when 0 < 1. Also system (3) always has a disease-free equilibrium 0 . Local and global asymptotic stability of the disease-free equilibrium are determined by 0 < 1 and 0 (1 + / ) < 1, respectively. Also we have studied the local and global asymptotic stability of the endemic equilibrium. Moreover, taking the diseasecaused death rate as bifurcation parameter, we discussed the Hopf bifurcation of system (3). We found that when 0 > 1, there is always a critical value * , such that system (3) exhibits Hopf bifurcation at * when passes through * .
From the sense of epidemiology, when 0 (1 + / ) < 1, if * 1 > 1 holds or * 1 < 1, 0 < * 0 hold; system (3) has one disease-free equilibrium which is globally stable. Namely, the disease will be eradicated. And when 0 > 1 and inequality (77) or (78) holds, system (3) has a unique endemic equilibrium * which is global asymptotically stable. Under this circumstance, the infectious disease becomes endemic disease. If 0 > 1, system (3) has a unique endemic equilibrium * and we found that when the rate becomes sufficiently large the disease will break out periodically. And differentiating the bifurcation coefficient partially with respect to , we can get / = − ( + ) < 0, which means that vaccinating more susceptible populations decreases the likelihood of the occurrence of backward bifurcation [15].