On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

We study a vector-borne disease with age of vaccination. A nonlinear incidence rate includingmass action and saturating incidence as special cases is considered. The global dynamics of the equilibria are investigated and we show that if the basic reproduction number is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out, while if the basic reproduction number is larger than 1, then the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Using the basic reproduction number, we derive a vaccination coverage rate that is required for disease control and elimination.


Introduction
Many of infections that have the important impact on human health in terms of mortality or morbidity are vector-borne disease.Mosquitoes [1] are perhaps the best known disease vectors, with various species playing a role in the transmission of infections such as malaria, yellow fever, dengue fever, and West Nile virus.One of the effective methods in disease prevention is the vaccination [2][3][4][5].Several studies in the literature have been carried out to investigate the role of treatment and vaccination of the spread of diseases ( [6][7][8] and the references therein).An epidemic model with vaccination for measles is derived by Linda [9].The effect of vaccination on the spread of periodic diseases, using discrete-time model, was studied by Mickens [10].
The impact of vaccination in two SVIR models with permanent immunity is studied by Liu et al. [11].Xiao and Tang [12] have shown from an SIV model that complex dynamics are induced by imperfect vaccination.Gumel and Moghadas [13] investigated a disease transmission model by considering the impact of a protective vaccine and found the optimal vaccine coverage threshold required for disease control and elimination.The eradicating of an SEIRS epidemic model by using vaccine was studied by Gao et al. [14].Yang et al. [8] derived a threshold value for the vaccination coverage of an SIVS epidemic model.Many previous studies have shown that the reemergence of some diseases is caused by the waning of vaccine-induced immunity [15][16][17].A consequence of this is that it is important for health authorities to take into account waning of vaccine-induced immunity in the disease control and elimination campaign.
In this paper, we consider a vector-borne disease model such as malaria that incorporates the waning of vaccineinduced immunity.Additionally, we use incidences with a nonlinear response to the number of infectious individuals and infectious vectors.The incidences take the form () and (), respectively, for the human and vector populations.We assume that  and  satisfy the following assumptions: (H1) For  ∈ R + , () ≥ 0 with equality if and only if  = 0,   () ≥ 0, and   () ≤ 0.

International Journal of Differential Equations
Let  ℎ ,  ℎ , and  ℎ denote, respectively, the number of susceptible, infectious, and removed host individuals and  V ,  V the number of susceptible and infectious vectors.The susceptible individuals are vaccinated at the rate  ≥ 0. V(, ) denotes the population size of the vaccinated compartment at time  with the vaccine age .Let () be the rate at which the vaccine-induced immunity wanes.We assume that () and the following assumption: (H3)  : [0, ∞) → [0, ∞) is bounded, nondecreasing, and piecewise continuous with possibly many finite jumps.
We consider a relatively isolated community where there is no immigration or emigration.Additionally, we assume that all the newly recruited, including the newborns, are susceptibles.Let, at any time , Λ ℎ and Λ V be the recruitment rate of host individuals and vectors, respectively. ℎ and  V are, respectively, the natural death rate of host individuals and vectors.Let  be the natural recovery rate from the infected population and  the disease induced death rate of host individuals.The number of individuals moving from the vaccinated class into the susceptible class at time  is ∫ +∞ 0 V(, ).From the above assumptions, we formulate our vector-borne epidemic model in the following way: where  1  + is the set of integrable functions from (0, ∞) into R + = [0, ∞).Since the removed host individual population does not appear in the remaining equations of system (2), it is sufficient to consider the following system: (3) From [18,19], we state that system (3) has a unique continuous solution if the initial conditions satisfy the compatibility condition In the remaining part of this paper, we always assume that condition (4) is satisfied.The existence and the nonnegativity of the solution of (3) can be reached in Browne and Pilyugin [20].We next introduce a semiflow solution of system (3).Define and consider the linear operator  : dom() ⊂  →  defined by with dom( We consider a nonlinear map  : dom() →  which is defined by and let Set Based on the above, we can reformulate system (3) as the following abstract Cauchy problem: for  ≥ 0, with  (0) ∈  + 0 .
By applying the results given in [19,21], we derive the existence and uniqueness of the semiflow {Φ()} ≥0 on  + 0 generated by system (3).By using the theory for dynamical system (see [19]), we can further obtain the following lemma.
The total population size of human hosts and vectors is, respectively, Then, from the time derivative of  ℎ () and  V (), we get which implies lim sup We hence restrict our attention to solutions of (3) with initial conditions in The rest of the paper is structured as follows.In Section 2, we study the existence and local stability of equilibria of system (3).In Section 3, we present the results for the global dynamics of equilibria of system (3).In Section 4, the paper closes with conclusion.

Existence and Local Stability of Equilibria
In this part, we state the result about the existence and local stability of equilibria of the model (3).We first start by the existence of equilibria.We define as Then, Let ( ℎ ,  ℎ ,  V ,  V , V(⋅)) be an equilibrium of (3).This implies International Journal of Differential Equations From the third and the sixth equations of ( 18), we deduce that By the first equation of ( 18), we get From the fourth equation of ( 18), we have Substituting  ℎ and  V into the second and the fifth equations of (18) gives From the second equation of ( 22), we obtain Replacing  V in the first equation of ( 22) yields By (H1) and (H2),  ℎ = 0 is a solution of the above equation.Thus, system (3) has a disease-free equilibrium Following the same method as [22], the basic reproduction number for model (3) is R() describes a threshold for endemic persistence/spread of the disease, the rate of increase in the number of cases during an epidemic.Its magnitude allows determining the effort necessary either to prevent an epidemic or to eliminate an infection from a population.Let ( * ℎ ,  * ℎ ,  * V ,  * V , V * (⋅)) be an endemic equilibrium.Then,  * ℎ ∈ (0, Λ ℎ / ℎ ) and ℎ( * ℎ ) = 0, where The function ℎ is continuous with ℎ(0) = 0 and ℎ(Λ ℎ / ℎ ) ≤ 0.
The sufficient condition for ℎ to have a zero in (0, Λ ℎ / ℎ ) is that ℎ is increasing at 0. Thus, there is an endemic equilibrium if which is equivalent to R() > 1.Let  * ℎ be a unique solution in (0, Λ ℎ / ℎ ) of ℎ( ℎ ) = 0.Then, system (3) admits a unique endemic equilibrium where We summarize the above analysis in the following result.
Theorem 2 (consider system (3)).If R() ≤ 1, then there is a unique equilibrium, which is the disease-free equilibrium E 0 .If R() > 1, then there are two equilibria, the disease-free equilibrium E 0 and the endemic equilibrium E * .
We now deal with the local stability of the disease-free equilibrium.We show the stability of E 0 by linearizing system (3) about E 0 .The result is stated as follows.
If R() > 1, the unique endemic equilibrium E * is locally asymptotically stable.
Proof.From the linearization of system (3) at E 0 , we deduce the following characteristic equation: where From (31), the eigenvalues are − V and solutions of All roots of (33) and (34) have negative real parts; otherwise let  0 be a root of (33) with Re( 0 ) ≥ 0.Then, we have This leads to a contradiction.Now, let  0 be a root of (34) with Re( 0 ) ≥ 0. From (26), we have This also leads to a contradiction by using (34) and then proves that E 0 is locally asymptotically stable.
The characteristic equation at By using We show that the characteristic equation has no eigenvalues with nonnegative real parts.The eigenvalues are − V and solutions of By way of contradiction, assume that there is one eigenvalue  1 with Re( 1 ) ≥ 0.Then, we have This leads to a contradiction.

Global Stability Analysis of Equilibria
In this section, we prove the global stability of the equilibria of model (3).We first start by the global stability of the diseasefree equilibrium E 0 .To attend this, we need the Fluctuation Lemma [23].
Let us introduce the notations The Fluctuation Lemma is stated as follows.
Lemma 4 (See [23]).Let  : R + → R be a bounded and continuously differentiable function.Then, there exist sequences {  } and We also need the following lemma for establishing the global stability of E 0 .
Proof.Using Theorem 3, it is sufficient to show that E 0 is attractive in Γ.
Let ( ℎ (),  ℎ (),  V (),  V (), V(, )) be a solution of ( 3) with ( ℎ0 ,  ℎ0 ,  V0 ,  V0 , V 0 (⋅)) ∈ Γ.We integrate the third equation of ( 3) with the boundary conditions to obtain Using the Fluctuation Lemma 4, we derive From ( 1) and ( 3), we get From (48), we have It is evident that all eigenvalues of the matrix have negative real parts when R() < 1.This leads to From Lemma 4, it follows that there exists a sequence Thus, Let  → ∞; then which gives Since  ∞ ℎ → 0 and  ∞ V → 0, we obtain That is, lim From ( 46), it follows that Therefore, We now deal with the global stability of the endemic equilibrium E * .
The result of the global stability of the endemic equilibrium is stated as follows.
Proof.Evaluating both sides of (3) at E * gives Let Then, International Journal of Differential Equations Let Define We study the behavior of the Lyapunov functional () given by (68).() is bounded and () ≥ 0 with equality if and For clarity, the derivatives of   ℎ (),   ℎ (),   V (),   V (),  V () will be calculated separately and then combined to obtain ()/.We first have Using (60) to replace Λ ℎ in (69) gives Next, we calculate   ℎ ()/.