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

Authors: Stanislas Ouaro; Ali Traoré
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

We study a vector-borne disease with age of vaccination. A nonlinear incidence rate including mass 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.


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## Body

## 1. 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 [2–5]. Several studies in the literature have been carried out to investigate the role of treatment and vaccination of the spread of diseases ([6–8] 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 [15–17]. 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 vaccine-induced immunity. Additionally, we use incidences with a nonlinear response to the number of infectious individuals and infectious vectors. The incidences take the formSf(I) and Sg(I), respectively, for the human and vector populations. We assume that f and g satisfy the following assumptions:( H 1 ) For x∈R+,f(x)≥0 with equality if and only if x=0, f′(x)≥0, and f′′(x)≤0.( H 2 ) For x∈R+,g(x)≥0 with equality if and only if x=0, g′(x)≥0, and g′′(x)≤0.From the above assumptions and the Mean Value Theorem, it follows that(1)f′xx≤fx≤f′0x,g′xx≤gx≤g′0x.Let Sh, Ih, and Rh denote, respectively, the number of susceptible, infectious, and removed host individuals and Sv, Iv the number of susceptible and infectious vectors. The susceptible individuals are vaccinated at the rate θ≥0. v(t,a) denotes the population size of the vaccinated compartment at time t with the vaccine age a. Let ϵ(a) be the rate at which the vaccine-induced immunity wanes. We assume that ϵ(a) and the following assumption:( H 3 )  ϵ : [ 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 timet, Λh and Λv be the recruitment rate of host individuals and vectors, respectively. μh and μv are, respectively, the natural death rate of host individuals and vectors. Let γ be the natural recovery rate from the infected population and δ the disease induced death rate of host individuals. The number of individuals moving from the vaccinated class into the susceptible class at time t is ∫0+∞v(t,a)da. From the above assumptions, we formulate our vector-borne epidemic model in the following way:(2)dShtdt=Λh-μhSht-ShtfIvt-θSht+∫0∞ϵavt,ada,dIhtdt=ShtfIvt-μh+γ+δIht,dRhtdt=γIht-μhRht,∂vt,a∂t+∂vt,a∂a=-μh+ϵavt,a,dSvtdt=Λv-μvSvt-SvtgIht,dIvtdt=SvtgIht-μvIvt,vt,0=θSht,Sh0=Sh0≥0,Ih0=Ih0>0,Rh0=Rh0≥0,v0,·=v0·∈L+1,Sv0=Sv0≥0,Iv0=Iv0>0,where L+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)dShtdt=Λh-μhSht-ShtfIvt-θSht+∫0∞ϵavt,ada,dIhtdt=ShtfIvt-μh+γ+δIht,∂vt,a∂t+∂vt,a∂a=-μh+ϵavt,a,dSvtdt=Λv-μvSvt-SvtgIht,dIvtdt=SvtgIht-μvIvt,vt,0=θSht,Sh0=Sh0≥0,Ih0=Ih0>0,Rh0=Rh0≥0,v0,·=v0·∈L+1,Sv0=Sv0≥0,Iv0=Iv0>0.From [18, 19], we state that system (3) has a unique continuous solution if the initial conditions satisfy the compatibility condition(4)v00=θSh0.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(5)χ=R×R×R×L10,+∞,R×R×R,χ+=R+×R+×R×L+10,+∞,R×R+×R+,and consider the linear operator A:dom⁡(A)⊂χ→χ defined by(6)AShIh0vSvIv=-μh+θSh-μh+γ+δIh-v0-v′-μh+ϵav-μvSv-μvIv,with dom⁡(A)=R×R×{0}×W1,1((0,+∞),R)×R×R, where W1,1 is a Sobolev space. Then, dom⁡(A)¯=R×R×{0}×L1((0,+∞),R)×R×R is not dense in χ. We consider a nonlinear map F:dom⁡(A)¯→χ which is defined by(7)FShIh0vSvIv=Λh-ShtfIvt+∫0∞ϵavt,adaShtfIvtθSht0L1Λv-SvtgIhtSvtgIht,and let(8)ut=Sht,Iht,0vt,·,Svt,IvtT.Set(9)χ0≔dom⁡A-=R×R×0×L10,+∞,R×R×R,χ0+≔dom⁡A-∩χ+=R+×R+×0×L+10,+∞,R×R+×R+.Based on the above, we can reformulate system (3) as the following abstract Cauchy problem:(10)dutdt=Aut+Fut,fort≥0,withu0∈χ0+.By applying the results given in [19, 21], we derive the existence and uniqueness of the semiflow {Φ(t)}t≥0 on χ0+ generated by system (3). By using the theory for dynamical system (see [19]), we can further obtain the following lemma.Lemma 1.
System (3) generates a unique continuous semiflow {Φ(t)}t≥0 on χ0+ that is asymptotically smooth and bounded dissipative. Furthermore, the semiflow {Φ(t)}t≥0 has a compact global attractor B⊂χ0+.The total population size of human hosts and vectors is, respectively,(11)Nht=Sht+Iht+∫0∞vt,ada,Nvt=Svt+Ivt.Then, from the time derivative of Nh(t) and Nv(t), we get(12)dNhtdt≤Λh-μhNht,dNvtdt=Λv-μvNvt,which implies(13)limsupt→∞Nht≤Λhμh,limsupt→∞Nvt≤Λvμv.We hence restrict our attention to solutions of (3) with initial conditions in(14)Γ=Sh0,Ih0,Sv0,Iv0,v0·∈R4+×L+1:v00=θSh0,Nh0≤Λhμh,Nv0≤Λvμv.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.


## 2. 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(15)ξ:R+⟶R+,as(16)ξa=e-μha-∫0aϵsds,A=∫0∞ϵaξada.Then,(17)A≤∫0∞ϵae-∫0aϵsdsda=1.Let (S-h,I-h,S-v,I-v,v-(·)) be an equilibrium of (3). This implies(18)Λh-μhS-h-S-hfI-v-θS-h+∫0∞ϵav-ada=0,S-hfI-v-μh+γ+δI-h=0,dv-ada=-μh+ϵav-a,Λv-μvS-v-S-vgI-h=0,S-vgI-h-μvI-v=0,v-0=θS-h.From the third and the sixth equations of (18), we deduce that(19)v-a=θS-hξa.By the first equation of (18), we get(20)S-h=ΛhfI-v+μh+θ1-A.From the fourth equation of (18), we have(21)S-v=ΛvgI-h+μv.Substituting S-h and S-v into the second and the fifth equations of (18) gives(22)ΛhfI-v-μh+γ+δμh+θ1-A+fI-vI-h=0,ΛvgI-h-μvI-vμv+gI-h=0.From the second equation of (22), we obtain(23)I-v=ΛvgI-hμvμv+gI-h.Replacing I-v in the first equation of (22) yields(24)Λh-μhI-hfΛvgI-hμvμv+gI-h-μh+γ+δμh+θ1-AI-h=0.By (H1) and (H2), I-h=0 is a solution of the above equation. Thus, system (3) has a disease-free equilibrium(25)E0=Λhμh+θ1-A,0,Λvμv,0,θΛhξaμh+θ1-A.Following the same method as [22], the basic reproduction number for model (3) is(26)Rθ=ΛhΛvf′0g′0μv2μh+γ+δμh+θ1-A.R(θ) 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(Sh∗,Ih∗,Sv∗,Iv∗,v∗(·)) be an endemic equilibrium. Then, Ih∗∈(0,Λh/μh) and h(Ih∗)=0, where(27)hIh=Λh-μhIhfΛvgIhμvμv+gIh-μh+γ+δμh+θ1-AIh.The function h is continuous with h(0)=0 and h(Λh/μh)≤0.Moreover, forIh∈(0,Λh/μh),(28)dhdIh=Λh-μhIhΛvg′Ihμv+gIh2f′ΛvgIhμvμv+gIh-μhfΛvgIhμvμv+gIh-μh+γ+δμh+θ1-A.The sufficient condition for h to have a zero in (0,Λh/μh) is that h is increasing at 0. Thus, there is an endemic equilibrium if(29)dhdIh0=μh+γ+δμh+θ1-AR2θ-1>0,which is equivalent to R(θ)>1. Let Ih∗ be a unique solution in (0,Λh/μh) of h(Ih)=0. Then, system (3) admits a unique endemic equilibrium E∗=(Sh∗,Ih∗,Sv∗,Iv∗,v∗(·)), where(30)Sh∗=ΛhfΛvgIh∗/μvμv+gIh∗+μh+θ1-A,Sv∗=Λvμv+gIh∗,Iv∗=ΛvgIh∗μvμv+gIh∗,v∗·=ΛhθξafΛvgIh∗/μvμv+gIh∗+μh+θ1-A.We summarize the above analysis in the following result.Theorem 2 (consider system (3)).
IfR(θ)≤1, then there is a unique equilibrium, which is the disease-free equilibrium E0.
IfR(θ)>1, then there are two equilibria, the disease-free equilibrium E0 and the endemic equilibrium E∗.We now deal with the local stability of the disease-free equilibrium. We show the stability ofE0 by linearizing system (3) about E0. The result is stated as follows.Theorem 3 (consider system (3)).
IfR(θ)<1, the disease-free equilibrium E0 is locally asymptotically stable.
IfR(θ)>1, the unique endemic equilibrium E∗ is locally asymptotically stable.Proof.
From the linearization of system (3) at E0, we deduce the following characteristic equation:(31)λ+μvλ+μh+θ1-A^λλ+μh+γ+δλ+μv-Λhμh+θ1-AΛvμvf′0g′0=0,where(32)A^λ=∫0∞ϵaξae-λada.From (31), the eigenvalues are -μv and solutions of(33)λ+μh+θ=θA^λ,(34)λ+μh+γ+δλ+μv=ΛhΛvf′0g′0μvμh+θ1-A.All roots of (33) and (34) have negative real parts; otherwise let λ0 be a root of (33) with Re⁡(λ0)≥0. Then, we have(35)λ0+μh+θ>θ,θA^λ0≤θA≤θ.This leads to a contradiction.
Now, letλ0 be a root of (34) with Re⁡(λ0)≥0. From (26), we have(36)ΛhΛvf′0g′0μvμh+θ1-A=Rθ2μvμh+γ+δ<μvμh+γ+δ≤λ0+μh+γ+δλ0+μv.This also leads to a contradiction by using (34) and then proves that E0 is locally asymptotically stable.
The characteristic equation atE∗ is(37)0=λ+μh+fIv∗+θ1-A^λ00Sh∗f′Iv∗-fIv∗λ+μh+γ+δ0-Sh∗f′Iv∗0Sv∗g′Ih∗λ+μv+gIh∗00-Sv∗g′Ih∗-gIh∗λ+μv,0=λ+μvλ+μh+fIv∗+θ1-A^λλ+μh+γ+δλ+μv+gIh∗+Sh∗Sv∗f′Iv∗fIv∗g′Ih∗gIh∗-λ+μh+fIv∗+θ1-A^λSh∗Sv∗f′Iv∗g′Ih∗.By using Sh∗=μh+γ+δIh∗/f(Iv∗) and Sv∗=μvIv∗/g(Ih∗), we get(38)0=λ+μvλ+μh+fIv∗+θ1-A^λλ+μh+γ+δλ+μv+gIh∗+μvμh+γ+δIh∗Iv∗f′Iv∗g′Ih∗-λ+μh+fIv∗+θ1-A^λ×μvμh+γ+δIh∗Iv∗fIv∗gIh∗f′Iv∗g′Ih∗.We show that the characteristic equation has no eigenvalues with nonnegative real parts. The eigenvalues are -μv and solutions of(39)λ+μh+fIv∗+θ1-A^λλ+μh+γ+δλ+μv+gIh∗+μvμh+γ+δIh∗Iv∗f′Iv∗g′Ih∗=λ+μh+fIv∗+θ1-A^λ×μvμh+γ+δIh∗Iv∗fIv∗gIh∗f′Iv∗g′Ih∗.By way of contradiction, assume that there is one eigenvalue λ1 with Re⁡(λ1)≥0. Then,(40)1+μvμh+γ+δIh∗Iv∗f′Iv∗g′Ih∗λ1+μh+fIv∗+θ1-A^λ1λ1+μh+γ+δλ1+μv+gIh∗×λ1+μh+γ+δλ1+μv+gIh∗=μvμh+γ+δIh∗Iv∗fIv∗gIh∗f′Iv∗g′Ih∗.From (1), it follows that(41)μvμh+γ+δIh∗Iv∗fIv∗gIh∗f′Iv∗g′Ih∗≤μvμh+γ+δ.Since (42)1+μvμh+γ+δIh∗Iv∗f′Iv∗g′Ih∗λ1+μh+fIv∗+θ1-A^λ1λ1+μh+γ+δλ1+μv+gIh∗>1,we have(43)1+μvμh+γ+δIh∗Iv∗f′Iv∗g′Ih∗λ1+μh+fIv∗+θ1-A^λ1λ1+μh+γ+δλ1+μv+gIh∗×λ1+μh+γ+δλ1+μv+gIh∗>μvμh+γ+δ.This leads to a contradiction.


## 3. 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 disease-free equilibrium E0. To attend this, we need the Fluctuation Lemma [23].Let us introduce the notations(44)ψ∞=liminft→∞ψt,ψ∞=limsupt→∞ψt.The Fluctuation Lemma is stated as follows.Lemma 4 (See [23]).
Letψ:R+→R be a bounded and continuously differentiable function. Then, there exist sequences {sn} and {tn} such that sn→∞, tn→∞, ψ(sn)→ψ∞, ψ′(sn)→0, ψ(tn)→ψ∞, and ψ′(tn)→0 as n→∞.We also need the following lemma for establishing the global stability ofE0.Lemma 5 (See [18]).
Suppose thatf:R+→R is a bounded function and k∈L1((0,+∞),R). Then,(45)limsupt→∞∫0tkζft-ζdζ≤f∞k1,wherek1=∫0+∞ksds.We state the stability result of the disease-free equilibriumE0 as follows.Theorem 6.
IfR(θ)<1, then the disease-free equilibrium E0 is globally asymptotically stable.Proof.
Using Theorem3, it is sufficient to show that E0 is attractive in Γ.
Let(Sh(t),Ih(t),Sv(t),Iv(t),v(t,a)) be a solution of (3) with (Sh0,Ih0,Sv0,Iv0,v0(·))∈Γ. We integrate the third equation of (3) with the boundary conditions to obtain(46)vt,a=θSht-aξa,t≥a,v0a-tξaξa-t,t<a.Using the Fluctuation Lemma 4, we derive(47)Sh∞≤Λhμh+θ1-A,Sv∞≤Λvμv.From (1) and (3), we get(48)dIhtdt=ShtfIvt-μh+γ+δIh≤Λhμh+θ1-Af′0Ivt-μh+γ+δIht,dIvtdt=SvtgIht-μvIvt≤Λvμvg′0Iht-μvIvt.From (48), we have(49)
d
I
h
d
t
d
I
v
d
t
≤
-
μ
h
+
γ
+
δ
Λ
h
f
′
0
μ
h
+
θ
1
-
A
Λ
v
g
′
0
μ
v
-
μ
v
I
h
I
v
.It is evident that all eigenvalues of the matrix (50)-μh+γ+δΛhf′0μh+θ1-AΛvg′0μv-μvhave negative real parts when R(θ)<1. This leads to(51)Ih∞⟶0,Iv∞⟶0.From Lemma 4, it follows that there exists a sequence {tn} such that tn→∞, Sh(tn)→Sh,∞, Sv(tn)→Sv,∞, and Sh′(tn)→0, Sv′(tn)→0 as n→∞.
Note that(52)limn→∞Ihtn=0,limn→∞Ivtn=0.Thus,(53)dShtndt=Λh-μh+θShtn-ShtnfIvtn+∫0tnϵaShtn-aξada+∫tn∞ϵav0a-tnξaξa-tnda,dSvtndt=Λv-μvSvtn-SvtngIhtn.Let n→∞; then(54)0≥Λh-μh+θSh,∞-Sh,∞fIv∞+∫0∞ϵaSh,∞ξada,0≥Λv-μvSv,∞-Sv,∞gIh∞,which gives(55)0≥Λh-μh+θ1-ASh,∞-Sh,∞fIv∞,0≥Λv-μvSv,∞-Sv,∞gIh∞.Since Ih∞→0 and Iv∞→0, we obtain(56)Λhμh+θ1-A≤Sh,∞≤Sh∞≤Λhμh+θ1-A,Λvμv≤Sv,∞≤Sv∞≤Λvμv.That is,(57)limt→∞Sht=Λhμh+θ1-A,limt→∞Svt=Λvμv.From (46), it follows that(58)limt→∞vt,a=Λhθξaμh+θ1-A.Therefore, (Sh(t),Ih(t),Sv(t),Iv(t),v(t,·))→E0 in R4+×L+1 as t→∞.We now deal with the global stability of the endemic equilibriumE∗.A total trajectory ofΦ is a function X:R→R4+×L+1 such that Φ(s,X(t))=X(t+s) for all t∈R and all s∈R+.We defineϕ:(0,∞)→R by ϕ(x)=x-1-ln⁡x. ϕ has a strict global minimum at 1 with ϕ(1)=0 and ϕ(x)>0, ∀x≠1.Lemma 7 (see [24]).
Define(59)FX=ϕfXfX∗-ϕXX∗.If assumptions (H1) and (H2) are satisfied, then F(X)≤0, ∀X>0.The result of the global stability of the endemic equilibrium is stated as follows.Theorem 8.
IfR(θ)>1, then the endemic equilibrium E∗ is globally asymptotically stable in Γ.Proof.
Evaluating both sides of (3) at E∗ gives(60)Λh+∫0∞ϵav∗ada=μh+fIv∗+θSh∗,(61)Sh∗fIv∗=μh+γ+δIh∗,(62)Λv=μv+gIh∗Sv∗,(63)gIh∗Sv∗=μvIv∗,(64)v∗a=θSh∗ξa.Let(65)αa=∫a∞ϵsv∗sds.Then,(66)dαada=-ϵav∗a.Let(67)VSht=ϕShtSh∗,VIht=ϕIhtIh∗,VSvt=ϕSvtSv∗,VIvt=ϕIvtIv∗,Vvt=∫0∞αaϕvt,av∗a.Define(68)Vt=Sv∗gIh∗Sh∗VSht+Ih∗VIht+Sh∗fIv∗Sv∗VSvt+Iv∗VIvt+Sv∗gIh∗Vvt.We study the behavior of the Lyapunov functional V(t) given by (68). V(t) is bounded and V(t)≥0 with equality if and only if Sh(t)/Sh∗=Ih(t)/Ih∗=Sv(t)/Sv∗=Iv(t)/Iv∗=v(t,a)/v∗(a)=1.
For clarity, the derivatives ofVSh(t),VIh(t),VSv(t),VIv(t),Vv(t) will be calculated separately and then combined to obtain dV(t)/dt. We first have(69)dVShtdt=1Sh∗1-Sh∗ShtdShtdt=1Sh∗1-Sh∗ShtΛh-μhSht-ShtfIvt-θSht+∫0∞ϵavt,ada.Using (60) to replace Λh in (69) gives(70)dVShtdt=1Sh∗1-Sh∗Shtμh+θSh∗+Sh∗fIv∗-∫0∞ϵav∗ada-μh+θSht-ShtfIvt+∫0∞ϵavt,ada=-μh+θSht-Sh∗2ShtSh∗+fIv∗1-ShtfIvtSh∗fIv∗-Sh∗Sht+fIvtfIv∗+∫0∞ϵav∗aSh∗vt,av∗a-Sh∗vt,aShtv∗a-1+Sh∗Shtda.Next, we calculate dVIh(t)/dt.(71)dVIhtdt=1Ih∗1-Ih∗IhtdIhdt=1Ih∗1-Ih∗IhtShtfIvt-μh+γ+δIht=1Ih∗1-Ih∗IhtShtfIvt-μh+γ+δIh∗IhtIh∗.Using (61) to replace (μh+γ+δ)Ih∗ in (71) gives(72)dVIhtdt=1Ih∗1-Ih∗IhtShtfIvt-Sh∗fIv∗IhtIh∗=Sh∗fIv∗Ih∗1+ShtfIvtSh∗fIv∗-ShtfIvtIh∗Sh∗fIv∗Iht-IhtIh∗.We now calculate the derivative of VSv(t).(73)dVSvtdt=1Sv∗1-Sv∗SvtdSvdt=1Sv∗1-Sv∗SvtΛv-μvSvt-SvtgIht.Using (62) to replace Λv in (73) gives(74)dVSvtdt=1Sv∗1-Sv∗SvtdSvdt=1Sv∗1-Sv∗SvtμSv∗+Sv∗gIh∗-μvSvt-SvtgIht=-μvSvt-Sv∗2SvtSv∗+gIh∗1-SvtgIhtSv∗gIh∗-Sv∗Svt+gIhtgIh∗.Differentiating VIv(t) with respect to t yields(75)dVIvtdt=1Iv∗1-Iv∗IvtdIvdt=1Iv∗1-Iv∗IvtSvtgIht-μvIvt=1Iv∗1-Iv∗IvtSvtgIht-μvIv∗IvtIv∗.Using (63) to replace μvIv∗ in (75) gives(76)dVIvtdt=1Iv∗1-Iv∗IvtSvtgIht-Sv∗gIh∗IvtIv∗=Sv∗gIh∗Iv∗1+SvtgIhtSv∗gIh∗-SvtgIhtIv∗Sv∗gIh∗Ivt-IvtIv∗.The derivative of Vv(t) is(77)dVvtdt=∫0∞αa∂ϕvt,a/v∗a∂tda=∫0∞αa1-v∗avt,a1v∗a∂vt,a∂tda=-∫0∞αa1-v∗avt,avt,av∗avat,avt,a+μh+ϵada=-∫0∞αavt,av∗a-1vat,avt,a+μh+ϵada,where va(t,a)=∂v(t,a)/∂a.
Using∂/∂aϕ(v(t,a)/v∗(a))=(v(t,a)/v∗(a)-1)(va(t,a)/v(t,a)+μh+ϵ(a)),dα(a)/da=-ϵ(a)v∗(a) and integration by parts, we get(78)dVvtdt=-∫0∞αa∂∂aϕvt,av∗ada=-αaϕvt,av∗aa=0a=∞+∫0∞ϕvt,av∗adαadada=-αaϕvt,av∗aa=∞+α0ϕvt,0v∗0-∫0∞ϵav∗aϕvt,av∗ada.From α(0)=∫0∞ϵ(a)v∗(a)da, we get(79)dVvtdt=-αaϕvt,av∗aa=∞+∫0∞ϵav∗aϕvt,0v∗0da-∫0∞ϵav∗aϕvt,av∗ada.Combining (70), (72), (74), (76), and (79) and multiplying appropriately by coefficients determined by (68), we obtain(80)dVtdt=-μh+θSv∗gIh∗Sh-Sh∗2Sht-μvSh∗fIv∗Sv-Sv∗2Svt+Sh∗Sv∗fIv∗gIh∗4-ShtfIvtIh∗Sh∗fIv∗Iht-Sh∗Sht+fIvtfIv∗-IhtIh∗-IvtIv∗-SvtgIhtIv∗Sv∗gIh∗Ivt-Sv∗Svt+gIhtgIh∗-αaSv∗gIh∗ϕvt,av∗aa=∞+Sv∗gIh∗∫0∞ϵav∗aϕvt,0v∗0-ϕvt,av∗a+vt,av∗a-Sh∗vt,aShtv∗a-1+Sh∗Shtda.By ∫0∞ϵ(a)v∗(a)da=θSh∗K, v(t,0)=θSh(t), and v∗(0)=θSh∗, it follows that(81)dVtdt=-μh+θ1-KSv∗gIh∗Sh-Sh∗2Sht-μvSh∗fIv∗Sv-Sv∗2Svt+Sh∗Sv∗fIv∗gIh∗ϕfIvtfIv∗-ϕIvtIv∗+ϕgIhtgIh∗-ϕIhtIh∗-ϕSv∗Svt-ϕSvtgIhtIv∗Sv∗gIh∗Ivt-αaSv∗gIh∗ϕvt,av∗aa=∞-Sv∗gIh∗∫0∞ϵav∗aϕSh∗vt,aShtv∗a.Thus, from Lemma 7, we deduce that(82)dVtdt≤0;that is, V is nonincreasing. Denote by M the largest invariant subset of {dV(t)/dt=0}.
SinceV is bounded on X(·), the ω-limit set of X(·) must be contained in M.
d V ( t ) / d t = 0 yields Sh(t)=Sh∗,Sv(t)=Sv∗, and v(t,a)=v∗(a).
Thus,dSh(t)/dt=dSv(t)/dt=0 in M. This implies that(83)Λh-μh+θSh∗-Sh∗fIvt+∫0∞ϵav∗ada=0,Λv-μvSv∗-Sv∗gIht=0,for t∈R, which gives f(Iv(t))=f(Iv∗) and g(Ih(t))=g(Ih∗)∀t∈R. The monotonicity of f and g stated in (H1) and (H2) implies that Iv(t)=Iv∗ and Ih(t)=Ih∗, ∀t∈R. Therefore, M={E∗}.
Then, theω-limit set of X(·) is the endemic equilibrium E∗ and hence V(X(t))≥V(E∗), ∀t∈R. Thus, B={E∗}.


## 4. Conclusion

We have analysed a vector-borne disease model with nonlinear incidences, in which we have incorporated the waning of vaccine-induced immunity. These nonlinear incidences rates include mass action and saturating incidence as special cases. The basic reproduction number denoted byR(θ) is derived. The model exhibits two equilibria, namely, the disease-free equilibrium E0 and the endemic equilibrium E∗. We have shown that if R(θ) is less than 1, then the disease-free equilibrium E0 is globally asymptotically stable; that is, the disease dies out and if R(θ) is larger than 1, then the endemic equilibrium E∗ is globally asymptotically stable; that is, the disease persists in the population.From these results, a critical vaccine coverage rate is obtained by solving the equationR(θ)=1, which yields(84)θ0=11-AΛhΛvf′0g′0-μhμv2μh+γ+δμv2μh+γ+δ.Then, if the vaccine coverage rate θ is greater than θ0, then R(θ)<1 and the disease will die out. The critical vaccine coverage rate θ0 is increasing in the waning of vaccine. Then, neglecting the waning of vaccine (i.e., A=0) when applying a vaccination for vector-borne disease will surely not be sufficient to make the disease die out in the population.



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*Source: 4168061-2018-03-15.xml*